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, which con- sists of a double-walled glass vessel, the space between the walls being exhausted as completely as possible. Traces of mercury vapor are left in this space ; and at low temperatures this, freezes, forming a metallic surface over the glass walls. Fio. 112. — Dewar flask. CHAPTER XI CHANGES IN VOLUME AND PRESSURE Introduction. — The fact that all bodies, with the excep- tion of water below 4° C. and one or two unimportant sub- stances, increase in volume when their temperature is raised, is most familiar to every one; and numerous measurements have been made of these changes. Naturally the amount of the increase in volume depends upon the external force act- ing, and upon whether this is constant or not. The mechan- ical force required to influence the expansion of a solid or a liquid is so great that ordinary changes in the atmospheric pressure have no measurable effect ; but this is not so in the case of a gas. It is therefore necessary, in studying those variations in volume of a body which accompany changes in temperature, to describe the external conditions, if one wishes to be definite. The condition which is always assumed, unless the contrary is stated, is that of constant external pressure. Solids Linear and Cubical Expansion. — In measuring the change in volume of a solid it is, as a rule, easier to measure the changes in length of certain linear dimensions of the body and from these to calculate the change in volume. It tin- body is isotropio, i.e. has the same properties in all direc- tions, this calculation is most simple. Imagine the body in the form of a cube, the length of whose edges at any one trm p. Mature tf is /, and at the temperature fa° is /3, then tin- voluin.- ;it /, ii /,:! and at tj is lf\ so the change in volume is I* — I*- It the body is not isotropic, but has no 230 HEAT different properties in different directions, three directions in it (which depend upon the arrangement of the molecules) may be determined, such that the changes along these are independent of one another — these are called "axes." (In crystals they are the crystallographic axes.) If, then, a rectangular solid is made of this body with its edges parallel to these directions, and if lv mv n^ are the lengths
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of the edges at t^ and?2, m2, n2 their lengths at £2°, the correspond- ing volumes are l^n^ and Z2w2w2; and the change in volume is?2W2W2 ~~ liminr The change in length of any straight line in the surface of a solid or of any straight edge may be measured by various means. The body is immersed in a bath of some fluid, whose temperature may be varied ; and the lengths are determined by a comparator of some kind. (Reference may be made to any laboratory manual.) Experiments show that, to a sufficient degree of accuracy, the change in length varies directly as the original length and as the change in temperature, but is different for bodies of different materials. If, as above, Zx is the length of a certain line of the solid body at ^°, and Z2 that at £2°, these facts may be expressed by the formula /,-*! = af, ft - Q, in which a is a factor of proportionality, which is different for different substances, a is called the "coefficient of linear expansion of the body referred to the temperature ^V Ordinarily, the temperature to which these coefficients refer is 0° C. ; and if 1Q is the length at 0° and I that at £°, the relation is I — Z0 = #(/o^' where a0 is the coefficient of linear expansion referred to 0° C. This formula may be written I = 1Q (1 + a0£). (It is evident, then, that a in the first formula is connected l-f-#0& with aQ by the relation a = ^ — - — ; and if aQ is an extremely small quantity, as it is for all solids, we may neglect the term a^t in comparison with 1, and write a = a0 approximately.) 1\ VOLl'MK AM) PRESSl'llE 231 If the solid is isotropic, and if we write v0 for the volume at 0° C. and v for that at t°, v0 = /0« and v = /» = /0» (1 4- <V)» = »0 (1 + a,/)'. IJut (1 +- a003= l + 3a0« + 3n0V + aoV; ancl if aois a 8ma11 quantity
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0.000019. 0.0000168. 0.0000083. 0.0000121 0.0000090 Copper Glass. Iron Platinum Steel (annealed).. 0.000011 Steel alloyed with 36% nickel... 0.00000087 Zinc, 0.0000292 Illustration of Expansion. — The fact that the coefficients of expansion of different bodies are different is often made use of to neutralize the expansion which ordinarily follows rise in temperature. Thus, the period of a pendulum clock would naturally increase with rise in temperature owing to the expansion of the pendulum rod ; but this may be avoided in several ways. One is to make the pen- dulum as shown in the cut, which illustrates Harrison's gridiron compensated pendulum. In it the rods which are shown as single black lines are of iron and the others of brass. It is seen that, as the temperature rises, the expansion of the iron rods lowers the pendulum bob, but that of the brass ones raises it. The linear expan- sion of brass is about 1£ times that of iron ; and so if the combined length of the two iron rods on each side and the middle one is 1* times that of the two brass ones on each side, there will be no change in length as the temperature varies. (A simpler method is used in the clock in the tower of the Houses of Parliament, London. The pendulum consists of an iron rod sur- rounded by a zinc cylinder, which in turn is surrounded by an iron one carrying the pendulum bob. The lower ends of the iron rod and the zinc cylinder are attached, and the upper ends of the two cylinders. Since the linear expansion of zinc is about 2£ times that of iron, the combined lengths of the iron rod and cylinder must equal 2i times that of the length of the zinc cylinder.) Graham's compensation pendulum consists of an iron rod whose lower end screws into a cast-iron cylindrical vessel partially filled with mercury. The quantity of mercury is so chosen that by its expansion combined with that of the iron rod and cylinder the centre of gravity of the whole does not move when the temperature is varied. FIG. 113. — Com- pensation "grid- iron " pendulum. The single black lines represent iron rods ; the double lines brass ones. Again, the period of a watch or of the ordinary spring clock is regu- lated by the vibrations of the balance wheel. This consists of a small Cll. 1
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with a tube attached, if the liquid tills the hull. ;md pan,,f the tube, it is observed that, when the l>ull> is heated suddenly hy immersing it in a basin of hot i «>r in any other \va\, the top of the column of liquid 234 UK AT in the tube immediately sinks and then rises gradually, ascending finally higher than it was originally. This is owing to the fact that the first effect of the application of the hot bath is to raise the temperature of the bulb, and it therefore expands before the liquid inside is affected ; as soon, however, as its temperature is raised, it expands and rises in the tube. If the bulb is chilled, instead of heated, the reverse of these changes takes place ; the liquid first rises in the tube and then sinks, falling below its original position. It is evident, then, that the apparent change in volume of the liquid is less than the real change by an amount equal to the change in volume of the containing solid. So, if the coefficient of expansion of the solid is known, the direct method for determining the change in volume of a liquid that accompanies a change in temperature is to inclose the liquid in a bulb with a finely divided stem whose volumes are known, and to measure the volume of the liquid at any one temperature and the apparent change in volume when the temperature is altered. Let vl be the initial volume at the temperature t±, and v the apparent increase in volume when the temperature is raised to £2° ; further, let the coefficient of cubical expansion of the solid have the value b. Then the increase in volume of the solid is v1 [1 + b (£2 — ^)] ; and hence the true change in volume of the liquid is v + vl [1. + 6(£2 — tfj)]. Experiments show that fyor nearly all liquids — water is an exception, as will be explained below — the relation between the change in volume and that in tempera- ture is of the same form as for solids, viz., v2 — v1 = vj) (£2 — ^), where b is the coefficient of cubical expansion of the liquid, referred to ^°. If the initial temperature is 0° C., this becomes, as before, v = vQ(l + bQt~) ; and it is to be noted that b = Q —, and therefore only if 50 is small can
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it 1 + bQt replace b. The coefficient of expansion is found to be different for CHANGES IN VOLUME AND PRESSURE 235 different liquids, as is shown in the following table. It should b»- noted that the expansion of liquids is, in general, m in -h greater than that of solids. Kthyl alcohol.. between 0° and 80° C. 0.00104 Kthyl ether... between - 15° and 38° C. 0.00215 (ilycerine... 0.000534 •ury... between 0° and 100° C. 0.000182 Turpentine... between - 9° and 106° C. 0.00105 It is evident that, if the coefficient of expansion of a liquid is known, tfl of observations and measurements gives a method for '{••termination of the coefficient of cubical expansion of the solid that contains the liquid. This method therefore can be used for all solids that can be formed into bulbs, e.g. glass or quartz. Mercury is the liquid that is, in general, used, because its expansion is known, and for r obvious reasons. Measurement of Coefficient of Expansion. — A better method, however, for the determination of the coefficient of expansion liquid, which does not involve a knowledge of that of solid vessel, was devised by Dulong and Petit and im- proved by Regnault. It depends upon the fact that the titfl at whieh two liquids of different densities stand when they balance each other in a (J-tube is independent of the material of the tube. If 7/t is the height of the column of LUC liquid of density dr and 7/.2 that of the column of the liquid whose density is dT -* == -2 (see page 176). But the formula for expansion, 2 l l>e replaced by v = *0(1 + /, 4 = ''0 + V). ii the density at £°, and </,, that at 0°; for tin- density of any ln»dy whose mass remains constant varies inversely as its volume (m = dv). Therefore, if the ratio of the densities of " liquid at <» and /' is known, its coefficient of expansion can be deduced at once. 'I'!"1 simplest i.»nu of the actual experiment is as follows : A tul..- ifl made
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in the form of a W- with a cross tube at the 236 HEAT top, as shown in the cut ; this last has a small opening -4 near its middle point and is kept horizontal ; there is a branch tube joined to the apparatus at B, which is con- nected with a reservoir K containing air that can be compressed to a pressure greater than that of the at- mosphere ; a quantity of the liquid whose expansion is to be studied is poured in, just sufficient to flow out of the opening A in the upper cross tube, thus insuring the con- dition that the upper level surfaces of the two columns are at the same height. These two separate portions of the liquid are then surrounded ==H: FIG. 115. — Apparatus for the determination of the coefficient of expansion of a liquid. by baths, one at 0°, the other at t°. When equilibrium is established, the difference in level of the two surfaces of one portion of the liquid is not the same as that of the other. Call this difference for the liquid at 0°, A0, and that for the one at t°, h. Then dQghQ = dyh, or ¥& = —. Consequently, if hQ and h are measured, or d'may be calculated ; for, ^ = 1 + bQt..-. 1 + b0t = n, Expansion of Water. — When water is studied, it is found that as the temperature rises from 0° C. to 4°C., it contracts; but above 4° it expands. These changes in volume of water (or of any substance) which accompany changes in tempera- ture may be best shown graphically. Lay off two axes, one to indicate temperatures; the other, volumes of a definite CHANGE* l.\ VOLUME.l.\/> 237 quantity of the substance. For water the curve giving the connection between v and t is as shown in the cut. (If the coefficient of expansion is a con-.t, the curve is a straight line sloping upward from left to right.) This fact, that the density of water is a maximum at 4° C., and that it decreases continuously from this temperature to 0° and to 100° is of great importance in nature ; for as the temperature of the water in a lake or a pond sinks in winter below 4° and reaches 0°, the water at 0° floats on top and does not sink to the bottom
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were to hold for a gas as the pressure became smaller and smaller and finally vanished without the volume at the same time being made infinitely great, the temperature of this condition would be given by T = 0, or t° C = - - = - 273° C. approximately. Similarly, if the same law could be applied to a gas as its volume was made less and less and finally vanished, the pressure remaining finite, the temperature would be given by the same value, viz., — 273° C. approximately. Of course the above law for a gas does not apply to one if greatly compressed, and the pressure of a gas can become zero only by its volume being infinite ; and so the above deductions have no physical meaning. We shall see later, however, by other reasoning, that this temperature --, or C, 273° — can reach. For that reason it is called the "absolute zero." (See page 308.) universe lowest which marks body any our the in temperature Change of Pressure at Constant Volume. — Another fact is apparent from this formula. If the changes take place in such a manner that the initial and final volumes are the same, V= i>0, and therefore P = p(\ + b0t). Therefore the pressure increases at the same rate with increase of tempera- ture when the volume is kept constant, as does the volume when the pressure is kept constant. (Actually, this is not exactly true of any gas ; but this discrepancy is a consequence of the fact that Boyle's law is not exact for an actual gas urn- is the coefficient of expansion a constant.) Laws of Gay-Lussac and Charles. — The fact that all gases have approximately the same coefficient for change of pressure when the volume is kept constant was discovered by the French physicist Charles ; while the corresponding one that all gases expand alike when the pressure is kept constant CHANGES IN VOU'Mi: AM) PEM88URM was discovered a few years later, ill 1802, by Gay-Lussac. se statements of fact are therefore called Charles's and -hussar's lav Other Forms of the Gas Law. — This formula for a gas, /' /'— HM'l\ ma\ be expressed differently and more simply if we assume the truth of Avogadro's hypothesis (see page I. If m is the mass of each molecule, and N is the num- ber in a unit volume, M= mNV\ consequently, on substit
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densities of gases, because P is the pressure of a gas at temperature T when the density is such that there are N1 mols in a unit volume. It is found to be 8.28 x 107 on the C. G. S. system. Energy Relations during Expansion Mechanical Expansion. — When the dimensions of a body are increased, either by the addition of heat energy or by mechanical forces, there is a certain amount of external work done and there are alterations in the kinetic and potential energies of the molecules. In the case of the thermal effect this is evident from what has been already said ; change in relative position of the molecules, change in temperature, and external work occur together. Similarly, when a brass wire is stretched, its temperature falls ; when a gas is com- pressed, its temperature is raised, etc. We might have predicted that these changes in tempera- ture would be as just stated. (See page 104.) If the tem- perature of a brass wire should rise when it is stretched, it would be in unstable equilibrium. If the wire hangs verti- cally under the stretching force of a heavy body, and if a sudden downward blow is given this weight, it will move down, stretching the wire still more, then come to rest and move up, etc., making harmonic vibrations, showing that it was in stable equilibrium. But if, owing to this stretching, as the hanging body moves down, the temperature of the wire were to rise, it would lengthen ; and this increase in length would cause another rise in temperature, etc., so the equilibrium would be unstable. Therefore, only if the wire CHANGES l.\ VOLUME AND PRESSURE 243 cools when stretched, is the equilibrium stable. The general law is that, if a body expands when its temperature is raised, its expansion by mechanical means will cool it, while mechan- ical compression will raise its temperature. The converse is true of those bodies which contract when their temperature is raised, e.g. water between 0° and 4° C. As the sun radiates energy, it contracts; and owing to this cause its parts are slowly coming closer together, and therefore energy is being liberated to make up for that loss. Thus, in ordinary language, the sun owes its heat to its slow contraction. Undoubtedly also, meteoric pieces of matter are falling into the sun, and their energy is thus also added to that of the sun. If the expanding body is a liquid or a gas
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, or if it is a solid immersed in a fluid, the amount of external work done equals the product of the increase in volume by the pressure. (See page 159.) The internal changes consist of change in kinetic energy and change in potential energy. The first of • •is connected intimately with changes in temperature, as has been already shown. (See pages 197 and 220.) Changes in potential energy occur if there are molecular forces. The fact that these exist in solids and liquids and are large is evident from the obvious properties of these forms <»f matter, e.g. they retain definite volumes; but noth- ing is known in regard to the amount or cause of these forces. The case is different with a gas, for in it the forces are extremely small. (Nothing, however, can be said as to their ran Internal Work in a Gas. — This fact that the internal forces in a gas are extremely small was first shown by Gay-Lussac and latrr by Joule, working independently and also in col- laboration \\ith Thomson (now Lord Kelvin). The earlv exporinirnts «,f.Iniilr arr perhaps the simplest to consider. II apparatus consisted of l\\o strong metal cylinders con- bed b\ ;, tube in which was a stopcock. In one of these cylinders quantities of the gas to be studied were compressed 244 II EAT until the pressure was as high as the apparatus would permit ; while from the other the gas was exhausted. The whole apparatus was then submerged in a tank containing water, and this was stirred until the temperature came to a steady state. Then the stopcock in the tube connecting the two cylinders was opened; and, when the gas had redistributed itself, occupying a greater volume and thus coming to a smaller pressure, but doing no external work, the temperature of the water in the tank after being well stirred was again observed. In no case was there any meas- urable change. If there had been molecular forces of attrac- tion between the molecules, it would have required work to separate them when the gas increased in volume, this energy would have necessarily come from the molecules themselves, which would have thus lost kinetic energy ; and therefore the temperature of the gas would have fallen. If there had been molecular forces of repulsion between the molecules, their potential energy would have been decreased by the increase in volume, their kinetic energy would then have increased an equal amount ; and therefore the temperature of the gas would have risen
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vessel to expand -udd.-nly. I f t li.-n- an- nuclei in tin- air. drops of water will!»«> loosed around them, thus forming a visil.li- mist and showing that the air has been chilled. (See page 186.) This is the explanation in many cases of the formation of clouds in the air. Expansion of a Gas in General. — When a compressed gas ilowed to expand out through a fine opening into a space 246 HEAT where its pressure is less, changes in temperature take place owing to numerous causes. We shall consider two cases of practical importance. Let a cloth with fine meshes — a piece of cheese cloth or toweling — be folded over the nozzle from which the gas is escaping ; it will expand with violence through these numerous openings into the air, thus forming a wind ; the gas that does not have time to escape during any small interval thus does work on that portion which it blows out and so experiences a fall of temperature itself. This may be sufficient to liquefy or even solidify it ; and the cloth will in this last case be found to contain the solidified gas. (This is the ordinary way of obtaining carbon dioxide in the solid form.) Again, let the opening or openings through which the compressed gas escapes be so fine and tortuous that the outcoming gas has no kinetic energy as a whole, i.e. there is no wind, it does noi flow ; in this case its temperature is lower than when in the compressed con- dition (except with hydrogen at ordinary temperatures), as shown by Thomson and Joule, owing to the fact that there are minute molecular forces of attraction, and as the poten- tial energy of the expanded gas is increased, its kinetic energy, and therefore temperature, must be diminished. This fall in temperature varies directly as the difference in pressure of the gas in its two conditions, and so may be considerable. In any ordinary expansion of a gas from a small opening, both of these actions take place. The gas that is escaping at any instant has done work in pushing out the portion that was just before it, and so its temperature is lowered, quite apart from the influences of the molecular forces. (Some distance away from the opening, however, in the case of a jet or a blast, the temperature of the gas is increased, owing to the friction of the moving currents.) This method of securing a decrease in temperature by allowing a gas to expand through a fine
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nozzle is being used practically in recent machines for the liquefaction of gases. (See page 280.) CHAPTER XII CHANGES IN TEMPERATURE Energy Relations when the Temperature is Raised. — It is 1)\ tin- change in temperature of bodies when exposed to some "source of heat" that our attention is directed to heat phenomena ; and it is in terms of changes of temperature that heat energy is measured, as we have already seen in the definition of the "calorie." If, however, we wish to deter- mine lm\v much energy goes to producing the rise in tempera- ture, it is necessary to ascertain how much is used in doing : nal work, and how much in overcoming the molecular forces. If there is a uniform pressure p over the body, and it its volume increases from v1 to vv the external work done against this pressure is p (v2 — Vj), and so may be calculated. It is only, however, in tin- case of a gas that anything is known <iuant it atively in regard to the molecular forces. 'Ill- '.hen the temperature of a solid or a liquid is raised, although tin- external work may be calculated if the external pressure is known, it is impossible to separate into its parts the energy that is spent in internal work ; and so do not know how much goes to producing rise in tem- r.ut tin; case of a gas is different. We can date the amount of external work done as well as meas- ure it. and we know further that the molecular forces are SO 1 that they can be neglected. ( '..us. -.jin -ntly when the temperature of a gas is raised owing to the addition of heat _ry, if we deduct the amount that is used in doing exter- nal work, the d nice is all spent in raising the tempera; i.e. in increasing the kinetic energy of the mole. ules. The Ml 248 HEAT amount of energy that must be added to the gas, then, in order to raise its temperature a definite number of degrees, depends upon the amount of external work done by the gas ; that is, upon the external conditions under which the gas is kept during the change in temperature. If this change is the same in different experiments, the energy that is spent in producing the increase in the kinetic energy of the mole- cules is the same whatever the external conditions ; and so the
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difference in the amounts of heat energy added must equal the difference in the amounts of the external work done. Special Case of a Gas. — For various reasons, partly prac- tical and partly theoretical, the change in temperature of a gas is considered, as a rule, under two different conditions : (1), when the volume is kept constant, and so no external work is done ; (2), when the pressure is maintained constant, so the external work equals the product of the values of this pressure and the change in volume. If Q1 is the heat energy added in one experiment when the volume is kept constant, Q2 that in another when the pressure is kept constant at a value p while the volume is increased from v1 to Vy, the change in temperature being the same in both' e^o, =,(.,-,>, where, of course, (?2 and Q1 are measured in mechanical units; i.e. in ergs or joules, since the product p(v2 — v^) is so ex- pressed. Reverse Changes. — When a body cools, it gives out heat energy ; but the external work done on it by the external forces is not necessarily the same as is done by the body against these forces when the body is heated and rises through the same range of temperature, because the external forces may have changed, or they may vary differently dur- ing the two processes. If, however, the pressure remains unchanged, or, when it is variable, if the series of changes is CHANGES 1\ il-MI'l-HMTRE tly reversed during the two processes, the external work done by the body during the rise in temperature equals that • on the body during the opposite change. If this is the case, the heat energy added in the former process must equal that LTiven out during the latter; for at the end of the two diaii^rs tin* body is hack in its original condition, and so there is no change in its internal energy, and there has also been no gain or loss of energy owing to external work. This fact serves as the basis of all methods for the measurement of quantities of heat energy. Many simple experiments show that, to produce the same change in temperature in equal quantities of different bodies, requires different amounts of heat energy. One of these, due to Tyndall, is to raise to the same temperature in a bath of heated oil several spheres, made of different materials, but
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be shown later that 536 calories, approximately, are given out by 1 g. of steam when it condenses into water at 100° C. So, if m grams are condensed by the body at this temperature, it must receive 536 m calories. "Specific Heat." — The number of calories that must be added to a body whose mass is 1 g. in order to raise its temperature through one degree Centigrade, from t° to (t + 1)°, is called the "specific heat" of that body at t° C. It should be noted that the specific heat of a substance is a number that is independent, in a way, of the heat unit used or of the tempera- ture scale adopted. We could adopt as a heat unit that quantity of heat energy required to raise the temperature of a unit mass (on any system) of water from t° to (t 4- 1)° (on any scale) ; and define the specific heat of a substance at t° as the number of these heat units that is required to raise the temperature of a unit mass of it from t° to (/ + 1)°. Or, if we assume that the specific heat is the same at all temperatures, we may define the specific heat of a body as the ratio of the amount of heat energy required to raise the temperature of the body through any range of temperature to that required to raise the temperature of an equal mass of water through the same range of temperature. With solids and liquids, as has been explained, we assume that the pressure is that of the atmosphere ; but with gases we distinguish two special conditions, constant pressure or constant volume, and so have two corresponding specific heats. For most bodies the specific heat is practically con- stant for all temperatures that are far removed from the melting points; and so, if c is the specific heat of a body whose mass is m, and if the temperature is raised through <2 — £j degrees, the number of calories added is mc(t^ — ^). C7/.l.V(;/:> /.V 'IKMrKHATURB 251 In iron, boron, carbon, and a few other substances the specific heat varies to a marked degree at different temperatures ; and with them c in the above expression is the "mean spe- cific heat " for the range from tl to tv Measurement of Specific Heat. — Corresponding to the three methods of measurement of heat energy
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referred to above there are, then, three methods for the measurement of specific heat of a given substance. We assume in each case that there is no loss of heat by radiation, etc. 1. Method of Mixtures. If iw, = mass of body, wij = mass of water, /, = original temperature of the body, fa = original temperature of the water, ts = final temperature of equilibrium, ~«ift-W OF m2(<3-<,) The water is contained in some solid vessel, called a " calo- rimeter " ; and its temperature is affected by the changes in that of the water. If mz is its mass and c1 its specific heat, and if we assume that the changes in its temperature are the same as those of the water, the number of calories it receives is w8c'(/8— £a); and, since this energy comes from the foreign body hit rod u« •« -d into the water, it is seen that, in the above formula t'.u- <?, wa must be increased by w8<?'. This quantity!lrd tlir M wuter equivalent" of the calorimeter. mrtlind was fust used by Joseph Black about 1760; and it is the one most generally used at the present time. Tli.-ivaiv in an v objections to it, however. Chief among these are tin- ditVn -ulties of determining the water equivalent of the calorimeter and of avoiding losses by radiation, etc. Most of these are overcome in a modification of the apparatus due to l'rof«->sor \Vah-nnan <>f Smith College. 252 HEAT 2. Method of Melting Ice. If w<! = mass of body, 7/i2 = mass of ice melted, t = initial temperature of the body, m^t = 80 w?2. This method was also first used by Joseph Black. The obvious difficulty in it is the measurement of the quantity of ice melted. This has been overcome most successfully in a form of apparatus due to Bunsen ; but great skill is required to use it properly. 3. Method of Condensation of Steam. If 7/jj = mass of the body, m2 = mass of steam condensed, /! = initial temperature of the body, t2 = final temperature of the body, which is never far from 100° C., mlc(f2
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~ ll) ~ 536 m2' With this method there is a correction for the water equivalent of the calorimeter; and one of the chief difficulties is to measure accurately the quantity of steam condensed. The apparatus used was invented by Professor Joly of Dublin. Specific Heats of a Gas. — Any one of these methods can be used to measure the specific heat of a solid or a liquid ; but with gases a difficulty enters owing to their small density — the correction for the calorimeter is the larger part of the heat energy. To measure the specific heat of a gas at con- stant volume, the third method may be used if. the gas is compressed into a hollow sphere; but this is not very satis- factory. To measure the specific heat at a constant pressure the best method is to pass a large quantity slowly through a spiral tube which is immersed in a bath at a high tempera- ture, then through another spiral surrounded by water which is cooler and whose temperature is thus raised, and finally r/M.y ';/•;> L\ I i-'.Mi '/•;/,'. i /T/;/-: 253 out into the air or into some large reservoir. If the gas is forced through very slowly, its pressure remains practically constant. There are two indirect methods by which the specific heat of a gas at constant volume may be determined : one depends upon a knowledge of the ratio of this to the specific heat at tant pressure ; the other, upon a knowledge of the differ- ence between these two quantities. It may be proved by higher mathematics that the ratio of the specific heat at constant pressure to that at constant volume equals the ratio of the adiabatic coefficient of elas- ticity to the one at constant temperature (see page 194). But this last ratio is a constant for any one gas, which may be determined by measuring the velocity of compressional waves in this gas, as will be proved in the next section of this book. (See page 337.) It will be shown there that this velocity is given by the following formula, V = where p is the pressure of the gas, d is its density, and c is the ratio of the two elasticities, and therefore of the two specific heats. It is not difficult, then, to determine <?, since v, p, and d can all be measured; and. if < \, and Ov are the two ilic heats, one at constant piv>surc the
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in a unit volume of the gas, the energy of the molecules in this volume is then \mW^N. But wiN is the mass of the molecules in this volume; so the internal energy of a unit mass of the gas is \ bVz. This can be expressed in terms of the temperature; for the pressure of a gas is given by the formula p = dRT, if d is the density; and on the kinetic theory p = ^dF2. Therefore V* = $RT\ and the internal energy per unit mass is J bR T. So, if the tempera- ture is raised one degree, this energy is increased by an amount f bR. This quantity, then, is the specific heat of the gas at constant volume expressed in mechanical units ; or, in CHANGES IN TEMPERATURE 255 symbols, Cp = f bR. But Cp - CV=R-, hence Cp = Cv + -f 1). Hence the ratio of the specific heats __ ~~ 21 + The least value of b is 1 ; and therefore the greatest possible value of c is 1 + f or 1.67. For all other values of b, c is less than t Ins. | 6 It is a most striking fact that for certain gases, viz., mer-^ cury vapor, argon, helium, and a few others, the value of c found by experiments on the velocity of waves in them is 1.67, while for all other gases it is less than this, being about 1.41 for air, hydrogen, and oxygen, 1.26 for carbonic acid etc. Those gases for which c equals 1.67 are called by i-li'-mists "monatomic"; and, whatever value may be attached to the above assumptions, it is certain that a large value of c indicates an extremely simple construction of the molecule or a molecule whose internal energy is small; while a small value «»f /• indicates the contrary. A few values of specific heats are given in the following tabi AVERAGE SPECIFIC HEATS Alcohol.. 0°-40°C. 0.597 Aluminium. 0°-100°C. 0.2185 Brass.... 0.09 Copper.. 0°-100°C. <> (crown).. 0.161 In.n.. 0°-100°C. 0.11:1, 0°-100°C. 0.
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031 Mercury. Paraffin. Platinum Bihra Tin r, Water. Turpenti iii-. 20°-50°C.. 0°-100°C.. 0°-100°C.. 0°-100°C.. 0.o:W 0.0:^:5 0.467 1.00 Air.... j Chlorine. Carbon dioxide (JASKS. o.i»:J7.. 0.1'Jl.. O.'JO'J Of 0.171 "iv.; RATIO. l.40i. 1.06. ; l.:to 256 HEAT GASES — Continued Cp Cv RATIO Helium Hydrogen.... 3.40.. 2.40 Mercury (vapor) Xitrogen 0.244 Oxygen 0.217.(17.41.41.41 Law of Dulong and Petit. — When the values of the spe-, cific heats of a great many substances are compared, a connec- tion becomes evident between them and the " atomic weights " of the substances. (For an explanation of this last quantity some book on chemistry should be consulted.) This was first noted by Dulong and Petit. It is found that the product of the value of the atomic weight of any solid substance and that of its specific heat is a quantity that is approximately the same for all substances, viz., 6.4, using the ordinary system of units; while the same constant for gases is 3.4. This means that the same amount of energy is required to raise the tem- perature of an atom, whatever solid substance it belongs to. This product is called the " atomic heat." Naturally, this law of Dulong and Petit is only approximately true ; for, as has been said, the specific heat of a substance varies with the temperature, and it is impossible to know when different sub- stances are at temperatures such that their conditions are comparable. CHAPTER XIII CHANGE OF STATE Introductory. — The fact that heat energy enters or leaves a body when it changes its state is familiar to every one. In order to melt ice or boil water it must be exposed to some source of heat ; if water evaporates from one's hand or from the surface of a porous jar, tin* latter is chilled, showing that heat energy has been taken from it ; as steam condenses into water in steam coils or " radiators," they are heated,
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showing they have received energy ; when an acid or salt is dissolved in water, its temperature is changed; when water freezes or dew is formed, the temperature of the surrounding air is raised slightly, etc. During these changes of state, not alone are there heat changes, but alteration in volume, and so external work is done ; and we shall see that the external conditions are of fundamental importance. We shall con- sider in detail a few of the most important cases of change of state : viz., fusion, evaporation, sublimation, solution, and chemical changes. Fusion Freezing and Melting Point. — •• Kusion " is the name «_ri\ en tin- prOOeSfl in which a solid lm.lv in. -Its and becomes liquid ; the reverse process is called ki solidification." It' a solid l>od\ that can form crystals, e.g. ice, i^ exposed to a smir. heat, its temperature will rise until a point is reached when it begins to melt; and then, so long as there is any solid to melt, the temperature of the mixture of the solid and its liquid remain^ unchanged; hut when the solid is entirch i m -i. * — 17 267 258 UK AT melted, the temperature again rises. Conversely, if the liquid thus formed is placed in such a condition as to lose heat energy, its temperature will fall until a point is reached at which some of the liquid solidifies ; then the temperature remains unchanged until the liquid entirely changes into the solid form ; and from then on the temperature falls again. This temperature which marks the transition from liquid to solid is the same as that which marks the reverse change. It is called the "melting point," or the "freezing point." If the solid and its liquid exist together in contact, and no energy is added or taken away from them, they will remain in equilibrium ; so the melting point may be described also as that temperature at which the solid and its liquid can exist together in equilibrium. Effect of Variations in the Pressure. — Experiments show that this temperature of equilibrium depends upon the external pressure, varying as it is changed. Thus, if ice and water are in equilibrium together, under ordinary atmospheric pressure and in a region where the temperature is therefore 0° C., an increase in pressure will cause some of the ice to melt, showing that the melting point is lowered and that heat energy flows into the ice from the surrounding region ; a
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decrease in pressure will cause some of the water to freeze, showing that the melting point has been raised and heat energy flows out from the water into the surround- ing region. The explanation of this variation of the melting point with the external pressure depends upon the fact that when a solid melts its volume changes, in some cases increas- ing, in others decreasing. If ice melts, its size decreases ; that is, the volume of a certain mass of water in the solid form is greater than in the liquid. (Blocks of ice float on the surface of lakes.) Those metals which are used to form castings also expand when solidified ; but in general bodies expand. Thus, gold and silver coins are stamped, not cast, because they expand when melted. When ice melts, the CHAXQB "/' -I ATE 259 change in volume is of the same kind as would be produced l»v an increase in the pressure by mechanical means. In other w.»r<ls, increase of pressure helps on the process of melting; so that if ice is being melted by the addition of heat energy, the temperature does not need to be so high in order to secure melting ; i.e. the melting point is lowered. If a hody expands on melting, an increase in pressure will, for similar reasons, raise the melting point. Regelation. — This pressure effect is small. In the case of ice and water, if the pressure is increased from one atmos- phere to two, that is, by 76 cm. of mercury, the change in the melting point is only 0°.0072 C.; and, consequently, ordinary barometric variations have no measurable effect on the melting point of ice. The fact, though, that the melting point is lowered by an increase of pressure is shown by many familiar illustrations. If two pieces of ice with sharp points are squeezed together, the pressure may be enormous because the area of contact may be extremely small ; so the melting point of the ice at the points where the pressure is great will be lowered and, if the surrounding bodies are at 0° C., some of the ice will melt, and the result- ing water will be pressed out, so that it is at atmospheric pressure, and its freezing point is again 0°. But in order to melt this ice, heat energy must be taken from portions of the body near it, and their temperature is reduced below 0° ; so the melted water, with a freezing point of 0°, is
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in contact with ice at a temperature below 0°, and it will there- fore immediately freeze again. This is the explanation of t he formation of snowballs ; and this action also plays a most important part in the motion of glaciers. The phenomenon is called "regelation." n<>ti<m <>f :i in : is due to the fact that the ice •• pressure of the edge of the skate, and so he is actually •!iin layer of water. As the skate moves on, this water freezes again. Another illustration is the formation of so-called " ground 260 in-: AT ice," which is ice formed at the bottom of streams where there are eddies. The ice crystals are whirled round in the current and stick against the bottom, owing to regelation ; then others stick to them, etc. Non-crystalline Substances. — Bodies that are not crystal- line, like waxes, plumbers' solder, etc., do not have a definite melting point ; but as they are exposed to a source of heat, their temperature rises continuously until they are entirely melted. They pass through a " pasty " condition ; and the temperature at which this begins is sometimes called the melting point. Similarly, as the liquid is cooled it begins to pass into the intermediate condition at a temperature called the freezing point. These two temperatures are not the same. These temperatures are affected by changes in pres- sure exactly in the same manner as those of crystalline bodies. Undercooling. — The transformation from the liquid into the solid condition does not always take place as described above ; for instance, if water is cooled gradually, its tem- perature will fall far below 0° and yet there is no ice formed. Its condition is, however, most unstable, because if it is shaken or if a minute piece of ice is thrown in, the liquid will solidify immediately and the temperature will rise to 0°. This phenomenon of a liquid existing below its freezing point, as above described, is called " undercooling " ; it was discovered by Fahrenheit. "Heat of Fusion. " — The number of calories that must be added to a solid body at its melting point in order to make 1 g. of it melt, is called the "heat of fusion" at that temperature. (It should be noted that this number has the same value if we define the heat unit to be such a quan- tity of heat energy as will raise the temperature
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of a unit mass of water one degree Centigrade, and define heat of fusion as that number of these heat units which is required to melt a unit mass of the substance, quite regardless of the size of the unit mass.) This energy is spent in overcoming molecular forces and in doing external work if the body expands on r//.i.v<;A' OF 8TATM 261 melting; if it contracts, the external forces also do work in overcoming the molecular forces. The exact way in which this work is done cannot be determined ; but, it is evident tiiat, if the reverse process is carried out, the same amount of heat energy is given out by the body as is received during the direct one. Thus, the temperature of the air in a closed room may be kept from falling far below 0°, if a tub of water is {-lured in it; for, as the water freezes, a definite amount of heat energy is given off to the air. Again, by placing a pail of water under a fruit tree on a cold night, the fruit may be kept from being injured by the cold. The heat of fusion of a substance may be determined by various methods, such as are used for the measurement of specific heats. Thus, a known number of grams of ice may immersed (not allowed to float) in a vessel containing a known mass of water at a known temperature, and the fall in temperature may be noted. If rw, = mass of ice, ms = mass of water, including the watt-r equivalent of the calorimeter, /, = initial temperature of water, /t = final temperature of water, L = heat of fusion of the ice ; Effect of Dissolved Substances : Freezing Mixtures. - freezing point of a liquid is affected it' tin-re i> ;i I'oivi-rn sub- stance dissolved in it. In every case the free/in- point is lowered : ami the change is, within certain limits, propor- tional t<> the amount of dissolved substance iii a jjiven quan- tity of the solvent, for en-tain substances. With others, the change is abnormally great ; and it is to l>e noted that these -'instances arc those which have an abnormal osmotic pressure. (See page 1M, 262 HEAT If the temperature of a solution is lowered to its freezing point, the solid formed is that of the
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pure solvent, in general ; so that the solution becomes more concentrated. (In certain cases some of the dissolved substance is caught in the meshes of the solid solvent ; but this is a mechanical process, not a thermal one.) Then, in order to freeze out more of the pure solvent, the temperature must be lowered still further ; for, as said above, the freezing point of the solution falls as its concentration increases. A condition is finally reached with certain solutions such that the solution is saturated ; if now heat energy is withdrawn, some of the solvent separates out in the solid form, and at the same time some of the dissolved substance is precipitated; the temperature remains unchanged, and as more and more heat energy is withdrawn, equivalent amounts of solid solvent and dissolved substance separate out. This complex solid mixture is called the " cryohydrate " of the two parts. It is in equilibrium with the solution of the same concentration, as we have just seen, at a definite temperature ; so, if a cryohydrate is placed in a region at a higher temperature, it will melt. Thus, the cryohydrate of common salt and water has a composition of 23.8 parts by weight of salt to 76.2 parts of water, and its equilibrium temperature is — 22° C. ; so, if salt and ice are mixed thor- oughly and are at a temperature greater than this, the ice will melt and dissolve the salt. In this process the tempera- ture of the mixture and of surrounding bodies falls, because heat energy must be supplied both to melt the ice and to dissolve the salt. (See page 283.) If the salt and ice are in exactly right proportions, this process will cease when the temperature — 22° C. is reached. Such a mixture of two bodies as this is called a " freezing mixture " ; and the above description explains the use of salt and ice in lowering the temperature of surrounding bodies in "freezers," and also the effect observed when salt is thrown on ice or snow. A freezing mixture of solid carbon dioxide and ordinary CHANGE OF STATE 268 sulphuric ether, known as Thirlorier's mixture, allows one to secure a temperature as low as — 77° C. The fusion constants of a few substances are given in the accompanying table : FUSION POINT HEAT or Fusion 1100° C. Ice Iron 1400°-1600°C. Lead B2PC. -39°C. iry Sulphur Zinc 0°C. llf,°
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C. 415° C. 80 23-33 5.86 2.82 9.37 28.1 Evaporation Boiling Point. — If a liquid stands in an open vessel ex- 1 to the air, it is observed that the quantity of liquid continually diminishes ; it is said to " evaporate " ; the sub- stance passes from the liquid to the gaseous condition. The gas rising from a liquid is called a " vapor," and an exact di >tinet ion between gases and vapors will be made later. (See page 278.) This process of evaporation requires that energy should be constantly added to the liquid, as may be proved by direct experiment, or as is seen by the fact that the hand is chilled when any liquid evaporates from it. ie the process may be hastened by applying some int. -use source of heat to the liquid. If this is done, its tem- perature rises until a point IN reached \\hen bubbles of the : (mm in the liquid, rise to the surface, and break. When this stage of "boiling," or "ebullition," is reached, the temperature ceases to rise, and remains constant until all the liquid is boiled away. This temperature is known as the " l><>iliii'_r point," and it is found to vary with the pressure of the m en the surface of the liquid. This is 264 HEAT what we should expect, because in order that the bubbles may form and rise to the surface, the pressure of the vapor in them must be at least as great as the pressure of the air on the surface, and the pressure of the vapor as it rises from the surface must equal this; so, if this pressure on the surface is increased, the liquid must be raised to a higher tempera- ture before it will boil. Similarly, if the external pressure is decreased, the boiling point is lowered. The process of boiling is one, then, of what may be called kinetic equilibrium, depending upon the equality of the pressure on the surface and that of the vapor as it rises from the surface. Saturated Vapor. — If the liquid is contained in a closed vessel, it is observed that after a time the evaporation appar- ently ceases. The vapor above the liquid is now said to be "saturated." If the pressure and temperature of this vapor are noted, it is found that if the temperature is raised, the pressure increases, and more liquid is evaporated
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; while, if the temperature is lowered, the pressure decreases, and some of the vapor condenses to form more liquid. If the tem- perature is kept constant, however, the pressure remains the same, entirely independent of whether there is a small or a large amount of liquid present. This condition may be called one of statical equilibrium. On the kinetic theory of matter it is easily explained. The molecules of the liquid may attain sufficient velocity to break through the surface, thus requiring work to be done upon them. Similarly, the molecules of the vapor may strike against the surface and become entangled, thus losing kinetic energy. So, if both these processes go on together, there will be equilibrium when the number of liquid molecules which escape in any interval of time equals the number of vapor molecules which are retained by the liquid surface in the same time. Spheroidal State. — A simple illustration of the kinetic nature of evaporation is afforded by what is called the " spheroidal state." If a small quantity of water is allowed CUAXGE OF STATE 265 t<» llow gently out of a tube or spoon on to a metal surface which is at a high temperature — far "above 100° C. — and which is slightly hollowed out so that the water will not run it is seen to collect in a flattened drop, which does not on the surface, and it now rapidly evaporates. The explanation is evident; for, owing to the rapid evaporation on the lower side occasioned by the heat energy received from the hot metal, the molecules are leaving tin- drop on this side with such velocity and in such quantity that their mechanical reaction holds up the drop. There is thus a layer of vapor between the drop and the hot plate. This spheroidal state can be noticed when water is spilled on a hot stove, and in fact the hotness of a stove or a flatiron is often tested in this manner by seeing if it can produce this state in small drops of water. (This condition of a drop is sometimes called " Leidenfrost's Phenomenon," because tin- first observation of it U attributed to him, 1766.) Vapor Pressure. - One of the simplest :i<»ds of obser \inir the phenomena of satu- d vapor is to intro- duce a small quantity of the liquid to be erimented <»n above the mercury column in a barometer which! a deep basin. Some of :i 190. — Experiment
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* illtintrmtlnjr tl preMare of Mtnrated vapor : » umall u ll.ini.l. t.g. water, to Introduced there tl the liquid will evaporate, and equilibrium will lx» reached at a pressure d. -pending upon the temperature. This ma\ 266 HEAT be varied at will by surrounding the tube with a bath of some liquid whose temperatures can be regulated. The pres- sure of this saturated vapor may be measured in terms of the atmospheric pressure by the ordinary law of hydrostat it- pressure. (If h is the difference in height of the mercury column in this tube and of that in a barometer, the pressuiv of the vapor is less than the atmospheric pressure by dgh, where d is the density of mercury.) If the temperature is kept constant and the ba- rometer tube slowly raised, some liquid will evaporate, but the pressure remains con- stant; similarly, if the tube is slowly pushed down, some of the vapor is condensed, but the pressure does not change. (If there were a gas above the mercury column in- stead of the vapor, its mass would not L00° 200° Fm. 121. — Curve showing connection between the tem- perature and the pressure of saturated water vapor. change, but its pres- sure would decrease and increase during the above changes, obeying Boyle's law : pv = constant. But when there is a vapor in contact with its liquid, there is no change in pressure, but the mass of the vapor increases and decreases.) If, however, the temperature is increased, some liquid evaporates and the pressure increases ; and, if it is lowered, some vapor condenses, and the pressure decreases. (If CHANGE OF STATE 267 there were a gas above the mercury column, its mass would remain constant during the above changes in temperature, a n« I its pressure would change, but at a different rate from that of the saturated vapor.) There is thus seen to be a delinite pressure of the saturated vapor which corresponds to a definite temperature when there is equilibrium, and conversely; the corresponding values may be found by either the statical method or the kinetic one, in which the boiling point at different pressures is determined. The results may be expressed by a curve drawn with axes of temperature and pressure. This curve for water vapor is given. That the boiling point varies directly with the pressure is shown l>y the fact that the
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temperature of boiling water is much less than 100° C. on a mountain top ; by the high tem- perature in steam boilers where the pressure is great, etc. From the description given above of the statical method, it is seen that there are two general methods available for con- dt nsiii'^ a vapor into a liquid : one is to lower the temperature ; the other is to decrease the volume. These will be discussed in full in a later section. If a vapor is not saturated, it obeys hiiyle's law quite closely ; and Dalton's law is also approximately exaet for the pressure produced by a mixture of vapors. A curve may be drawn that will >s this law of a vapor in con- tact with its liquid when the tem- i tu re is constant. Lay off axes « of pressure and volume; then, since a vapor keeps its pressure constant so long as the temperature is con- VOLUMES. the isothermal is a straight Fi«.i«. line parallel!•• theaxis of volumes. Formation of Dew, Clouds, etc. — We have seen that, as the temporal UP- d, the corresponding vapor pressure of -a i u rated vapor becomes less ; that is, if there is a certain 268 HEAT amount of unsaturated vapor in a closed vessel, it will become saturated if the temperature is lowered sufficiently ; and then, if it is lowered still more, some of the vapor will condense. This is illustrated by the formation of dew, of clouds, etc. There is always a certain amount of moisture in the air, and the method of expressing it is as follows : we measure the temperature of the air and then by experiment find what temperature some solid body — like a metal can — must have in order to make moisture condense on it. This is called the " dew-point." The vapor pressure corresponding to these two temperatures is found from tables or from the curve given on page 266 : one of these expresses the pressure that the water vapor in the air might have if it were saturated at the existing temperature ; the other gives the pressure that the water vapor in the air actually has. The ratio of the latter to the former gives what is called the "humidity." If we assume the truth of Dalton's law, we can easily calculate the mass of a unit volume of ordinary damp air. This equals the sum of the masses of the air itself and of the water vapor in the space. The quan
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- tity of water vapor in a unit volume corresponding to various dew-points is given in tables ; e.g. if the dew-point is 10° C., the mass per cubic metre is 9.3 g. The pressure of this vapor corresponding to 10° C. is 0.914 cm. of mercury ; thus, if the barometric pressure is 76 cm., the pres- sure due to the air is 75.086 cm.; and, if the temperature of the air is known, e.g. let it be 20° C., the mass of the air can be calculated, since the density of dry air at 0° C. and 76 cm. pressure is known to be 0.00129. Thus, using the gas law, the density at 20° C. and at 75.086 cm. pressure equals |Z?. 0.00129, or 0.00119. Therefore the mass of the air in a cubic metre is 1190 g. ; and the total mass of the cubic metre of damp air is 1190 + 9.3 = 1199.3 g. If the air were perfectly dry and at 20° and 76 cm. pressure, its density would be |^ x 0.00129, or 0.00120 ; and so the mass of a cubic metre of it would be 1200 g. It is seen, then, that the mass of dry air is greater than that of damp. Boiling. — As already explained, the process of boiling consists in the formation of bubbles of the vapor in the interior of the liquid. Nuclei of some kind are required in order for CHANGE OF STATE these to form, such as sharp points or minute bubbles of some foreign gas, like air. As a liquid boils, the supply of such nuclei is used up, unless in some way it is renewed constantly, and it becomes more and more difficult for the liquid to boil. The temperature rises above the boiling point until the molec- ular forces are sufficient to form the bubble; there is a miniature explosion; and the temperature falls back to the boiling point. If the nuclei are removed from the liquid as completely as possible and if the walls of the containing vessel are smooth, the temperature of the liquid may be raided far above the boiling point; but this condition is of course unstable. If the liquid were entirely free from nuclei, it would never boil; but,! temperature were gradually raided, it would finally explode. Heat
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attached to an air CIIAM.I: <>r >TATE 271 care being exercised to guard against any possible conduc- tion of heat to the water; as the air is exhausted, thus diminishing the pressure on the water and removing the vapor, the water evaporates so rapidly that the heat energy required is taken from the water left behind, and its temperature falls until it freezes. A somewhat similar experi- ment is one that involves the use of the *' cryophorus," an instrument invented l»y Wolluston. This consists of two glass bulbs connected by a bent glass tube, as shown in the cut in two forms, a and b. There is sufficient water inside to half fill one of the bulbs. The experiment con- sists in placing the instrument a in a vertical position, with the curved portion uppermost, or the instrument b horizontal ; the water is poured into one bulb, which is carefully shielded against heat loss or gain, and the other is surrounded by a freezing mixture of salt and ice. After a short time the water will be found to be frozen, owing to rapid evapora- tion at its surface, which is caused by the continuous condensation and ing of the vapor in the other bulb. The action, then, is the same as if a substance " cold " were carried from one bulb to the other ; hence th»- name •• rryophorus," which means "carrier of cold." Some forms of apparatus for making artificial ice depend upon the same fact, that when a liquid evaporates, heat energy is required. In most cases the liquid which is evaporated is ammonia. This is placed in the space between the two walls of a double-walled vessel which contains water, and as the ammonia is evaporated, the water is frozen. Steam Engine. — As a further illustration of the proper- ol a vapor, the steam engine may be mentioned — a dia- gram of a simple form of which is given in the cut. The principle of its action is as follows: Steam is produced in a "boiler" under hi^h pressure, and therefore at high tem- perature; this steam is allowed at regular intervals to enter the "cylinder." in which there is a movable piston, at the instants when the piston reaches one end of the cylinder, and in such a manner as to exert a pressure on this piston, pushing it away from this end. Steam continues to enter, and the pressure is that
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of the steam in the boiler as Imi^ as the connection is maintained; but after the supply of steam is cut off, the steam, as it expands, decreases in pres- sure. In the meantime the pressure in the cylinder on the other side of the piston has been made as small as possible 272 HEAT by one of three methods: (1) by opening it to the atmos- phere— this makes a kt non-condensing" engine; (2) by join- ing it to a large vessel, which is kept as nearly exhausted as possible by means of pumps and as cool as possible by means of coils or jets of water — this makes a "condens- ing " engine ; (3) by join- ing it to another cylinder, exactly like the first one, only larger, and allowing the steam ejected from the first cylinder to work the piston FIG. 125. — Steam Engine. CHANGE OF 8TATM 273 in the second — this makes a "double, triple, etc., expan- sion" engine. When the piston reaches the other end of the cylinder, connection is made with the boiler, as above described, at this end, and the expanded steam is expelled, as the piston retraces its path, in one of the ways just mentioned. In the case of the non-condensing engine the steam escapes to the air and is lost; in that of the condensing engine it is condensed into water in the con- denser and pumped back into the boiler; in the expan- sion or compound engine it is used over again, expanding more and more until, finally, it is condensed in a condenser and pumped back to the boiler. These changes are made automatically by certain valves ; the " sliding valve " opens and closes the inlet pipes and also the exhaust pipe to the condenser. In all cases a certain quantity of saturated vapor is received at a definite pressure (and corresponding temperature) ; this expands, doing work on the piston, until a certain lower pressure is reached, then — in the condens- ini: engine — it is condensed to water at the pressure corre- sponding to the temperature of the condenser; this water is pumped into the boiler, its temperature is raised until it boils, and the process begins again. There is t hus a " cycle " of changes. While in the boiler, the water receives heat energy; and when water is lu-ing formed in the condenser,
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heat energy is given out by the steam. The steam does work in pushing on the piston, and work is done on it and on the water formed from it when the piston performs its reverse motion. If H is the heat energy received at the temperature of the boiler, and W is the net external work W H. done, the ratio -= is called the "efficiency" of the process. The amount of work done may he measured by a simple mechanical device. We have seen (page 161) that on a pressure- \o] u in. diagram, a closed curve.describing the series of changes tlin>u<_:h whieh a tin id passes indicates by its area AMES'S PHYSICS — 18 274 HEAT the total net work done. The curve giving the cycle of changes just described for the steam leaving the boiler, ex- panding, etc., must be somewhat as shown in the cut. A marks the instant when the water in the boiler begins to be changed into steam ; B, when this process is finished, the pressure and temperature remaining constant; this steam can be imagined as formed directly back of the piston and exert- ing a pressure on it; so the instant marked by B is that of the " cut-off "; the steam now expands, and both temperature and pressure fall; at (7, connection is made with the condenser, in which the pressure is that marked by the horizontal line ED\ so the pressure falls to D, and the steam condenses to water as marked by E\ this water is forced by a pump into the boiler, and its temperature and pressure both increase until the point A VOLUMES v is again reached. Looked at from another point of view, this curve indicates very approxi- mately the changes in volume and pressure of the space in the cylinder which is open to the steam as the piston moves to and fro : when the piston is close to one end, the volume is small, and as steam is admitted, the pressure rises from E to A ; then it remains constant as the piston is pushed out, thus increasing the volume, until the cut-off is reached at B ; then, as the piston continues to move forward, the volume increases and the pressure falls; O marks the end of the motion of the piston ; as it moves back, the volume decreases, but the pressure remains unchanged, being that of the condenser ; etc. These pressure and volume changes may be recorded automatically by the engine
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, as the isothermals at higher and higher temperature are drawn, they have the same general shape, but the horizontal portions become shorter, until a temperature is reached, the isothermal for which has no horizontal portion. This is called the "critical" temperature. For temperatures higher than it, the isothermals approximate more and more closely to those of a gas. The Critical Temperature. — The only points on the dia- gram for which the matter is in the form of a liquid are along the horizontal portions of the isothermals, where there is a free surface separating the liquid and its vapor, and along the continuations of these lines to the left, where the liquid completely fills the vessel. Consequently, whenever we see a liquid partially filling a vessel, it is represented on the diagram by a point on a horizontal portion of an iso- thermal. In other words, if a vapor is to be liquefied, it must be at a temperature whose isothermal has a horizontal portion, that is, it must be at a temperature lower than the critical one. Bearing this fact in mind, all gases, with the possible exception of helium, have been liquefied. The name "vapor" may then be restricted to a body in the gaseous condition which is at 'a temperature below the critical one ; while the name " gas " may be limited to temperatures above this. The exact point on the diagram, marked by A, where the critical isothermal has its point of inflection, that is, the point where an isothermal infinitely near the critical one, but below it, has a minute horizontal portion, is called the "critical point." If we have matter in this condition filling a vessel, and if the temperature is lowered, even the least amount, the liquid will separate out and sink to the bottom, filling approximately half the space and leaving the rest full of vapor. At the critical point, then, the surface of separa- tion disappears, and the liquid and vapor dissolved in each CHAtfQM OF -/.I//-: 'J7'.' other make a homogeneous form of matter. The pressure corresponding to the critical point is called the "critical pressure," and the volume of one gram of the substance in this critical condition is called the "critical volume." In order to liquefy a gas, then, two steps are necessary ; the temperature must be lowered below the critical tempera- ture, and the volume must be decreased until the pressure
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is reached that corresponds to the state of saturation of the vapor for the temperature. After this, any further decrease in volume or temperature will cause the vapor to condense. The values of the critical temperatures for various gases are given in the accompanying table : CRITICAL TEMPERATUI:I - Alcohol... 243°.6C. Ammonia... 130° C. Argon. - 120° C. Carbon dioxide.. 30°.9 C. Chloroform 260° C. Hydrogen... - 242° C. Nitrogen. - 146° C. Oxygen... - 119° C. Sulphur dioxide.. 156° C. Water. 365° C. Liquefaction of Gases. — The critical temperatures of such gases as hydrogen, oxygen, nitrogen, etc., are seen to be extremely low ; so that special means must be adopted in order to liquefy them. There are three methods in use for the production of low temperature ; application of freezing mixtures, rapid evaporation of a liquid, expansion of a gas from lii^h to low pressure. (See page 246.) The standard method for liquefying gases is a combination of these to a certain degree. The gas to be liquefied is compressed by pumps to a high pressure and is cooled by a freezing mixture or l»y evaporation of a liquid : it is then allowed to expand through a small opening, is again compressed ; and tin- process is repeated. This expanded • older than it was : iind 1). -fore iM-injr compressed again, as.is it expandfl through the opening, it i^> drawn i the mi IQ unexpand : hus chilling 280 HEAT it. As the process continues, the temperature of the com- pressed gas gets lower and lower, until finally, on expansion, the critical temperature is passed and drops of the liquefied gas fall to the bottom of a Dewar bulb. Two forms of apparatus are shown in the cut on page 281 ; one due to Linde, the other to Dewar. The most important parts of the former are the two-cylinder air compressor and the "counter-current interchanger." This last consists of a triple spiral of three copper tubes wound one inside the other. The cycle of operation is performed in such a manner that compressed air at the temperature of the coil g and at about 200 Atm. flows through the inmost
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., etl, - a thlrtl ring the g«* thr»ui;li //.''. • Nrft|K- tliniinrh hjr the rod f. As the gas escni <lrawn out through t a-« It rises it c<>. 282 HEAT The melting point of hydrogen is estimated at — 257° C. ; and the max- imum density of liquid hydrogen is 0.086. By allowing a quantity of liquid air to evaporate slowly, and using the methods of fractional distillation, Ramsay discovered three new constituents of our atmos- phere, which he called Krypton, Xenon, and Neon. Continuity of Matter. — It is seen from the experiments of Andrews that it is possible to make a body pass from the state of vapor to that of liquid, and vice versa, by a series of continuous changes, because a path can be drawn from a point on the pressure-volume diagram where the body is in the form of a vapor to one where it is liquid, which does not pass through the region where the isothermals are horizontal, and where therefore the liquid and vapor exist separately, but in contact. The isothermal has two points of discon- tinuity, at the two ends of its horizontal portion, which correspond to molecular • rearrangements ; but a series of changes can be imagined, as above described, during which there will be no sudden molecular changes, but by which a vapor can be gradually and continuously changed into a liquid. This is ordinarily expressed by saying that matter is continuous from the liquid to the vapor state. Similarly, matter is continuous from the solid to the liquid state. Sublimation In many cases a solid body evaporates directly without passing through the liquid condition. This process is called " sublimation " ; and it is illustrated by camphor, arsenic, iodine, carbon, many metals, snow or ice, etc. It is found by experiment that, if this process takes place in a closed vessel, there will be equilibrium between the solid and the vapor when a definite pressure is reached, which depends upon the temperature. If the latter is increased, the equi- librium pressure is higher, and conversely. The reverse of this process of sublimation is seen in the formation of frost. It is found that all these substances which sublime under • ll^^<,l•: or STA n-: ordinary conditions may be obtained in the liquid condition it'
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a suitable pressure ami temperature are applied. (This i> one step in Moissan's method of making artificial diamonds. ) The number of calories required to make one gram of a solid sublime at a definite temperature is called the "Heat of Sublimation." It equals the sum of the Heats of Fusion and of Evaporation at that temperature, in accordance with the principle of the Conservation of Energy. Solution Heat of Solution. — When a body — solid, liquid, or gas — is • Ivi-d in a liquid, both being at the same temperature, tin-re is a change in temperature, showing that heat changes are involved. If the temperature falls, it shows that work is required to make the substance dissolve, if we assume that tin-re are no secondary molecular changes such as the forma- tion of ne\v molecules or dissociation. The heat energy that is gained or lost is measured by the product of the mass of the solution, its specific heat and the increase or decrease of temperature. If the temperature rises, heat energy is said to be " evolved " ; if it falls, the energy is said to be "absorbed." If one gram of the substance is dissolved in a eertain quantity of solvent, the heat energy thus involved will, in general, vary with the quantity of solvent ; but, by continually increasing this, it is found that after a certain point the heat changes are independent of the quantity of solvent. The heat energy gained or lost, when one gram of a substance is dissolved in such a large quantity of the solvent as this, is called the " Heat of Solution " of the substance. Some values are given in the following table, the solvent being water. A plus sign indicates a rise in temperature, or "evolution of heat," and a mi mis si LTD, the opposite. Ammonia ffaa. +495.6 calories. Caustic potash. •• ones. Ethyl alcohol. + 55.3 calories. Sodium Chloride. - 1 XL' calories. Sulphuric ;i<i<l. + 182.5 calories. Silver chloride. —110. calories. IIKAT Effect of Rise in Temperature upon Solution. — We can from these facts predict whether raising the temperature of a solu- tion increases its solubility or not. Let us consider a saturated solution with an excess of dissolved substance precipitated ; and let it be one in which heat energy is absorbed when solution takes place. If
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now heat energy is added, the sol- vent will dissolve more of the substance ; and if heat energy is withdrawn, the solvent will precipitate some of its dissolved substance ; because in this last case, for instance, if the effect were to make the solution more soluble, this act of solution would withdraw some heat energy, etc., and the condition would be unstable. Therefore, with a solution of this kind, an increase in temperature makes it capable of dissolving more ; a decrease, less. Just the converse is true of those solutions in whose formation heat energy is evolved. Dissociation ; Ions. — The fact that certain solutions have an abnormally great osmotic pressure, and also have freezing and boiling points that differ abnormally from those of the pure solvent, can all be explained if it is assumed that in these solutions a certain proportion of the dissolved molecules are broken up into smaller parts. This has the immediate effect of increasing abnormally the number of moving particles due to the dissolved substance. If we assume, further, that these fragments of the molecules are electrically charged, we can explain the phenomena of electrolysis, as we shall see later ; and all solutions of this kind do conduct electric currents. The whole science of Physical Chemistry is based upon these two assumptions, and they may be regarded as justified by experiments. When such a solution is formed, some of the molecules dissociate into their parts since thereby the poten- tial energy is made less ; or, as we may express it, there is a force producing dissociation. The process ceases — or equi- librium is reached — when this "solution pressure" is balanced by the electrical forces that are called into action. The equilibrium is not one in which there is no further dissocia- tion, luit one in which for each molecule dissociated there is formed. The dissociated parts, called "ions," are mov- ing t<> and fro in the solvent, and combinations and dissocia- tions are taking place continually but at the same rate. Chemical Reactions Heat of Combination. — In all chemical reactions there are molecular changes, and consequent heat changes. If the ;in^ bodies are gases, these changes depend to a marked degree upon the external conditions that are maintained ; for these determine the amount of external work. It is entirely immaterial, however, whether the change takes place in one or more stages. Thus, if one gram of carbon in the form of a diamond is converted into carbon
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monoxide (CO), -11<» calories are evolved: and if this is converted into carbon dioxide (CO2), 5720 more calories are involved — 7860 in all. And, if the same amount of carbon is oxidized at once into carbon dioxide, the heat involved is the same. This is an illustration of the Conservation of Energy. A few illustrations of these heat changes may be given. : <j. of hydrogen L^as and 16 g. of oxygen, at 0° C. and 76 cm. pressure, combine to form 18 g. of water at 0°, the heat energy evolved is 68,834 calories. If 65 g. of zinc are •1 ved in dilute sulphuric acid, 38,066 calories are evolved ; while if 63 g. of copper are dissolved in dilute sulphuric acfd, 12,500 calories are absorbed. Dissociation. — One of the most interesting chemical changes is the dissoci.it inn of a '_r;ls i,,((, other gases. Thus some gases with complex molecules break down into others with simpler molecules when the temperature is raised to a high degree. In many cases it is observed that for a definite temperature equilibrium is reached at a definite pre- nd, if the temperature is increased. H Lfl the •ndiiiLf pressure. This condition of equilibrium is one of continual recombination and d "ii. CHAPTER XIV CONVECTION, CONDUCTION, AND RADIATION It has been shown that there are three general methods by which heat energy is added to or taken from a body : Con- vection, Conduction, and Radiation. Each will now be dis- cussed in turn. Convection When a vessel containing a liquid is placed on a hot stove, the upper layers of the liquid receive heat energy from the lower ones by the process known as " convection " ; the por- tions of the liquid in the lower layers have their temperature raised and therefore expand ; and, since their density is thus diminished, they are forced upward by gravity, the cooler upper portions sinking. Thus the temperature of the whole liquid is made uniform. The mechanics of the phenomenon is not difficult to understand, because the molecules of the hot upward-moving portions of the fluid communicate by their impacts some of their energy to the other molecules ; and thus the internal energy is distributed. It is evident that convection processes can occur only in fluids under the action of gravity, and
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when the heat energy is applied to its lower portions. It should be noted that the energy is gained by the portions at low temperature and is lost by those at a higher temperature. This method of distribution of heat energy is the one that forms the basis of the ordinary means of heating houses, — hot-air furnaces and stoves, hot-water systems, steam pipes, etc. Convection is of funda- mental importance in the economy of nature, as is explained in Physical Geography, in connection with winds, ocean currents, etc. The metal 286 CONVECTION, CONDUCTION, AND RADIAtlON 287 •in of a tea kettle (or of a steam boiler) does not become unduly hot so long as there is water in it, localise, owing to convection, the cooler portions of the water are being continuously brought down to the >m. Conduction General Description. — When one end of a inetal rod, like a poker, is put into a fire, the temperature of this end rises, and in a short time that of the other portions not too far from the fire rises also. The temperature of the end in the fire is the highest; and that of the other points of the rod decreases gradually as one passes from the fire, until a point in the rod is reached that is at the temperature of the sur- rounding air. If a thin transverse portion or slab of the rod is considered after it has come to a steady state, it is evident that its side near the flame is at a higher temperature than the one away from it; the molecules at the former section have more energy of motion than those at the latter. As a consequence, the former molecules give up some of their energy to the molecules of the slab ; and its temperature would rise were it not for the fact that the slab is losing lit at energy by convection in the surrounding air (and by radiation, also, a process to be described presently), and that tin- molecules in the cooler end of the slab are themselves handing on energy to the other portions of the rod. This process by which molecules give up some of their energy to iguous molecules, there being no actual displacement as in convection, is called "conduction." Thus, considering lah across the rod, we say that it gains heat energy at the hot face and loses it at the cooler one by conduction : and the difference between the «|uantities gained and lost must e<|i
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ial that lost at the surface of the rod by convention 1 radiation), -iuce the rod i- in a steady state. It is important to note that the heat energy is conducted from the hotter portions of matter to the colder ones. When the >t in a steady state, e.g. immediately after one end is 288 BEAT put in the fire, part of the energy that enters the slab to raising its ti-inperature, to doing external work, etc. " Conductivity " for Heat. — If the rod is in a vacuum, there is very little energy lost, in general, from the surface, because there is now no convection ; and, when the bar is in a steady state the energy conducted in at one section of a slab equals that conducted out at the other. If t± is the temperature of one section and t2 that of the other, if A is the area of each section and a the thickness of the slab, ex- periments show that the quantity of heat energy conducted through from the former section to the latter, if t± is greater than £2, is proportional to — ^—, but is different for rods Sf £ \^ of different material. This fact may be expressed by the following equation, in which Q is the quantity of heat energy conducted by the slab, Q = k— 2A, where k is a factor of proportionality, which is different for different bodies. It is called the " conductivity " for heat. If the conductivity of one body is greater than that of another, it is said to "conduct better." Thus silver conducts better than copper; copper, better than iron ; all metals, better than wood and other non-metals ; etc. The conductivity of any one body varies slightly with its temperature. If the conductivity of a fluid is to be determined, the upper surface must be made the hotter, so as to avoid con- vection. All liquids conduct poorly, with the exception* of fused metals ; and all gases conduct still worse. Thus loss of energy from a body by the processes of conduction and convection may be avoided by inclosing it in a quantity of eider down, feathers, or loose wool or felt; because these solids are poor conductors and motion of the air inclosed by them is prevented, as it is contained in small cavities. The best method of all, however, for avoiding these losses is to have the body inclosed by another and to have the
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space CONVECTION, CONDUCTION. AND HA />/ J77OJV 289 between completely exhausted of air. I>\ using a Dewar tlask (sec page -'28), liquid air and hydrogen may be kept for hours in a room at ordinary temperatures. Illustrations. — The fact that metals conduct well is shown by count- less experiments. Thus, if a piece of wire gauze with fine meshes is lowered over a flame, e.g. one from ;i Bunsen burner, the latter burns below the gauze only ; because the molecules of the gas as they pass through the wire meshes lose so much of their heat energy by conduction to the outer portions of the gauze beyond the flame that the temperature of the gas as it rises through the -;ui/. • is lower than that at which if burns. However, the temperature of the gauze gradually rises, owing to the Maine, and as soon as the temperature of combustion is reached, t In- flame will Imrn on both sides of the gauze. Or, if the gas is turned on through the burner, but is not lighted, and the gauze is held close to the burner, the gas rising through the gauze may be ignited by a match, but the flame will not strike back below it. (This is the principle of the miner's safety lamp invented by Sir Humphry Davy.) Again, a bright luminous flame may be made smoky by bringing a large piece of metal close to it, so as to conduct off the heat energy and thus lower the temj>erature. The cracking of a tea cup or tumbler when hot water is poured into it is due to the sudden expansion of the inner surface before the outer one has time to be affected; this may often be prevented by putting a silver spoon (not a plated one) in the cup or tumbler and pouring in the water along it ; the silver is such a good conductor that it prevents the temperature from rising too high at once. Conduction in a Gas. — The process of conduction in a gas is evidently simply the redistribution of the kinetic energy of the particles. When the temperature is high at one point in the gas, the kinetic energy there is great; and SO, owiiiLT t() the increased velocity of these particles, this energy is communicated to the neighboring ones. On the assumption that a gas behaves like a set of elastic spin we can deduce a value for the conduct i
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\ it \ in terms of the mean free path, etc. (See page 202.) The conduetU ;i few bodies at 0° C. are given in the following tahle, in which the heat unit is a calorie and the C.G.S. system is \\«-<\: AMES'S PHYSICS — 10 HEAT Silver 1.096 Copper 0.82 Aluminium 0.34 Zinc 0.307 Iron 0.16 Mercury 0.0148 Water 0.0012 Radiation Radiation as a Wave Motion. — When one's hand is ex- posed to sunlight, a sensation of hotness is perceived ; simi- larly, if a body is brought near a flame, — even when not above it, — its temperature rises, or if brought near a block of ice, its temperature falls. There is neither convection nor conduction involved in these changes of temperature, yet heat energy is being gained or lost. The process is called "radiation." Boyle noted as early as the seventeenth century that it went on through a vacuum, and this fact is proved also by the heating action of the sun which we observe here on the earth. When we discuss, in Chapters XVI et seq., the phenomena of waves and show how wave motions may be detected, it will be proved that this process of radia- tion consists in the motion of waves in the ether, i.e. in the medium which occupies space when ordinary matter is removed and which permeates ordinary matter as water does a sponge or as air does the stream of motes revealed by a beam of sunlight entering a darkened room. Without going into details in regard to waves, several facts may be men- tioned which are familiar to every one from observations of waves on lakes or the ocean, or of waves along a rope. One is that in wave-motion we do not have the advance of matter, but the propagation of a certain disturbance or condition; each particle of matter makes oscillations about its centre of equilibrium, but does not move away from this as the wave itself advances ; and therefore by the " velocity of waves " is meant the distance this disturbance advances in a unit of CONVECTION, CONDUCTION, AND RADIATION 291 time. (Consider the waves produced in a long, stretched rope when one end is shaken sidewise.) Thus, in order to produce waves, there must be some centre of disturbance or vibration, and this centre is giving out energy,
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for it is evi- dent that a medium through which waves are passing has both kinetic and potential energy. Thus, as waves advance into a medium, energy is carried forward, owing to the action of the particles of the medium on each other. We say, then, that "waves carry energy," although of course this energy is associated with the material particles. If wave motion ceases gradually as the waves enter a different medium (e.g. if a stretched rope is so arranged as to pass through some viscous liquid, waves sent along it will cease when they enter the liquid), this medium is said to "absorb" the waves; it gains the energy which the waves carry. Further, waves t different lengths, depending upon the nature of the dis- t iii-bin^ vibration ; to produce short waves, that is, waves in which the distance from crest to crest is short, requires very rapid vibrations; while long waves are due to slow vibra- tions. We know, too, that waves suffer reflection, as is seen when water waves strike a large pier with a solid wall. If several waves are passing through the same medium at the same time, the resulting motion is the geometrical sum of idividnal waves. Production of Radiation. — When ether waves are discussed, it will be shown that they have a velocity of 8 x 1010 cm. per second, or about 187,000 mi. per second, and that they are known to have lengths varying from -m-fc^ cm. up to man\ kilometres, depending upon the frequency of the vibrating centre where they are produced. It will be shown presently that all portions of matter, whatever their temper- ature, are producing spontaneously waves in the ether, whose lengths are so small as to be comparable with the size <>f HIM], The exact mechanism of this is not kno\Mi : but it is clear that there must be some mechanical connection 292 HEAT between the ether and the minute particles of matter, and that these last must be making exceedingly rapid vibrations. If the wave length is called I and the velocity of the waves v, the number of vibrations in a unit of time is -, because L during each vibration the waves advance a distance I ; and so, if there are n vibrations in a unit of time, the waves advance a distance nl in that time, or v = ril. If, then, there are waves in the ether whose length is y^ol7 cm'> the
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num- ber of vibrations per second is 3 x 1010 x 104 or 3 x 1014, i.e. 300 trillions. Consequently these vibrating particles are thought to be the infinitesimal parts of a molecule. Our conception, then, of the structure of ordinary matter is as follows : it consists of molecules which are moving to and fro, vibrating about centres of equilibrium in solids, or traveling from point to point in a fluid, and at the same time the parts of the molecule are vibrating and producing the waves in the ether — this is similar to the case of a man moving a ringing bell, for the bell moves as a whole and its parts are making vibrations. Ether waves are also produced during certain electrical changes, as will be shown later ; and they are short if the body experiencing this change is small, but long if the body is large. Measurement of Radiation. — In order to study the nature of these waves in the ether and the connection between them and the material bodies which produce them, it is necessary to have some instrument which will detect their presence and measure the energy they carry. To do this some body must be found which absorbs the waves, for then some change which can be observed will be produced in it, depending upon its own properties and the length of the waves. Thus, if the waves are short, they may produce vibrations in the parti- cles of the molecules in accordance with a simple mechanical principle known as that of "resonance." This is illustrated by a boy setting in motion another who sits in a swing ; this CONVECTION, <<).\1>I < 770.V, AND RADIATION 203 has a natural period of vibration, and may receive a large amplitude if a series of pushes are given each time the swing passes through its lowest point, going in the same direction ; hut if the pushes are given at irregular intervals, one may neutralize another ; so the force applied must have the same period as the natural period of the swing. Similarly, if the waves of a certain period enter a material body, the particles <>t' whose molecules have a natural period the saint- as this, they will be set in vibration by the waves, and will therefore absorb them, gaining their energy. (If the waves have a much longer period, they may produce electrical changes in the body.) If the energy of the waves is absorbed by the particles of the molecules, further changes will occur, deter- mined by the nature of the molecules
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glass is not, — and containing a fine vertical quartz fibre which carries a horizontal arm; to each end of this is attached a thin piece of mica, polished on one side and blackened on the other. The blackened face of one mica disk comes opposite and parallel to the fluorite window; so, if radiation enters this, it falls upon the blackened face of the mica, whose temperature therefore rises and which then moves backward. This motion twists the quartz fibre; and when the torsional moment of reaction of the fibre equals the moment due to the "repulsion" of the blackened disk occasioned by its rise in temperature, everything comes to rest. The angle of deflection of the horizontal arm measures, then, the intensity of the radiation. Another class of instrument must also be used in order to describe radiation from any source; this is one which ana- lyzes it, and so distributes it that waves having different wave lengths proceed in different directions and may be studied separately. This process is called " dispersion " and is illus- trated by the action of a glass prism on the light from a lamp. Radiation Spectra; Energy Curves. — Using these instru- ments, certain facts have been established. All material bodies in the universe, so far as we know, are producing waves in the ether. Solid and liquid bodies emit waves of all wave lengths between certain limits, whereas gases emit trains of waves of definite wave lengths. The emission of a solid or liquid depends largely upon the condition of its < »\VECT1ON, CONDUCTION, AND RADIATION 295 surface, other things being the same. A polished metallic sin face emits very little radiation, i.e. the energy of the radia- tion is small; whereas, rough or blackened surfaces emit a great deal. ( This is the reason why stoves, steam pipes, etc. are blackened.) Again, the amount of the radiation from a body depends largely upon its temperature. As this is raised, the energy carried by each train of waves of a definite wave length increases; but this increase is greater for the short than for the long waves. This fact can be represented by a graphical method. Let two axes be drawn, distances SHORT WAVE-LENGTHS LONG WAVE-LENGTHS Fio. 181. — Radiation from blackened copj.. r. along the hori/.ontal one to represent the wave lengths of the eon
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ipMnent \\a\cs, vertieal distances to represent quantities "f energy carried by the individual trains of waves. Several eurves are given of the radiation from blackened copper at different temperatures. It is seen that these curves are in •!-d with tin- statements made above. If we consider any individual wave length at the extreme ends of the curves, it marks evidently tin* limiting power of the measuring instru- ment used ; ;md therefore waves whose energy at one tem- perature of the l)od\ is so small that they cannot be detected may be so intense at a higher temperature as to permit of 296 HEAT observation. For instance, the longest waves which affect the human eye in such a manner as to produce the sensation of light are those that cause the sensation we call red; therefore, if the temperature of a solid body, e.g. a piece of iron, is raised, the experiment being performed in a darkened room, the solid is invisible (except for stray re- flected light) until the temperature becomes so high that the energy of the waves whose length corresponds to " red light " is sufficiently intense to affect our eyes. (Actually, the fact must be taken into account that the human eye is not sensitive to all colors alike, and that if the light of any color is feeble, the eye perceives " gray.") The body is now " red hot " ; and as the temperature rises still higher, its color changes continually, and finally it appears white and is said to be "white hot." Laws of Radiation. — Careful observations upon the radia- tion of a blackened body have shown a most intimate connection between the total quantity of energy emitted and the temperature. If Q is the energy emitted at a given temperature, and T the absolute temperature, i.e. T=t°C. +273, Q=cT^ in which c is a factor of propor- tionality. This statement is called " Stefan's law," having been first proposed by him. Again, there is a connection observed between the temperature of a blackened body and the wave length of the train of waves which carries more energy than any other train, i.e. the wave length which corresponds to the highest point of an energy curve for that temperature, as shown on page 295. This relation is due to Weber ; and calling T the absolute temperature of the body and Lm the wave length just defined, TLm = «, where a is
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a constant. These two laws, which have been verified over wide ranges of temperature by most painstaking investigations, offer con- venient means of obtaining the temperature of bodies when they are so hot as to render it inconvenient to use ordinary CO.\T7-:<T/M.\. CQNDUi rfOJT, -i.v/; HAhiATioy 297 means. If we assume that the laws are true for temperatures which are higher than those for which they have been veri- fied, we may assign them numbers. (In this way, making the above assumption, the temperature of the sun is observed to be about 5700° C.) Pre vest's Law of Exchanges. — It is thus seen that the radiation of any body is independent of other neighboring bodies, because it depends upon the vibrations of its own minute particles. So, if two bodies are associated in such a manner that one receives the radiation of the other, each radiates independently; and the temperature of either one will fall if it radiates more energy than it absorbs. This principle of the complete independence of the radiation of two bodies was first stated by Prevost (1792) in his "Law • •I' Kxchanges," which is equivalent to the above. Newton's Law of Cooling. — If a body is surrounded by an inclosure at a lower temperature, it loses more heat energy than it receives; and, if the radiation from the two bodies is that which is characteristic of a blackened body, this net loss may be expressed at once. Let Tj and T.2 l>e the absolute temperatures of the body and the inclosure: then the heat energy lost by the former diminished by that received from tin- inclosure is c(Tf — TJ). This equals and. if T{ is only slightly greater than 7!r it may be written f\\Tl — 7!,). So the net loss in heat energy varies as the ditl ere nee in temperature between the body and the inel(,siire. This is called "Newt«m's law of cooling." and it is trin- of other bodies than "black" ones for small differ- ences in temperature. Absorption Reflection and Absorption. When radiation falls upon a l"«dy, SOUL- i> tboorbed, BOUM i* n-lle.-trd. and some is trans-
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mit ted. A body which allows waves of a certain wave length 298 HEAT to pass through it is said to be "transparent" to them/ but no body is perfectly transparent to any waves ; if it is suffi- ciently thick, it will absorb them. In a thin layer, however, a body may absorb certain waves completely and may trans- mit others comparatively freely. Thus, ordinary glass permits those waves to pass which affect our sense of sight, but either absorbs or reflects other waves which are shorter or longer. This is the explanation of the action of the glass roof of a greenhouse. The " visible waves " from the sun are transmitted through the glass and are then absorbed by the black earth or the green leaves. The tempera- ture of these is raised, — but not sufficiently to make them self-luminous, — and they radiate waves which are so long that they are reflected by the glass. Thus the energy which enters through the glass is trapped and stays inside; consequently the temperature is raised. A body which transmits comparatively freely those waves which carry the greatest amount of energy is called " diathermous " ; but the word is not often used, because of its indefinite character. If we wish to compare the properties of reflection and absorption, it is best to consider a body which is so thick as to transmit no waves. It is then at once evident that if a body absorbs well it must be a poor reflector, and conversely. Thus, a blackened surface absorbs well and reflects poorly ; while a polished metal reflects well and absorbs hardly at all. In order to secure what is called "regular" reflection, as from a mirror, not alone must the body be itself large in compari- son with the length of the waves, but its surface must be smooth to such an extent as to have no irregularities so large ; otherwise the different portions of 'the surface reflect the waves in different directions and so scatter them. Under the above conditions a body reflects at its surface waves of all lengths to a greater or less extent ; but in every case certain waves enter the body, although their intensity may be very small. Absorption. — Let us consider the process of absorption more closely. When ether waves fall upon a body, certain CONVECTION, CONDUCTION, AND RADIATION 299 partirK's in the molecules are set in more violent vibration bv ivsnnaiK-e, and thus the waves lose energy. la some bodies
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t IK-SI- vibrating particles emit waves immediately, without the temperature sensibly rising. This is the case with pieces of fluor spar, thin layers of kerosene oil, and with a few other bodies, as will be shown later under Fluores- cence and Phosphorescence. In other bodies the absorbed • •ii'Tgy is distributed among the molecules and becomes ap- parent in heat effects. This absorption, where the energy goes into heat effects, is called "body absorption." Many I xxlies absorb only waves of definite wave-length, and trans- mit others. 'I' bey are said to have "selective" absorption. Metals and substances that have strong selective absorp- tion reflect certain waves more intensely than others ; they are said to have "selective" reflection. Thus, the reason why gold appears yellow to our eyes is because, when viewed in ordinary white light, besides the waves that are reflected at the surface and that would make the gold appear white, thriv art- certain waves, of such a wave length as to produce the sensation of yellow, that are reflected more intensely than the others. Those waves which enter the gold are absorbed in the surface layer of molecules and produce heat effects. It is evident, then, that if white light is reflected again and again from a series of gold surfaces, in the end the only waves which will leave the last surface will be those which produce in our eyes the sensation of yellow. The waves which leave the last surface after a great number of reflections from the same material are called the " residual " ones. It one looks at a bundle of nonll.- in white light, the points being t ii mod toward the eye, they appear black, because the waves are reflected to and fro from needle to needle, but are continually getting weaker and weaker and being deflected down the needle ; thus no waves come back to the eye, and the points appear black. Finely divided silver and plati- num appear black for the same reason. Connection between Radiating and Absorbing Powers. — abruption i> due to resonance, it is simply a restate- oOO HEAT ment of this to say that a body absorbs to a marked extent waves of the same period as those which it has the power of emitting. But we can say more, if we consider the intensi- ties of the waves absorbed and emitted, and if we assume that there are no chemical or other molecular changes in the body. This
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excludes fluorescence, phosphorescence, etc. If several bodies at different temperatures could be inclosed inside a vessel which absolutely prevents any heat energy from entering or leaving, and which keeps a constant volume (so that no external work is done), there is every reason for believing that equilibrium would finally be reached, but not until the temperature of all the bodies inside was the same as that of the walls of the vessel. When this is the case, each body must be absorbing and turning into heat effects as much energy as it emits, provided there are no chemical or other energy changes, otherwise its temperature would change. That is, the absorbing power of a certain body at a definite temperature exactly equals its emissive power at that same temperature; where by absorbing power we refer to body absorption. (In general language, a body which absorbs well, in the sense of transforming radiant energy into heat energy, radiates well; e.g. a blackened surface.) We can imagine, moreover, a body in the vessel described above, which is entirely inclosed by some envelope which allows to pass through it waves of only one wave length ; therefore, when equilibrium is reached, the body inside must radiate as much energy in the form of waves of this wave length as it absorbs. Consequently, the amount of energy of a definite wave length which a body emits at a given temperature equals exactly the amount of energy in the form of waves of this same wave length which it absorbs at that temperature. In other words, the absorptive power of a body at a certain tempera- ture equals both in quantity and quality its emissive power at that same temperature. If at ordinary temperatures a body appears black when viewed in white light, it is owing to the CONVECTION, COMJUCTIO.\. AM) i;Al>IATK)\ 301 fact that it absorbs those waves which affect our sense of : : and. if raised to such a temperature as will enable a body to emit such short waves, it will emit them and so shine iitly in a darkened room. Similarly, if a body appears ivd at ordinary temperatures when. viewed in white light, it is because it absorbs all waves except those which produce in our eyes the sensation of red ; these are either transmitted or are reflected out from the interior by some small foreign particles. (Thus, a colored liquid appears perfectly black j.t by transmitted light, if it is entirely free from small solid particles; but, if a minute quantity of dust is
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stirred in it. it appears colored when viewed from any direction.) Then, if such a red body is heated until its temperature is suiliciently high, it will emit all the waves except those which < -spend to the sensation of red, and so, if viewed in a dark room, will appear bluish green. This law connecting radiation and absorption was fir>t stated by Balfour Stewart, but was discovered inde- pendently by Kirchhoft'. The latter, however, in expressing it, did so in a more mathematical form. He took as a stand- ard of absorption a hypothetical body, which is called a "perfectly black body" and which is defined to be such a body as will absorb and turn completely into heat effects all radiations which fall upon it. (Any non-reflecting body, if sufficiently thick, is such a body.) We can approximate to such bodies experimentally by usin<j lampblack or platinum black as the surface layer. (It is seen, too, that the radiation inside an\ hollow inclosure, provided it is rou^h, is that which is characteristic «.f a perfectly black body at thai perat ure, because after a sufficient number of irregular reflec- 18 any train of waves will he totally absorbed, SO the walls ic inclosure finally produce the effect of a black body.) It will be shown later that radiation may be produced and 1.1 m. ins than by raising or altering the tem- perature of the radiating h..d\ ; but this law of Stewart and 302 HEAT Kirchhoff refers only to radiation that is due to the same cause which conditions the temperature of bodies, and to absorption that results in heat effects. Atmospheric Absorption. — One important case of absorp- tion of radiation is that of the solar rays incident upon the earth. As these pass through the earth's atmosphere, a cer- tain percentage of their energy is reflected by the floating particles and drops in the air, and also by the molecules of the air themselves ; and so does not reach the earth. There is also true body absorption as the waves traverse the atmos- phere. The energy that is not reflected or absorbed reaches the earth and is there almost completely absorbed and spent in producing heat effects. The earth itself is therefore also radiating energy, and this again passes through the atmos- phere and is partially absorbed. If there are clouds, this radiation from the earth
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itself is absorbed by them, and they radiate a certain proportion back toward the earth. Our appreciation of the temperature of the air in which we live depends largely upon the quantity of heat energy absorbed by the air, not so much upon the radiation received by us directly from the sun. Thus we see the reason why the temperature is lower upon mountain tops where the air is rare and so absorbs poorly than at sea level where the air is denser and absorbs more. CHAPTER XV THERMODYNAMICS Nature of Heat Effects. -- Throughout the previous chap- ters we have assumed that heat effects are due to work done against the molecular forces of a body and that for a definite amount of energy received the same effect is produced regardless of the source of the energy. These assumptions are justified by countless experiments, some direet and some indirect. Thus Joule, in a series of in- vestigations beginning about 1843 and lasting over forty : s, caused heat effects to be produced in many different ways ; compression of gases, friction of various kinds, con- duction of an electric current through a wire, chemical re- actions, etc. The only possible explanation of the results of these experiments is the assumption that heat effects are due to energy being given the molecules of the body and are proportional to the amount received. Thus, if a certain amount of hoat energy received from friction is used to boil some water or to melt some ice or to raise the temperature rater, we ean. by allowing this water to cool through tins temperature range, obtain an amount of energy which will melt an equal amount of ice, etc. — it being rem- that heat energy passes from high to low temperature. Again, when the energy received from any two source heat — for instance, a candle and the sun — is compared, if under any condition they raise the temperature of the same amount of water through the same range of temperature, 11 melt the same amount of ice, or boil the same water, et< •. Therefore the production of heat m 304 HEAT effects depends upon the energy received, not upon the tem- perature or condition of the source of the energy. Mechanical Equivalent of Heat. — As stated before, the practical unit of heat energy is that required to raise the temperature of one gram of water from 15° to 16° C. The value of this in ergs, or the " Mechanical Equivalent of Heat," has been determined by different observers. For many years it was thought that
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the specific heat of water was the same at all temperatures ; and in the early work no distinction was made between the amounts of energy required to raise the temperature of water between different degrees. Robert Mayer calculated the mechanical equivalent from experimental data secured by other observers on the heat energy required to raise the temperature of a gas at constant volume and at constant pressure. We have shown that, on the assumption of no internal forces, if J is the " mechanical equivalent," J (<7p — •#„) = R for a gas ; and hence knowing 0^ Cv, and R for a gas, J may be calculated. Joule, and afterwards Rowland, used the method of turn- ing a paddle rapidly in water, and measuring the mechanical work, the quantity of water, and the rise in temperature. The mechanical work was measured by a simple dynamom- eter method (see page 121) ; and in Rowland's work most accurate results were obtained. In fact, it was he who first measured the variations in the specific heat of water. This method was later modified by Reynolds and Moorby, who used revolving paddles to raise the temperature of a known quantity of water from 0° to 100° C. ; and so their results are independent of thermometers. The most recent work, and the best, has been done by Griffiths, Schuster and Gannon, and Callendar and Barnes, using electrical methods. It will be shown later that when an electric current passes through a conductor, heat-energy is produced ; the amount of this depends upon the electrical quantities involved, all of which can be measured with great TIlEHMnl) YNAM1CS o< I.J •tness. much more so than mechanical power. If i is the iLTth of the current; R, the resistance of the conductor; t. the time the current flows; the heat energy produced is expressed in ergs, if the electrical units are properly 11. Therefore, calling the measured heat energy IT, JH = i*Rt. In practice the current is passed through a wire which is in contact with water ; so H is measured directly. Thus J may be determined. (The weak point in the method is the uncertainty as to the value of the electrical units in terms of mechanical ones.) As a result of all of these experiments \ve know that to a high degree of approximation the work required to raise the temperature of one gram of water from o!6°C. is 4
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energy received, and Hv that lost, and W the ex- ternal work done, W= Hl — N2 by the conservation of energy (using the same units for heat energy and external H\ work), and the efficiency is * — 2. In discussing the TT __ TT action of this engine Carnot was led to several most impor- tant conclusions which Clausius and Lord Kelvin have shown to be rigid consequences of the two principles of thermo- dynamics. One of these was that the efficiency of his ideal engine was independent of the nature of the substance used to work it : steam, water, alcohol, etc. Another was that the efficiency of his engine varied directly as the difference in temperature between the two reservoirs referred to above. Absolute Thermometry. — This last fact led Lord Kelvin, then William Thomson, in the year 1848 to propose a system 307 of thermoraetry depending upon the use of Carnot's engine; this has the great merit of being independent of the sub- stance used in the thermometer (or engine). Thomson's system of " absolute " thermometry, as it is therefore called, is equivalent to defining the ratio of the temperatures of two bodies as equal to the ratio of the quantities of heat energy received and given out by a Carnot engine working between these two temperatures. Thus, if in the above description of an ideal Carnot's engine the ratio of the tem- peratures on Thomson's scale of the hot and cold reservoirs T *i is written -=±t T H /2 //, 2 rI = 771' Hence the efficiency is 1 —?. Since it is impossible to T — T have an efficiency greater than unity, there is a minimum value of trmpfi-ature, that for which T2 = 0 ; for, if T2 had a negative value, the efficiency would be greater than unity. This minimum temperature is called "absolute zero." Thomson's definition of absolute temperature does not specify the " size " of a degree, but simply the ratio of two temperatures : we can choose the degree to suit ourselves. Let us agree to use the Centigrade scale, so that if T is the temperature of melting ice, T + 100 is that of boiling water. By a series of most ingenious experiments Thomson showed that this system of temperature agrees most approximately with that which we have been using, namely, that of a con- s tai it-pressure hydrogen thermometer on the Centigrade scale, if \ve add
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to each temperature reading the reciprocal of the coefficient of expansion of the gas, i.e. 273 approximately. (This is what we have called on page 240 "absolute gas temperature.") Thus, if large quantities of boiling water and in» -It ing ice could be used as the reservoirs between which a Carnot engine worked, the quantities of heat energy received and given out, ITj and Hv would have such values 308 11KAT TT that their ratio ~~i almost exactly equaled. It can be x/2 2*1 & proved that, if the gas had no internal forces and obeyed Boyle's law exactly, the agreement between Thomson's ab- solute system of thermometry and that of a gas thermometer as above described would be perfect. Therefore, in the formula for a gas, pv = RmT, T is very nearly equal to Thomson's absolute temperature. The slight differences be- tween this system and that we have been using were deter- mined by Thomson and Joule ; and in all exact work in Heat this correction is made to the gas temperatures. There are several ways of determining the absolute tem- perature of a body and of calculating the value of absolute zero on the system of thermometry ordinarily used. The accepted value is -273°. 10 C. Historical Sketch of Heat Phenomena It was believed by the Greek philosophers that all the phenomena of a material body which we associate with the word " heat," such as expansion, change in temperature, boiling, etc., were due to the addition to the body of a sub- stance; but to Newton and his immediate predecessors and associates it seemed clear that in some way they were due to motion of the parts of the body. Thus Boyle gave a correct explanation of the heat effects observed when a hammer strikes a nail. The materialistic theory of heat, however, was again proposed, and prevailed for nearly two hundred years. Its great defenders were Gassendi (1592- 1655), Euler (1707-1783), and Black (1728-1799). Even as late as 1856, in the eighth edition of the Encyclopaedia Britannica, this theory is offered as the accepted explana- tion of heat phenomena. One great reason why this theory was so universally accepted was because it was so analogous to the accepted explanation of combustion. Stahl (1660- 1734), who was professor at Halle, advanced the theory that Til Kit Unit VXAMICS 309 •l
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uring the process of combustion a material substance, called "phlogiston," was given off; and this idea persisted until the \\ork of Lavoisier, about 1800, and even later. Fan- tastic theories in regard to the properties of phlogiston and of the substance heat (or "caloric") were of necessity brought forward in order to account for the observed facts. Joseph Black showed that when two bodies at different temperatures were brought together he could speak of '•quantities of heat" leaving or entering the bodies; and we owe to him our ideas and methods in regard to specific heats. Black considered two kinds of effects when " heat " was added to a body : if the temperature was raised, the heat was called " sensible," and it was supposed to be free in the body; but if the temperature did not change, the heat was said to be "latent," and it was supposed to form some kind of a compound with the molecules of the body changing their state. But the experiments of Rumford, in 1798, and of Davy, in 1799, convinced nearly every one that heat effects in a body were due primarily to the transmission of motion to its minute parts. Rum ford showed that in such a process as that of boring out a brass cannon, " quantities of heat " could be produced, limited only by the amount of work done. Similarly, Davy arranged an apparatus which caused one block of ice to rub violently against another, and showed that the quantity of ice melted varied directly with the work done. Tin- first one, however, to express clearly the belief that heat effects were due entirely to the addition of energy to tin- small parts of a body was Rolx-rt Mayer, in lsj-J. He followed hy Joule, in 1848, and later by I Irlmholt/., in 1M7. i;\ the epoch-making research.- ••!' Joule, the prin- « -iplo of the conservation of energy — a phrase nf Rankine's -was soon extended so as to cover all heat phenomena. The fact that -radiation" is a phenomenon due to wave motion in thr « th« i, of exactly the same nature as that win »-h 310 UEAT produces the sensation of light, has been established by a long line of investigators. William Herschel showed, in 1800, that there were rays in the solar spectrum invisible to the eye and yet having the power of affecting
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a ther- mometer. Herschel speaks of these rays as subject to the laws of reflection and refraction; and this fact was fully established by Melloni some thirty years later. It was proved in the following years that these rays could be dif- fracted, be made to interfere, be polarized, etc.; and that, in short, they were due to waves in the ether. BOOKS OF REFERENCE EDSER. Heat for Advanced Students. London. 1901. PRESTON. The Theory of Heat. London. 1894. An advanced text-book, containing extended descriptions of the experi- ments which form the basis of our knowledge of heat quantities. TYNDALL. Heat a Mode of Motion. London. 1887. In this the author describes a series of experiments illustrating the fact that heat effects are due to energy changes. BRACE. The Law of Radiation and Absorption. New York. 1901. This contains the original papers of Prevost, of Balfour Stewart, and of Kirchhoff. AMES. The Free Expansion of Gases. New York. 1898. This contains the papers of Gay-Lussac and Joule on the expansion of gases through small openings. RANDALL. The Expansion of Gases by Heat. New York. 1902. TAIT. Heat. London. 1884. MAXWELL. Theory of Heat. London. 1892. YOUMANS. The Correlation and Conservation of Forces. New York. 1876. This contains reprints of papers by Helmholtz, Mayer, and others on the Conservation of Energy. STEWART. The Conservation of Energy. New York. 1874. GRIFFITHS. Thermal Measurement of Energy. Cambridge. 1901. This gives a most interesting description of various methods for the measurement of heat energy. VIBRATIONS AND WAVES CHAPTER XVI WAVE MOTION General Description. One of the most important phe- nomena in nature, and one that is of most frequent occur- rence, is the transmission of energy from one point to another by what is called " wave motion." This motion is illustrated in many ways: if a stone is dropped in a pond, waves are produced on its surface, which do work on any movable object which they meet; a vibrating bell produces waves in the air, which do work in a similar way, or by bending a suitably stretched membrane such as the drum of the human ear or the diaph
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ragm of a telephone instrument ; if one end of a long, stretched rope is fastened to a movable object and if the other is given a sudden side wise or lentil i\\ ise motion, a disturbance will pass along the rope and will do work on the object; etc. In all these cases it is evident that there is a vibrating centre which produces motions in those portions of the surrounding medium immediately in contact with it; portions affect those next them, etc. The vibrations at th< centre of disturbance must be of such a nature as to produce in tin- medium a displacement or change which can be propagated l»\ it. Thus, if the hand is moved through air or \\at.-r, <.r if a pendulum vibrate* in air or water, waves ai« not pn.duerd, because the fluid flows around the obstacle as it moves through it. and is not Q pressed; in order to Loot waves, the vibration must be so rapid or the motion so sudden that the fluid,t have time to flow, and is 312 VIBRATIONS AND WAVES therefore compressed on one side and expanded on the other. Again, any sidewise motion in a fluid of an object like a thin board, however sudden, would not produce waves, because in a fluid there is no elastic force of restitution when one layer is moved over another. In order that a medium should carry waves, there must be forces of restitution called into action when its parts are displaced ; for these forces are due to the action of the neighboring parts on each other ; and owing to the reaction of the displaced parts on those in contact with them the latter are displaced also, and so waves are produced and propagated. If these conditions as to the medium and the centre of disturbance are satisfied, and if the motion of this centre is a vibration, or a series of vibrations, the various portions of the surrounding medium will in turn be set in vibration. If the motion of the vibrating centre ceases, that in the medium will persist until forces of friction (or other causes) bring it to rest. In all these cases it is clear that the particles of the medium vibrate, but do not advance with the waves ; a certain " condition " moves out from the centre. " Water Waves " and Elastic Waves. — There are several kinds of forces of restitution that enable a medium to propagate waves. If the surface of a pond is disturbed at one point by the motion up
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and down of a stick dipping in the water, waves spread out, owing to the fact that the force of gravity tends to maintain a level surface. Again, if a vertical cord, like a fishing line, is moved sidewise through the surface of a pond (or, if a quietly flowing river flows past a stationary vertical cord), short waves called " ripples " may be observed on the side toward which the cord is mov- ing (or up stream in the other case). These waves are due to the force of restitution of surface tension. The waves just described occur on the surface of a liquid; but any portion of matter that is elastic can also carry waves through its interior. Thus, fluids can propagate com- pressional waves, that is, waves produced by having a centre WAVE MOTION 313 of disturbance where the fluid is compressed or expanded : as is illustrated by a bell vibrating in air or under the surface of a lake. Fluids cannot, however, propagate a distortional disturbance. An elastic solid body can carry both com- pivssiunal and distortional waves, as is illustrated by a long, stivtehed wire. If this is disturbed at some point by a transverse vibration, or if it is twisted back and forward, tin-re will be distortional waves; if it is pulled to and fro longitudinally, in the direction of its own length, at some point, there will be compressional waves. (This fact is also illustrated by the disturbances that we call earthquakes; for these are due to some great disturbance in the interior of the earth that produces both kinds of waves in the body of the earth. As we shall soon see, these two types of waves in a solid travel with different velocities ; and this fact is ob- served in all earthquakes.) Compressional waves are often called "longitudinal," and distortional ones, "transverse," •bvions reasons. Polarization. — One distinction between longitudinal and transverse waves, other than the one which gives rise to the names, is worth noting. A longitudinal wave, in whieh the particles of the medium move to and fro along the line of advance of the waves, appears the same to the eye from what- ever side this line is viewed. But, since in a transverse wave the particles are vibrating in planes that are at right angles to the direction of propagation, it will as a rule appear different when viewed from different directions. Thus, if a t raus
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verse t rain of waves is produced in a long, stretched rope by n loving one end up and down in a vertical line, the rope will at any instant have a sinuous form when viewed from ide, but will appear straight when viewed from above. In this case all the particles are vibrating in straight lines through which a plane can be drawn; and such a train of '1 to be " linearly " or - plane polarized." Simi- larly, if the end of the rope is moved rapidly in a circle or in 314 VIBRATIONS AND WAVES an ellipse, all the particles of the rope will in turn move in circles or ellipses ; and the waves are said to be " circularly " or " elliptically polarized." Thus only transverse waves can be polarized ; and, conversely, if in any wave motion we can detect polarization phenomena, we know that the waves must be transverse. Intensity. — In all classes of waves it is at once evident that we are dealing with the propagation of energy. For wherever there is wave motion the moving parts have kinetic energy, and the existence of the waves presupposes forces of restitution, so when the parts are displaced there is potential energy. Therefore, as the waves spread out from a centre of disturbance, the medium into which they advance gains energy. If the cause of the waves is a temporary disturb- ance, any portion of the medium gains energy when the waves reach it and loses it again when the motion ceases, owing to the waves passing on. Thus a portion of the medium simply transmits the energy. It gains none perma- nently unless there are forces of friction when the particles of the medium move relatively to each other as the wave passes. (Of course if there are foreign bodies immersed in the medium carrying the waves, they may be set in vibration and may continue to vibrate after the wave passes ; in which case this portion of space — not the medium itself — has an increased amount of energy in it afterward.) If at any point in a medium a plane having a unit area be imagined described at right angles to the direction in which the waves are prop- agated, the amount of energy transmitted through it in a unit time is called the " intensity " of the waves at that point. (If the waves are varying, the exact definition is as follows : if E is the energy transmitted in time t through an area A, the intensity is the limiting value of the ratio — -, as
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t and A are taken smaller and smaller.) Detection of Waves. — The effect of the waves is perceived in two ways. If the medium is limited in one direction by \\ ATE MOTION 315 e object, with which it is connected in such a mannei that the vibration of this object would produce waves in the medium, this will be set in motion by the waves unless it is restrained by mechanical means. Thus, waves in the air set in motion the drum of the ear or the diaphragm of a tele- phone receiver. If the object is surrounded by the medium, l)ii t cannot move bodily with the waves, it may be set in vibration by them, owing to " resonance," a process whieh will l)e described in detail later. Thus, waves produced in tin- air by one tuning fork may set in vibration another one, if the periods of vibration of the two are identical ; waves in the ether may set in vibration particles of matter, as described in the chapter on Radiation. Other Kinds of Waves. — So far we have spoken of mechan- ical waves only, that is, waves in matt-rial media or in the ether, in which the disturbances are displacements of par- ti, hs. lint we can have many other kinds of waves, de- pending upon the property of the medium that is varied. Tli us, if the temperature of one end of a metal rod is first d gradually, then lowered, raised again, etc., we have a vibration of temperature; and, if we observe the tempera- tun; of any point in the rod not too far away from this end, we shall find that its temperature also rises and falls. Since it takes time for the conduction of heat, the temperatures at rent points of the rod will not have their maximum values at the same time, so there is a wave of temperature in the rod. This is illustrated in the daily heating and cool- ing of the earth's surface as it is tunu-d t.. \\ard and a\\.i\ ill. sun, also by the seasonal heating and cooling owiiiLr to th«- revolution of the earth around the sun. There are temperature waves, then, going down into the earth for a short distance. The daily wave is appreciable for a depth of about 2 or 3 ft. ; the annual one for about 50 i|Hrarup <>t th. * M it }i'-cru8
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t increases gradually but OOD- tn.'i..-. :\- with the depth; the rate of increase varies greatly with the VIBRATION H AND WAVES geological conditions, but is on the average about 1° C. for a depth of 28 m. This condition requires that heat energy should be continually flowing from the interior of the earth to the surface. From considera- tions based on this fact it is possible to make an estimate of the " age of the earth " ; that is, the interval of time since the earth was in a liquid condition. This is probably about 50,000,000 years. Similarly, if one end of a long electrical conductor, e.g. an ocean cable or a telephone wire, has its end suddenly joined to an electric batter}^ the effect is gradually felt along the conductor ; and, if the electric battery at the end is varied, there is an " electric wave " in the conductor. Again, if a metallic body is charged electrically, there are electric forces, so called, at points in space near by ; so, if this electric charge is varied, these forces will vary ; and, as they change, variations are produced at neighboring points. Therefore, when the electric charge on a body varies, that is, when there are "electric oscillations," waves of electric force are produced in the surrounding medium. The medium which carries these waves has been proved to be the same as that which carries the waves that affect our sense of sight ; namely, the "luminiferous ether." It has been proved, too, that, wherever there are variations in the electric force, there are also variations in the magnetic force ; so these waves are called " electro-magnetic." (It should be borne in mind that if we look upon matter as the fundamental concept in nature, then as soon as we are able to explain electric and magnetic forces as in some way due to the motion of matter, we shall be able to describe electro-magnetic waves in the ether as dis- placements of material portions of the ether. But, if an elec- tric charge is the fundamental concept, as soon as we can explain the properties of matter as due to the motion of charges, we shall be able to describe all waves in matter in terms of electric forces.) These electro-magnetic waves may be detected by suitable means, as is shown by the various systems of "wireless telegraphy." CHAPTER XVII HARMONIC
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AND COMPLEX VIBRATIONS \Vi: shall now proceed to discuss in detail the two funda- mental features of waves : liist. the properties of the centre of disturbance; second, those of the medium through which the waves pass. The effects produced when waves in the air or in the ether are perceived by our senses of hearing or of vision will be considered later in the sections devoted to Sound and to Light. The Kinematics of Vibrations Simple Harmonic Vibration. — A disturbance may be peri- odic <>r not ; that is, it may after a definite period of time called the "period" be repeated identically, and again at the end of another period, etc.; or it may be irregular. Thus, the vibrations of a pendulum, of a tuning fork, of a violin string. «-tc., are periodic; while the motion of a piece of tin as it is " crackled," of two stones when struck together, or of one's hand ; s it is moved to and fro at random, are not peri- The simplest case of periodic motion is that which is called "simple harmonic." and which is discussed on page 48. The period of this has been defined above; and the numher «.f \il.rati..us in a unit of time is called the u fre- quency"; this is, of course, the reciprocal of the period, nplitude " has been defined aa one half tin- length of the suing, or as the value of the maximum displacement. • harmonic motions having the same period and ampli- tude may yet differ in "phase"; that is. t h<- infant s at which they pass through their on ;ins ma\ lie different. 317 318 VIBRATIONS AND WAVES Composition of Harmonic Motions. — 1. In the Same Direc- tion. If a point is subjected to two harmonic motions, the resulting motion may be found by compounding the dis- placements geometrically at consecutive instants of time, or by simple algebraic processes. Thus, if the two harmonic motions have the same period and are in the same direc- tion, they may be represented by xl = A1 cos (nt — ax) and #2 = A2 cos (nt — a%) (see page 51) ; and the resultant mo- tion is x = xl + 2r2 = Al cos (
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nt — ctj) -f A2 cos (nt — a2). By ordinary trigonometrical formulae this takes the form x = A cos (nt — a), where A2 = Af + A22 + 2 AlAl cos (a, - a2), A, sin a, + A „ sin a9 and tan a = -^ — • A l cos ax + A 2 cos a2 This shows that the resulting motion has the same period as that of its two components, but a different amplitude and phase. Graphical Methods. — If the two component harmonic motions are in the same direction but have different periods, the algebraic formulae are much more complicated, but their resultant may always be found by a simple graphical method. FIG. 133. — Graphical representation of harmonic motion. Any harmonic motion may be represented by a curve drawn on a diagram whose axes are intervals of time and displacements. Thus, the motion x = A cos (vit — a) will be given by a curve, a portion of which is shown in the cut —.\\i> COMPLEX \'ii;i;.\Tif)N8 319 wliirh i> ealled the "sine curve." Thus, let. as mi the harmoiiie motion be that of the point (J. the projection on the diameter of a point P, which is moving in a circle with e< distant speed. Jf, when we begin to count time, the point Pis at S, the "initial" value of the displacement is the projection of OS on the diameter. From this time on the displacement assumes different values; OA is the great- est, OB is the least ; when P is at M, the displacement is Therefore, if we erect at each point of the "axis of time " a line whose length equals the displacement of Q at that instant, and if we remember that displacements in one direction are positive, but in the other negative, we obtain a curve like that shown in the cut, which is known as a "sine curve." Sv Av Pr Mv Bv etc., indicate points corresponding to points S, A, P, Af, J5, etc., in the circular diagram. As the motion is periodic-, the curve repeats itself. This curve can be obtained practically hv fastening a wire to the bottom of a heavy pendulum, ami ilra\\in«,r under it, at right angles
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phase, l><»th will he repre- sented by the point 0 at the same instant; then, when one vil. ration has reached /',. the nth.-r.ha8 reached Ql ; and the geometrical sum is given by Rr When the fnrmer \ il»i at inn reaches PT the latter rca< -In •- '/, : and the geometrical sum is givm l>\ /{., : etc. It is fvid.-nt that the resulting motion is a Straight line. If the \ihrati.ms dilT.-r in phase by a quarter of a period, i.e. by in angular measure, one vihratinn will be at the 324 VIBRATIONS AND WAVES end of its path, J., when the other is at 0. The geometrical sura is given by A. When the former vibration reaches P2, the latter is at Q1 ; and the geometrical sum is given by S1 ; etc. The resulting vibration is evidently a circle. FIG. 139. — Periods equal ; difference in phase 0, -,-,—, IT. If the difference of phase is one eighth of a period, i.e. j, the resulting vibration is an ellipse. The curves are shown for a number of different differences in phase. If the amplitudes of the two vibrations are not the same, the geometrical methods are exactly similar. Lay off two lines, AB and CD, perpendicular to each other at their middle points; divide them into lengths that corre- spond to equal intervals of time ; at these points draw lines parallel to AB and CD. If the phase of vibra- tion is the same, the result- ing motion is in a straight line. If the difference in phase is one eighth of a FIG. 140. — Periods equal; amplitudes unequal; period, the vibration is in difference in phase one eighth of a period. an ellipse, as shown, etc. The simplest way, however, of compounding the vibrations is to eliminate t from the equations, and plot the resulting equation. Different curves will be obtained by giving a different values. The curves of Fig. 139 may be obtained by several physical processes. One is to use, as described on page 319, a pendulum AMt <-<>Mi>LEX VIBRATIONS 826 that can trace a path on a piece
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of glass ; but in this case the kept stationary and the pendulum is set swinging, not in a plane through its origin, but in a cone, as a result of a sidewise push given it when it is held out at the end of its swing. Another method, due to Lissajous, is to use two large tuning forks whose frequencies are the same, and to phice them, with their vibra- tion planes perpendicular to each other, in such a position that a pencil of light from a small source incident on the end of a prong of one fork is reflected to the end of a prong of the other and thence to a screen, or into the eye of the ver. This arrangement is.shown in the cut. If only one fork is vibrating, a straight line is seen; but if this fork is quiet and the other is vibrating, another straight line at right angles to the first is seen. These lines are caused by the rapid harmonic motions of the two forks. It now both forks are set vibrating, the path of li^ht seen is an ellipse. It' the forks are started again at NT I,.: Kio. 148. - LlM«Jou.' «rrmn««.ment of two tunlnjr millC tO rest, the shape of the ellipse will be different. in general, owing to the fact that their difference in phase i> not the same.1- 1.. This is on the assumption 326 VIBRATIONS AND WAVES that the frequencies of the two forks are exactly equal ; if they are not, the shape of the ellipse will change as one looks at it, showing that the difference of phase between the vibrations has changed. The reason for this is seen at once if one considers the two equations for the forks. If their periods are not quite the same, these may be written x = y = A2 cos (n^ - a), where n1 = n + bt and b is a small quantity. Therefore sub- stituting for Wj its value, y = A2 cos [n* — (a — &£)]. Comparing this with the equation for #, the difference in phase is seen to be a — bt\ and this is different for different Periods in the ratio 1 : 2. Periods In the ratio 8 : 4. FIG. 143. — Lissajous' figures. HARMONIC AND COMPLEX
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rapid, as for in- stance if a pendu- F.G. 145. - A damped vibration. lum has a plCCC of paper fastened to it, the period is sensibly increased. This decrease in ampli- tude is due to loss of energy by the vibrating body, generally by friction as it moves through the air or at the pivot. This is evident if we calculate the energy of the vibrating particle. If its motion is given by x = A cos (nt — a), its speed at any instant is 8 = An sin (nt — a) (see page 51) ; and so if its mass is m, its kinetic energy at this instant is £ mAW sin2 (nt — a). This varies at different instants of the vibration, but is always proportional to A2. (The mean value over one period may be proved by the infinitesimal cal- culus to be %mAW.) During the motion, as fast as kinetic energy is gained, potential energy is lost ; and so the mean total energy during' a vibration is twice the mean kinetic energy ; and this, from what has just been said, varies directly as the square of the amplitude. Therefore as the amplitude of vibration decreases, the particle loses energy. Forced Vibrations; Resonance. — Even though a body which can make vibrations like a simple pendulum has a definite period of its own, it may be given a different period, if it is attached to some other vibrating system ; thus, if the point of support of a pendulum is moved to and fro by a hand making harmonic motion, the motion of the pendulum is HARMONIC AND COMTLKX VIBRATIONS 329 i In.' resultant of two, one due to the force applied by the hand, the other its own natural motion. If this last is greatly damped by attaching a sheet of paper to the pendulum, it soon dies down ; and the iinal motion of the pendulum is that dui' to the harmonic force of the hand. This motion has the same period as that of the harmonic force; and is called a ••forced vibration," to distinguish it from the natural free vibration. The amplitude of this forced vibration depends to a great extent upon how closely the period of the force agrees with that of the natural vibration ; if they are exactly equal, the amplitude is very large. This condition is called Mmance," and is illustrated in many ways. The case of
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a child in a swing being set in motion by a series of pushes given at intervals agreeing exactly with the natural period of the swing has been mentioned already. In a similar manner a heavy church bell may be set swinging. If a tun- ing fork is vibrating near another one of the same frequency, tin- latter will be set in vihr.it ion. Many other simple mechanical cases of resonance are given in Rowland's Physi- cal Papers, page 28. If the ju-riod of the force varies slightly from the natural period of the vibrating body, the amplitude is not so great as when there is resonance ; and in most cases one can tell with considerable accuracy when the resonance is exact. The fad that a harmonic force produces in a system whose own vibrations are greatly damped a vibration whose period is the same as its own is of great importance. If the force is periodic, but complex, each of the component harmonic forces produces a corresponding harmonic vibration having its period ; but the phases and amplitudes of these component vibrations bear relations to each other that are not the same as for the component forces. Then-Tore, die resultant complex \ibration is different from the complex force in "form." Il Is Only a harmonic force that can - reproduce ". of OOUlVe \\ ith variations in the amplitude. CHAPTER XVIII VELOCITY OF WAVES OF DIFFERENT TYPES Wave Front. — When waves spread out from a centre of disturbance, a surface can be described that marks at any instant the points which the disturbance has reached. This is called the "wave front." Thus, if a stone is dropped in a pond, or if a raindrop falls on a pool of water, the wave front is marked by an ever-expanding circle. If waves are pro- duced in air by a vibrating tuning fork, the wave front at some distance from the fork is very approximately a sphere. These are called " spherical " waves. The waves in the ether that reach us from a distant star, or from any distant terres- trial source of light which is small, have a spherical wave front ; but this sphere has such a large radius that the portion of the wave front that affects us is practically a plane; so we call these "plane waves." Intensity of Spherical Waves. — If we consider a point source, which therefore produces spherical waves, we can easily calculate the relative intensities (see page 314) at
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different distances from the source. Let us describe two spherical surfaces of radii rl and r2 around the source ; their areas are 4<7rr12 and 4?rr22. So if the source emits in a unit of time an amount of energy equal to E, and if there is no absorption by the medium, the intensity at any point of the first surface Tfl TJ1 is 2 2' an^ ^at a^ any P0^ °f the other surface is 2- If the former intensity is called 7X and the latter J2, it is seen that l 1 '• 2"^V' 330 VI:L<>< ITY OF WAVES OF DIFFERENT TYPES 331 Or, in words, the intensity varies inversely as the square of the distance from the source. (If the source is of such a kind as to produce harmonic motions at all points in the surrounding medium, we see at once that the amplitude of tin- vibration at a point in the first spherical surface bears a ratio to that at a point in the second surface given by ri ri A1 : A% = — : — ; for we have shown on page 328 that the energy of a harmonic vibration is proportional to the square of the amplitude, i.e. /j : J2 = Af : A£ ; and therefore by the formula just deduced for the intensity, the one for the ampli- tude follows at once.) Velocity of Waves. — The rate at which the wave front ad van * < > is called the "velocity" of the waves. In the next article we shall deduce its value in certain simple cases in trims of the physical properties of the medium; but from general considerations it is evident that in an elastic medium the velocity will be increased if the elastic force of restitu- tion of the medium is increased, and will be decreased if the inertia of the medium is increased; and conversely. In fact, we can prove without difficulty that the velocity of compres- sional waves in a homogeneous fluid whose density is d and whose coefficient of elasticity is E^ is given by the formula Tli is formula is due to Newton, and is deduced in the Principia. WHY.- front of waves in the air is affected naturally by winds. if plane waves are advancing in a direction opposite to the \vin.l. the iij>|"T portions of the wave front will be more retarded than the
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res- sion in the springs of the slow set. An expansion, then, propagated along the slow set produces an expansion in the fast one, but a compression is reflected back along the slow one. If a series of waves consisting of alternate compressions and expansions, such as would be produced in this model by giving harmonic motion to the first ball of the slow set, meets a set of " faster " balls and springs, the reflected waves are of the same nature ; but whenever a com- pression reaches the boundary, an expansion is reflected, and conversely. In an exactly similar manner it may be shown that, if such a series of waves in a set of fast balls and springs is incident upon a boundary beyond which there is a slow set, a similar series of waves is reflected ; but in this case a com- pression produces a compression, and an expansion an expan- sion at the boundary. There is,a difference, then, in the reflection at the boundary between the fast and slow sets, depending upon the direction from which the incident waves come ; and this difference is equivalent to a substitution of a compression for an expansion, or vice versa. It should be particularly noted that if the velocities of the waves in the two media are the same, there is no reflection. (It is assumed that the waves are not damped, that is, that there is no absorption.) Therefore, in order to have reflec- tion of waves at a boundary separating two media, these must be such that the velocity of the waves is different in the two. (The "rolling" of thunder is an obvious illustra- tion of the reflection of air waves owing to the presence in the air of foreign bodies, namely clouds, or of regions in or M-.ir/-> Of i>in-i-:i;!-:.\T TYPES 335 which the velocity is different.) Another obvious condition for securing reflection from un obstacle is that its area should hi- large compared with the length of the waves; otherwise they will puss around it. Velocity of Transverse Waves along a Flexible Stretched Cord. — It is not difficult to calculate the velocities of certain ses of waves. This is true of the propagation of trans- verse disturbances along a stretched but perfectly flexible cord in which the tension is constant. Imagine a tube, which is straight except for a circular portion near its middle, slipped over this cord and moved rapidly along it with a con-
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st ant velocity v. Let us consider the motion of the particles of the cord as the curved portion of the tube reaches it, and the forces which the tube exerts on the cord. As this curved P Q F i... 149. — A cord, over which has been slipped a bent tube, is stretched between Pand Q. portion of the tube reaches any particle of the cord, it gives the particle a motion which may be resolved into two com- ponents : uniform motion in a circle with constant speed v, and uniform motion parallel t<» that of the tube along the cord with constant speed v. The former motion will be con- si ( In -i •« I presently. As the particle enters the curved portion, it is given, therefore, a momentum along the line of the cord, whirl i it keeps until it leaves the other end of the curved portion, when it is given an e<|iial momentum in the opposite (liivriinM and brought to rest. The two forces that pro- duce these changes in momentum are due to the tube; hut one balances the other exactly; so then- is no resultant action or reaction due to them. The only other acceleration is that occasion. <1 by the particle being made to move in a circle with constant speed. It' / is the length of any minute portion of the ronl. </ its mass per unit length, an<l r 336 VIBRATIONS AND WAVES tin- radius of the circular portion of the tube, the force which must be exerted on this portion of the cord to make 2 it move in the circle is dl - - There are three forces acting on it, the tension in the cord acting at its two ends and the reaction of the side of the tube against which it presses. Let us calculate the former. In the cut let A, B, O represent three consecutive points in the cord, drawn on an immense scale ; let the length of the arc ABO be Z, the portion of the cord which we are considering; let 0 be the centre of the circle drawn through these three points ; and T be the ten- sion in the cord. Owing to this tension B there are two forces acting on this por- tion of the cord, as shown in the cut. Calling the angle between AO and OB T (and also between OB and 00) N, the component of each of these forces along BO is T sin N, and their components perpendicular
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defined under which the change in volume occurs (see page 194). In the present case, the number of compressions and expansions in one second is so great that the change in volume of the fluid takes place adiabatically ; for there is not time for heat transfer to occur. Consequently in the above formula E is the adiabatic coefficient <>f elasticity. Its value for any gas has been given already (page 253). If c is the ratio of the two specific heats, and p the pressure of the gas, E equals the product cp. Therefore, for a gas, and this can be simplified by using the gas law p- RdT. Thus, V=^Tftft Then-fore, the velocity varies directly as the square mot of the absolute temperature; and AME*' -- 22 338 VIBRATIONS AND WAVES further, if F", 72, and T are known for a gas, c may be calculated. Laplace was the first to see (1816) that the coefficient of elasticity in this formula was the adiabatic one. Newton, who was the first to derive and apply the formula, used the isothermal coefficient, whose value equals p, and thus made an error. Assuming the value of c for air to be 1.40, and substituting proper values of p and d, the velocity of air waves may be calculated at any temperature. The value of this velocity at 0° C. is thus equal to 33,170 cm. per second, if the C. G. S. system is used. The fact that the velocity varies with the temperature is illustrated by the observations of arctic travelers who have noticed that the so-called " velocity of sound " is less at low temperatures. The velocity of waves in air is also seriously aifected by the presence of moisture, because the density of the air is changed. The velocity is seen to be independent of the nature of the disturbance propagated, and also of the pressure. When there is a violent explosion in the air, there are slight variations in the velocity near the centre of disturbance, owing to the fact that the value of the elasticity of the gas, as given above, is true for small variations in the pressure only. At some distance away from the centre, however, the velocity becomes normal. * a Similarly, the velocity in other gases may be calculated. The velocity of waves in liquids may also be deduced from the original formula V=\— - For water
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at 8° C. it is found, as stated before (page 172), that an increase in pressure of one atmosphere, i.e. of 76 cm. of mercury, decreases a unit volume by 0.000047 of its value. If the thermal effects may be neglected, „ 76 x 981 x 13.59, 7, therefore, V= 145,000 cm. per second. -£666617 — 'and'/ = There are also experimental methods for the determination of these velocities in gases and liquids ; the details of which VEL»< ILY or WAVES <>r Dirri:nEM TTPMS 339 are described in larger text-books such as Poynting and Thomson, Sound. These methods may be divided into two classes : direct and indirect. In the former, a disturb- ance is produced at some point and the time taken for the waves to reach a point at a measured distance away is accu- rately measured. The disturbance may be the ringing of a bell, a mild explosion, etc. ; and the instant of arrival of the waves may be determined by the mechanical motion of a diaphragm or by the perception of the sound. In the indirect methods, the gas or the liquid is set in vibration by some periodic disturbance whose frequency is known : since this is due to resonance, the frequency of the vibration of the gas or liquid is known ; and, as will be shown presently, the velocity of waves in it may be at once calculated. Velocity of Waves in a Solid. — In solids a purely compres- sional wave cannot exist, because when there is a compression produced by two opposite forces there is at the same time a distortion. The velocity of longitudinal waves in a solid that extends in all directions is given by the formula when- /r is the coefficient of elasticity for a change in volume, and // the one for a chanty in shape. A train of waves can, however, be produced where there is only distortion, as when one end of a long wire is rapidly twisted to and fro. In this case, if n is the coefficient of rigidity for the solid, V— *\pj- If longitudinal waves are produced in a wire or a rod by stroking it lengthwise with a resined or damp cloth, the ve- locity of the waves is given by V=^—, where E is Young's modulus. ( S66 page I •"> I
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. ) Water Waves. — Tin- velocity of waves upon the surface of a liquid di-j.rmls upon many quantities, and we can do no 340 r//;/,M770ATs AMD WAVES more here than state certain facts in regard to liquids whose viscosity may be neglected. These statements involve the quantity known as the " wave length," which in the case of waves on liquids may be defined to be the distance from crest to crest or trough to trough. If this quantity has the value Z, we have the following formulae for the velocity : If the liquid is deep, V—\—, where g has its usual 2 7T meaning. If the liquid is shallow, V= Vfy/, where h is the —, surface tension and d the density. The general formula for waves on liquids which are deep is Id 2* r y o V and it is clear that, if the waves are long, the first term is negligible, while, if they are short, the second one is. It is seen by calculation that in the case of water if Z>10 cm., the first term may be neglected, and if Z<0.3 cm., the second. For intermediate values of Z, the full expression must be used. There are many most interesting applications of these formulae. The fact that, if one sailing boat has a longer water-line than another, the latter is given a " time allowance " in a race, is due to an attempt to equalize the advantage of the longer boat ; for a boat moving through the water produces waves that are comparable in length with its own ; and as the boat is helped on by these waves, the longer boat is helped the more because the velocity of the waves it produces is greater. Again, as waves approach a shelving shore, if they are oblique to- the shore line, they will gradually turn so as to approach parallel to it, owing to the fact that in shallow water the waves are faster in the deeper portions than in the ones less so. The motion of the individual particle of a liquid as a wave passes over its surface is in general an elliptical path ; and the effect of the waves is felt only a short distance down VELOCITY <>F \VAVES OF DIFFERENT TYPES 341 from the surface, as the amplitude of the vibration decreases rapidly \\ith the depth. This is not the place to discuss the velocity of temperature waves or of
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electric waves along wires. But it may be stated that in both these cases part of the energy of the waves is dissipated in heat effects throughout the media and in other ways, and as a consequence of this the waves die down and are not propagated as far as they otherwise would be. The FIG. 151. — A drawing1 of Lyman's wave model for water waves, showing the form of the wave, the motions of the indi- vidual particles, etc. waves are said to become "attenuated." It may be proved also that long waves persist for a greater distance than short ones ; and this fact is of fundamental importance in tele- phone service, as will be shown later. CHAPTER XIX HARMONIC AND COMPLEX WAVES — « STATIONARY WAVES" Trains of Waves and Pulses. — In discussing many of the properties of wave motion it is essential to distinguish two types of waves : one is produced by a sudden irregular dis- turbance, and may be called a " pulse " ; the other is pro- duced by a periodic disturbance, and is called a " train of waves." Harmonic Waves : Wave Length, Wave Number, Amplitude. — The simplest type of a train of waves is one produced by a centre of disturbance whose motion is harmonic. This is called a "train of harmonic waves," or a "harmonic train." As a consequence of this disturbance, each particle of the medium will be set in harmonic motion, but the phase of the vibration varies from point to point at any one instant. (This may be illustrated on the ball-and-spring model.) The simplest mode of representing waves graphically is to choose two axes, one giving the distance the wave front advances in any direction, the other the displacement at any instant at points along a line drawn in this direction. A curve on this diagram gives the displacements at FIG. 152. — Distances along a line in the direction of prop-.,,,, agation. A harmonic train of waves. any instant Ot all the particles in the medium along a line in the direction of advance of the waves. 342 HARMONIC AND COMPLEX WAVES 343 I III! II I ILL LL ILL ILL I MM II i i in INI III I II III III Hill If the waves are due to a harmonic disturbance, the curve is as shown, where the line PQ indicates the displacement of the particle whose posi
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wave length, and so in a unit of time the waves advance a distance equal to the product of the wave number and the wave length. Therefore, if V is the velocity of the waves, I the wave length, and ^V the wave number, V— Nl. Or, if T is the period of the waves, V= —. (The velocity of an individual particle depends upon the instant we consider it, for it is making harmonic vibrations ; and so the velocity varies from zero to a maximum value, then decreases to zero, etc. It is clear that there is not the faintest connection between this varying velocity and the constant velocity of the waves.) //.l/M/o.y/r AM) ro.u/'/./;.v \\\l\ 345 Doppler's Principle. — If we speak of that portion of a train of waves whieh is a wave length long as a " wave," we may say that, if the wave number is 2\T, the source emits N waves in a unit of time ; and, in general, N waves pass any point in the medium in a unit of time. This is true if the vibrating source, the point in the medium, and the medium as a whole are not moving. It is interesting, however, to consider the two cases, when the source is moving and when the point in the medium where the waves are counted is moving, the medium not being in motion in either. If the vibrating source is at rest, and the point in the me- dium is moving toward it in a straight line, let N be the frequency of the source, V the velocity of the waves in the stationary medium, I their wave length, and v the velocity of the moving point. Owing to the motion of this point it would pass - waves in a time if the waves were stationary in the medium ; but since they are moving toward the point at such a rate that N waves would pass a fixed point in a unit of time, the total number that passes the moving point in that time is the sum JV+y. (But-ZV=-; and so this number jr. \ may be written — j^-. J For similar reasons, if the point is moving away from tin- fixed source with a velocity v, the number of waves which pass it in a unit of time is N—'/ T *" \ * If the point in the medium is fixed, and tin
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- vibrating source is approaching it in a straight line with the velocity v, the case is entirely different. If tin- source were n«>i the length «,f ;v wave would be I; but. when it is the wave fnmt advances a distance T from the in,i unit time, and in this same time the source advances a diMam-e v; so the X \\aves that have been 346 VIBRATIONS AND WAVES emitted in this time are crowded together in the interval of space V— v, and the length of a wave is now — ~^- The new wave number, or the number of these waves that pass a fixed point in a unit time, is the velocity V divided by this wave length, or — — N. Similarly, if the vibrating source v — v is receding from the fixed point with the velocity v, the new wave number is - — N. y V+v If v is small compared with FJ these last two expressions may be written fl + iW and (l - £\flT, or N+ - and N— - ; so the formulae for this and the preceding case agree under this condition. It is thus seen that when the source of waves and the point under consideration are approaching each other, the wave number is apparently increased ; while, if they are receding from each other, it is apparently decreased. These formulae express what is called Doppler's Principle. It is illustrated in the case of a star approaching or receding from the earth, in a whistling locomotive approaching or receding from a station, etc. Attenuation of Waves. — The energy of a harmonic vibra- tion varies as the square of the amplitude, and hence the intensity of harmonic waves varies as the square of their amplitude. This amplitude decreases as the waves advance, owing to various causes. One case has been considered already on page 331, where it was proved that in spherical waves the amplitude varies inversely as the distance from the source. Further, there may be friction involved in the relative displacements of the particles of the medium, as is the case to a greater or less degree with all waves in all forms of matter ; or, motion may be given particles of foreign matter immersed in the medium (see page 311); in both of HARMONIC AND COMPLEX WAVES 347 which cast's the amplitude of the waves decreases and they lose energy. Similarly, if waves in one
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medium are inci- dent upon a boundary separating it from another in which the velocity is different (see page 333), waves are transmitted into the latter medium, and waves are also reflected back into the former. The combined intensity of these two trains of waves must equal that of the incident train ; and so the amplitude of the transmitted waves and that of the reflected waves must both be less than that of the incident waves. (An obvious law connects them.) Superposition of Waves; Complex Waves. — There can be two harmonic trains of waves of the same wave length and amplitude, but differing in phase at any instant, depending upon when or how their motion was begun. Similarly, we may have in any medium waves of different wave length, different amplitude, etc. ; and their combined action may be found by compounding them as was done for the vibrations of a particle, for it may be proved that this is allowable, pro- vided the individual displacements of the particles of the medium are small in comparison with the wave length. The best method of considering the superposition is a graphical one. Thus, two plane polarized transverse waves (see page 313) which are harmonic, and whose directions of vibrations are at right angles, may be compounded as shown in Lissa- jous' figures. In particular, two such waves having the same period would combine to produce an elliptically polar- ized train of waves, or a circularly polarized train if their amplitudes are equal and their difference in phase ^- Con- versely, an elliptically or circularly polarized train of waves may be resolved into two plane polarized trains of waves which are harmonic and whose vibrations are at right angles to each other. Two plane polarized transverse waves which are harmonic and whose directions of vibration an- the same, or two 348 VIBRATIONS AND WAVES tudinal harmonic trains of waves, may be compounded in the same manner as were two vibrations in the same direc- tion ; and any of the illustrations given on page 321 may be applied to the case of waves. Conversely, by Fourier's theorem, any complex train of waves may be resolved into trains of harmonic waves whose wave lengths are in the ratio of 1:2:3: etc. There are other modes of resolution, also, which often are more convenient. We see, then, that a harmonic vibration produces a har- monic train of waves ; a complex vibration, a complex train. A special case of this last is a non-
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periodic or a confused vibration ; it will produce a corresponding wave. If the disturbance is intense, but lasts only a short time, it produces in the medium what we have called a "pulse." Its effect when it reaches a portion of the medium containing foreign matter is naturally different from that of a long train of harmonic waves ; because owing to these last there are peri- odic forces brought into action on the foreign particles, and resonance may follow. Distortion of Complex Waves. — As we have seen in speak- ing of the attenuation of waves owing to their decrease in amplitude, there are cases in which long waves are less affected than short ones. (See page 341.) In such cases, if a complex vibration is producing a train of complex waves, its harmonic components of long wave length will persist longer than those of short wave length ; and so the char- acter of the complex vibration at different points in the medium will vary. This phenomenon of the change in the "form" of the wave is called "distortion." The further one is from the vibrating source, the more nearly does the vibration approach that of being simple harmonic. This fact is illustrated by ocean cables and by long-distance telephone wires. However complex the electrical disturbance at one end of a cable, that at the other is nearly, if not quite, har- monic. In using a telephone over a long distance the qual- 349 ity of the sound is entirely changed, only the graver notes l>cin«r heard. The great merit of Pupin's new system of constructing cables and telephone lines is that it not alone il< -creases the attenuation, but also diminishes greatly the distortion by making the attenuation of all the waves, long and short, the same. Nodes and Loops " Stationary Waves. " — A most important effect of wave motion is illustrated by a simple experiment which may be performed with a long flexible cord; e.g. a long spiral spring or a long rubber tube, one of whose ends is fastened to a fixed support and the other is held in the hand. If the cord is stretched fairly tight and the free end moved sidewise with a rapid harmonic motion, waves will be produced in the cord which will be propagated up to the fixed end, and will there suffer reflection and be propagated back to the hand, etc. Consequently, at any instant there are in the cord two trains of waves traveling in opposite directions. If the frequency of
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the motion of the hand is exactly right, it will be observed that the cord ceases to have the appear- ance of being traversed by waves, and vibrates transversely in one or two or more portions or " segments." That is, there are certain joints in the cord where there is very little, almost no, motion, which are called "nodes" ; and the cord in between these vibrates just like a short cord whose two ends are fastened. The points halfway het ween the nodes, where therefore the motion is greatest, are called "loops." Tli is type of vibration is sometimes, but improperly, called "stationary" or "standing" Craves, In reality, the waves have disappeared in the production of a vibration. Such vibrations as this are extremely common; and may occur with waves of all kinds. The explanation i^ evident Consider a medium through which are passing in opposite directions two harmonic- in in- 350 VIBRATIONS AND WAVES of waves which are not suffering attenuation and whose wave lengths and amplitudes are equal. At any instant they may be represented by curves as in the cut. Since the actual motion in the medium is found by superimposing the two wave motions, there will be certain points, one of which is P2 in the cut, at which the displacement is zero, owing to one wave neutralizing the effect of the other. As the waves ad- vance— in opposite directions — they continue to neutralize x...../ \. p FIG. 155. — Formation of nodes and loops by two trains of waves advancing in opposite direc- tions. PU P2, PSt etc., are nodes. each other at this point, as is seen from the cut; therefore this is a node. The importance of the conditions that the waves should not be damped and that the amplitudes of the two trains should be the same as well as their wave lengths is evident. Further, since in a train of waves at a distance of half a wave length from any point the displacement is exactly reversed, if two trains of waves neutralize each other at a point JP, they will also do so at points distant from P by half a wave length, a whole wave length, etc. Therefore, the distance apart of two nodes equals one half the wave length of either of the component trains of waves. In the case of the cord which was first considered, the fixed end is obviously a node ; and the one held in the
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hand is approximately one, as is evident if one observes the motion. If the long flexible cord is held suspended vertically from a balcony so that the lower end hangs free, a vibration of the same kind can be produced ; only in this case, since reflection takes place at a " free end," this point is a loop. As the frequency of vibration of the hand is increased gradually, it is found that there are certain definite fre- " STATION ABT \\.\VES" 351 quencies for which the cord separates into vibrating seg- ments ; ami the number of these segments increases, that is, the distance apart of the nodes decreases, as these critical frequencies increase. Similarly, if the tension of the cord is increased, the critical frequency must be increased also. The explanation of these facts is not difficult. Let us consider the first case, that of the cord with its end fixed ; and let the length of the cord be L, the wave length of the component waves be Z, the distance apart of two nodes be c?, the velocity of the waves in the cord be K, the frequency of the vibration be N, and the number of segments be n (necessarily a whole number). Then, we have the relations: c? = -, V •= Nl, L = nd; and therefore, on substitution, N= —. So if the tuision is kept constant, i.e. if V is constant, and if the length L of the cord is not varied, n varies directly as JV; 1 1 ml is, tin- frequency of the vibration must have a definite value, since n must be an integer, and if the number of seg- ments is increased from 1 to 2 to 3, etc., the frequency must be increased in an equal ratio. Again, if the tension in the cord is increased, the velocity V is increased ; and therefore, if // U tin- same, iV, the frequency, must be increased. Vibration of a Stretched Cord. — These vibrations in a stretched flexible cord are not always produced by the method described. If the cord is stretched between two fixed points, it may be set vibrating by using a violin bow, by plucking it with a tinker, by striking it a blow, etc. The vibrations are exactly like those just described. By lightly touching the middle point of the cord, so as to hold it nearly at rest, and bowing the cord or plucking it
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formula, * d IT r=\' —, where T is the tension in the cord and d is the mass per unit length. Substituting this in the general for- mula, we have N= ^'^\—', which shows that if the tension of cord is increased, the frequency is increased ; if the length of the cord is increased, the frequency is decreased; etc. All of these facts are illustrated in various musical instru- 2.L * a ments, as will be noted later. Vibrations of a Column of Gas. — The same type of vibra- tion can be produced in a column of gas, such as we have in the case of an organ pipe, a flute, a horn, etc. There are many ways in which the vibrations may be produced: by blowing a blast of air over the sharp edge of an orifice open- ing into the column ; by making some solid body in contact with the gas at one end vibrate harmonically with a suitable frequency; etc. (The first is illustrated when one blows an ordinary whistle or when one blows over the end of a hollow key; the latter when one blows a horn by means of the vibrations of the lips or holds a tuning fork over the mouth of a bottle in which water is poured until there is resonance.) If a particle in a column of gas is at a node, that is, if its motion is a minimum, it must be in contact with the end wall or it must be held stationary by symmetrical conditions on its two sides, up and down the column. Thus, as the particles in the gas vibrate, there are the greatest fluctua- tions in pressure and density at the nodes. Similarly, at a loop there is the greatest motion, but tl <- least change in pressure and density. Tin- vibration..f a column of gas is illustrated in t he accompany MILT cut, in which the transverse lines indicate the positions of 1 1 -a us verse layers of gas. If the column <>f gas is closed by a solid partition, this point is a node; while if the en. I ix opm to thr air, so that AMES'S PHYSICS — 28 354 VIBRATIONS AM) WAVES the pressure there cannot change greatly, this point is approx- imately a loop. (Actually, the loop is a short distance beyond the open end in the air outside. If the tube contain- I 1 FIG
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. 157. — Stationary vibration in a column of gas. Vertical lines represent positions of layers of gas. Curves represent by their vertical displacements the horizontal displacements of the layers of gas from their positions of equilibrium. Arrows represent the directions of motion of the layers of gas. ing the gas is a circular cylinder with a radius R, the loop is at a point beyond the end at a distance given approxi- mately by 0.57J2.) As in the case of a stretched cord, the column of air can vibrate in different ways, depending upon the number of seg- ments. We shall consider several special cases : "STATION A It Y II.1 I'A'.s 855 1. A column of gas closed at both ends. — There is there- fore a node at each end. The simplest mode of vibration is wht'ii there is only one loop, which must then be at the mid- dle point : in this case the distance from node to node is the length of the column L. This must equal half the wave length of the component waves, J ; and, if 2 is the fre- quency of the vibration and V the velocity of the compres- sional waves in the gas, ^ = —. Hence, L = A y *i JVj = ^-y. The next simplest mode of vibration is when the column of gas is divided into two segments by a node at its middle point. In this case the distance between two nodes V or is half the length of the column, —. Hence, f Tin- analogy with the transverse vibrations of a cord stretched between two fixed points is complete. The possible vibrations make a complete harmonic series. 2. A column of gas open at both ends. — There is then a loop at each rnd. The simplest moil,- ut' vibration is wlu-n tin -re is only one node, which must be at the middle point. We saw in our general discus- sion that a loop came halfv 1..-I vreen two nodes; so the distance from loop to loop equals tl ii'Mlu to node. In this rasr. thr L N L N L N L '•*. — Vibration* of A column of RM open »t both en.U : (1) ftimhuncnul ; (2) flrtt p*rtUl ; (.1) Mcond partial.
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356 \'li;ilATlONS AND WAVES distance from loop to loop is the length of the column, neglect- 2 2JV, ing the slight correction for the ends ; and so L = -1 = -—, The next simplest case is when there is a loop at the middle point and a node at each of the points halfway from it to the ends. The distance from loop to loop is one half the length of the column. Hence, So this case is the same as the previous one ; the vibrations form a complete harmonic series. This kind of a column of gas is called an " open " one ; and, as in previous cases, if it is set vibrating by some random or indefinite means, the resulting motion is a complex vibra- tion equivalent to the addition of these simple ones. 3. A column of gas open at one end and closed at the other. — There is thus a loop at one end and a node at the other. The simplest mode of vibration is when there are no other nodes or loops. The distance from node to loop is one half the distance from node to node ; so in this case 2 L= J = — —, v 2 2JV] orai-Jl The next simplest mode of vibration is when there is a node at a distance — from the open end, and a loop at a dis- 2 L 2 L tance — • The distance from node to node is then — ; and we have the relation =£ = * = -^-, or N2 = 2±- Simi- 9 T 7 V ^ V o 22 J\/ 4 Jj larly, the next mode of vibration will give a frequency JV3 = — r ; etc. So in this case the vibrations have fre- quencies in the ratios 1:3:5: etc. ; a series of the odd har- monics only. It should be noted that the fundamental in "8TAT10\Ai;r 357 this column of gas has a frequency one half that of the fun- damental in the other two cases, when their lengths are the same. This kind of a column is called a "stopped" one. If it is set vibrating at random, its complex vibration is com- pounded of these simpler ones just described. Manometric Flame. — A simple method of studying the vibrations of the column of gas in a tube is to pierce openings at intervals along the tube and close them with flexible rubber di
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is bowed or struck so as to be set in vibration. The frequency of a fork may be measured with great accuracy by comparing its period with that of a standard clock ; but for details of the methods of comparison, reference should be made to some larger text-book, such as Poynting and Thomson, Sound, Chapter III, or to some laboratory manual for advanced students. (The frequencies of two forks may be compared by Lissajous' figures, or by " beats," if the frequencies are close together. See pages 327 and 412.).Vibrations of Metal Plates. — Thin metal plates cut in squares or circles can be made to vibrate transversely by clamping them at some point — generally the centre — and drawing a violin bow across their edges. When this is done, it is easy to show that there are certain lines along the plates where there is comparatively no motion ; these are called "nodal lines." The simplest method of proving their existence is to scatter fine dry sand over the plates before they are set vibrating; when the vibrations begin, the sand collects along the nodal lines, being thrown there by the other vibrating parts of the plates. There are thus formed most beautiful, regular geometrical figures, which are "STATIONARY WAVM&" 361 called "Chladni's figures," after the physicist who first sys- tematically investigated them. Their shape and complica- tions depend upon the point of support of the plate, the point where it is touched with the finger so as to make a node, the point of bowing, and the manner of bowing, which. to a certain degree, determines the number of vibrations, besides the fundamental, which are present. Similar nodal lines may be observed on stretched mem- branes, such as drumheads, when they are set in vibration in a proper manner. If some extremely light powder is used instead of the sand, with which to cover the plate, it is observed that it gathers, not at the nodal lines, but over those portions of the plate which are moving with the greatest amplitude. The reason for this was discovered by Faraday, who showed that small whirls were formed in the air near the vibrating portions of the plate, and that these carry the light powder off the nodal lines on to those portions. Vibrations of Bells. — Metal bells or bell-shaped objects, like glass tumblers or bowls, may be set in either
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transverse or longitudinal vibrations. The simplest mode of transverse vibration is shown in the cut. The position of the loops is fixed by the I MI i 1 1 1 where the clapper strikes. Many other vibrations than the fundamental are always present, but there is no simple relation between their frequen- It there ll an irregularity in the thickness of the rim at some point, this will produce comparatively little FIG. 168. - Ono mode of vibr*. effect if it comes at a node; but if it is at a loop, it will result in a change of the frequency. So a hell like this.-an vibrate in two ways, giving frequencies that are n..t very different : and if the bell is Struck a random l»lo\\, both these vibrations will occur. This is the 362 V1BRATH).\* AM) }\'A\rES cause of the "beating" of large church bells, as will be explained in a later chapter. (See page 412.) Other Illustrations of Stationary Waves. — The illustrations so far given of these vibrations due to the superposition of two trains of waves in opposite directions in a medium have been purely mechanical ones, the medium being either a solid or a fluid. But waves in the ether can produce these vibra- tions also, as is shown by the experiments of Wiener. A node in this case corresponds to a point where there is no motion in the ether (if we retain our mechanical concept of the ether) ; and a loop, to a point where there is the greatest motion. Rapid vibrations in the ether produce in the sur- rounding matter various effects, such as chemical changes which may be shown by photographic processes, fluores- cence, etc. ; and by all of these the existence of nodes and loops in the ether has been proved, when waves in it fall upon a mirror and suffer reflection. The distance apart of the nodes equals half the wave length of the incident waves. Electrical waves along wires and electromagnetic waves, produced by ordinary electric oscillations, cannot produce these vibrations with nodes and loops, because these waves are all "damped"; and owing to this fact the amplitudes of the reflected and incident waves are not equal. Measurement of the Velocity of Waves. — These facts in regard to the vibrations of stretched cords, of wires and rods, and of columns of gas (or, in fact, liquids also), lead at once to obvious methods of comparing the velocities of
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waves in different media. Thus, suppose two stretched cords have the same frequency of vibration when vibrating transversely in their fundamental modes ; then if L^ and L^ are their lengths, and Vl and F^ are the velocities of transverse waves along them, •• BTA /vo.v.i/;)' HMFJ&S" 363 Similarly, if the fundamental longitudinal vibrations of two \viivs «'i rods of different material are the same, and if Zj and LI are their lengths, we have the same relation, I \ : ] 2 = Ll : Ly for the velocities of compressional waves. Or. it t\ 11 pipes contain different gases, and if their fundamental vibrations have the same frequency, the same formula applies for the velocities of waves in these gases. There an- two general methods for determining when two vibrations have the same frequency if the medium is a ma- il-rial one. As we shall see later, when two vibrating bodies that are not moving bodily are producing sounds, they have the same frequency if the " pitches " of the two sounds are the same; and this can be told with great exactness by a trained ear; or, if they differ slightly, the exact difference in frequency can be determined by the beats. Again, the vibrations may both be produced by resonance from a third vibration. Thus, if a vibrating tuning fork is held at the opening of a column of a gas whose length can be varied, resonance may be secured by noticing for what length the sound i> most reenforced ; and similarly with the column of a second gas. (A column of gas can vibrate, as has been shown, in many ways, breaking up into a different number of segments. So with a Lfiven tuning fork, resonance may be secured for several different lengths of the column of gas. The distance from node to node is the same, however, for all these different nodes of vibration ; and it is the quant it I. in the previous formula. If the frequency of the fork Nis known, and this length L is m.-asm-.-d, the velocity of tin- waves in the gas is obviously *2LN.) Again, stretched • •ords or wires may have their lengths.-han^i-d until they are in resonance with a given tuning fork by the following mechanical method : stretch the cord or wire from two pegs or over two knife edges which are fastened
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to a large box open at its ends ; place loosely on the cord or wire a light saddle of paper ; set the stem of the vibrating tuning fork 364 VIBRATIONS AND WAVES on the box ; if there is resonance, the wire or cord will be set in vibration and the paper saddle will be thrown off. Kundt's Method. — We can also compare the velocity of congressional waves in a solid rod with their velocity in a gas or liquid by a method devised by Kundt. The gas is contained in a long tube, which is closed at one end by a tightly fitting piston and at the other by a very light one, which can move to and fro easily, but which nearly fits the tube. This last piston is attached rigidly to one end of the rod which is along the axis of the tube but projects beyond it, and which is clamped at its middle point. This rod is set in longitudinal vibration by stroking its free end with a damp cloth, and so the piston attached to its other end FIG. 164. — Kundt's apparatus for measuring the velocity of waves. vibrates and produces waves in the gas contained in the tube. The length of this column of gas is altered by means of the piston at its further end until it is vibrating with nodes and loops. The method of determining this condition is to sprinkle through the tube some light powder, such as is obtained from the finest cork dust ; when nodes and loops are formed, the powder collects in ridges across the bottom of the tube, leaving, however, the nodes perfectly bare. The frequency of the vibration of the rod is the same as that of the column of air, because the latter is " forced " by the former ; so, if L^ is the distance from loop to loop in the rod, that is, its length if we neglect the mass of the vibrating piston ; and if Lz is the distance from node to node in the column of air, the velocity of compressional waves in the rod is to the velocity of waves in the gas as Ll : Z2. (This method may also be used for a liquid instead of a gas by using suitable powder to mark the nodes.) 365 It should be observed that the piston on the end of the rod is at a loop for the rod, but a node for the gas; exactly as when the long elastic cord is set in vibration by the hand, as d---n il»ed on page 350, — the motion at a loop in
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the solid rod is extremely small compared with that at a loop in the gas. The explanation of the formation of the transverse ridges of the pow- der depends upon the fact that, as the particles of gas away from the nodes vibrate back and forward between the particles of powder, their mean velocity depends upon the arrangement of the dust particles, and varies at different points, thus producing pressures in certain directions. (See page 169.) The full description of the process is long, and will not be given here. It may be found in Rayleigh's Theory of Sound, Vol. II, page 46. Therefore, if we know by direct experiment, or otherwise, the velocity of waves in air, we may by Kundt's method de- termine the velocity of compressional waves in any solid mate- rial out of which a rod can be made ; and then, by replacing the air in the tube by some other gas (or by a liquid), we may determine the velocity of waves in it. (This is the.standard method for all ^ascs which can be secured in a small quantity only, such as helium, argon, and the other new gas« The values of the velocity of compressional waves in a few substances is given in the following table : Air • • •. 0° C.... 33,140 cm. per second Hydrogen... 0° C.... 128,600 cm. per second I Humiliating gas.. 0° C.... 49,040 cm. per second Oxygen. 0° C.... 31,7i>0 cm. per second...| (abx.lnt.-). 8°.4C.... 126,400 em. IMT. second :-nm... 73.4C.. i Water.. 8° C.... 1 1.V)<H> em. i>er second Braes 350.000 cm. per second < "pper... 20° C.... 356,000 cm. per second 600,000 to 600,000 cm. per second. 20° C.... f> 13.000 cm. IMT second tfin... 16° C.... 130,400 cm. per second CHAPTER XX HUYGENS'S PRINCIPLE. REFLECTION AND REFRACTION Huygens's Principle. — One of the most important theorems in regard to wave
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motion is in part due to Huygens, and it is called by his name. In its most general form it is com- plicated, and can be demonstrated only by the aid of the infinitesimal calculus. We shall give certain special applica- tions of it ; and, although the statements to follow are not rigorous, they are sufficiently so for all present purposes. We cannot do better than to use Huygens's own language, as it appears in the translation of his TraitS de la Lumiere by Crew in The Wave Theory of Light (New York, 1900). Huygens's treatise was written in 1678, but was not pub- lished until 1690. " In considering the propagation of waves, we must remem- ber that each particle of the medium through which the wave spreads does not communicate its motion only to that neigh- bor which lies in the straight line drawn from the luminous point, but shares it with all the particles which touch it and resist its motion. Each particle is thus to be considered as the centre of a wave. Thus, if DCF is a wave whose centre and origin is the luminous point A, a particle at B, inside the sphere DCF, will give rise to its own individual [secondary] wave, KCL, which will touch the wave DCF in the point (7, at the same instant in which the principal wave, origi- nating at A, reaches the position DCF. And it is clear that there will be only one point of the wave KCL which will touch the wave DCF, viz., the point which lies in the straight line from A drawn through B. In like manner, each of the 366 REFLECTION AND REFRACTION 867 other particles, bbbb, etc., lying within the sphere DCF, gives \ n wave. The intensity of each of these waves may, however, be infinitesimal com- pared with that of I) OF, which is the resultant of all those parts of the other waves which are at a maxi- mum distance from the centre A. " We see, moreover, that the wave DCF is determined by the extreme limit to which the motion has traveled from the point A within a certain interval of time. For there is no motion beyond this wave, whatever may have been produce (I inside by those parts of the secondary waves FIG. 165. which do not touch the sphere DCF."
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We shall also quote Huygens in his explanation of reflec- tion and n-fractioii. 1. Reflection of Plane Waves by a Plane Mirror. — " Having explained the effects produced by light waves in a homogene- ous medium, we shall next consider what happens when they impinge upon other bodies. First of all we shall see how reflection is explained by these waves, and how the equality of angles fol- lows as a consequence. l.«-t AB represent a plane polished surface of some metal, glass, or other sub- stance, which, for the | Ft... 1M. ent, we shall consider as perfectly smooth (concern- in^ iiT("_nil;int i> - which are unavoidable, \\.- shall have some- thing to sav at the close of this demonstration); and let the liii..I/..', inclined to AB, represent a part of a light 368 VIBRATIONS AND \YAVES wave whose centre is so far away that this part, AC, may be considered as a straight line. It may be mentioned here, once for all, that we shall limit our consideration to a single plane, viz., the plane of the figure, which passes through the centre of the spherical wave and cuts the plane AB at right angles. " The region immediately about 0 on the wave A 0 will, after a certain interval of time, reach the point B in the plane AB, traveling along the straight line CB, which we may think of as drawn from the source of light, and hence drawn perpendicular to AC. Now, in this same interval of time, the region about A on the same wave is unable to transmit its entire motion beyond the plane AB ; it must, therefore, continue its motion on this side of the plane to a distance equal to CB, sending out a secondary spherical wave in the manner described above. This secondary wave is here rep- resented by the circle SNR, drawn with its centre at A and with its radius AN equal to CB. " So, also, if we consider in turn the remaining parts, H, of the wave AC^ it will be seen that they not only reach the surface AB along the straight lines HK parallel to CB, but they will produce, at the centres, K, their own spherical waves in the transparent medium. These secondary waves are here represented by circles whose radii are equal to KM, that is, equal to the prolongations of HK to the
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straight line BGr, which is drawn parallel to AC. But, as is easily seen, all these circles have a common tangent in the straight line BN, viz., the same line which passes through B and is tangent to the first circle having A as centre and AN, equal to BC, as radius. " This line BN (lying between B and the point N, the foot of the perpendicular let fall from A) is the envelope of all these circles, and marks the limit of the motion produced by the reflection of the wave AC. It is here that the motion is more intense than at any other point, because, as has been RE I '!.!•:< T/o.Y l.\/> UEFRACTION 369 explained. HX is tlie new position which tin? wave AC has assumed at the instant when the point C has reached B. For there is no other line which, like BN, is a common tan- gent to these circles... " It is now evident that the angle of reflection is equal to the angle of incidence. For the right-angled triangles ABC and BXA have the side AB in common, and the side CB equal to the side NA, whence it follows that the angles oppo- site th.-se sides are equal, and hence also the angles CBA and NAB. But CB, perpendicular to CA, is the direction of the incident ray, while AN, perpendicular to the wave BN, has the direction of the reflected ray. These rays are, therefore, equally inclined to the plane AB. " I remark, then, that the wave A C, so long as it is con- sidered merely a line, can produce no light. For a ray of light, however slender, must have a finite thickness in order to be visible. In order, therefore, to represent a wave whose path is along this ray, it is necessary to replace the straight line AC by a plane area... where the luminous point is supposed to be infinitely distant. From the preceding proof it is easily seen that each element of area on the wave [front], having reached the plane AB, will there give rise to its own secondary wave ; and when C reaches the point B, these will all have acommon tangent plane, vi/.., [a plane through BN~\. This [plane] \\ill he cut... at right angles hy the same plane which thus cuts the [wave front at A C at
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right angles, i.e. the plane of incidence]. "It is thus seen that the spherical second a r\ waves can have no common tangent plane other than BN. In this plane will he located more of the reflected motion than in any other, and it will therefore receive the light transmitted from the \\.ive CH. ******* VM! -'« I II •, - i 370 VIBRATIONS AND WAVES "We must emphasize the fact that in our demonstration there is no need that the reflecting surface be considered a perfectly smooth plane, as has been assumed by all those who have attempted to explain the phenomena of reflection. All that is called for is a degree of smoothness such as would be produced by the particles of the reflecting medium being placed one near another. These particles are much larger than those of the ether, as will be shown later when we come to treat of the transparency and opacity of bodies. Since, now, the surface consists thus of particles assembled together, the ether particles being above and smaller, it is evident that one cannot demonstrate the equality of the angles of inci- dence and reflection from the time-worn analogy with that which happens when a ball is thrown against a wall. By our method, on the other hand, the fact is explained without difficulty. " Take particles of mercury, for instance, for they are so small that we must think of the least visible portion of sur- face as containing millions, arranged like the grains in a heap of sand which one has smoothed out as much as possible ; this surface for our purpose is equal to polished glass. And, though such a surface may be always rough compared with ether particles, it is evident that the centres of all the second- ary waves of reflection which we have described above lie practically in one plane. Accordingly, a single tangent comes as near touching them all as is necessary for the pro- duction of light. And this is all that is required in our demonstration to explain the equality of angles without allowing the rest of the motion, reflected in various direc- tions, to produce any disturbing effect." The law of reflection in regard to the equality of the angles of incidence and reflection was known to the ancients, and was made use of by Euclid as early as 300 B.C. He also deduced some of the properties of concave mirrors. The law stating that the normals to the two waves and the surface lie Hl-:i-'LK< TIO
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ii would equal the whole lengths of the various lines KM. " But all these circles have a common tangent in the line BN\ viz., the same line which we drew from the point B tangent to the circle SNR first considered. For it is easy to see that all the other circles from B up to the point of contact N touch, in the same manner, the line BN, where N is also the foot of the perpendicular let fall from A upon BN. " We may, therefore, say that BN is made up of small arcs of these circles, and that it marks the limits which the motion from the wave AC has reached in the transparent medium, and the region where this motion is much greater than anywhere else. And, furthermore, that this line, as already indicated, is the position assumed by the wave AC at the instant when the region C has reached the point B. For there is no other line below the plane AB, which, like BN, is a common tangent to all these secondary waves.... " If, now, using the same figure, we draw EAF normal to the plane AB at the point A, and draw DA at right angles to the wave AC, the incident ray of light will then be repre- sented by DA ; and AN, which is drawn perpendicular to BN, will be the refracted ray; for these rays are merely the straight lines along which the parts of the waves travel. "From the foregoing, it is easy to deduce the principal law of refraction; viz., that the sine of the angle DAE always bears a constant ratio to the sine of the angle NAF, ui:rii:moN AND i;i-:n;.i< TION 373 whatever may be the direction of the incident ray, and that tin- ratio is the same as that which the speed of the waves in the medium on the side AE bears to their speed on the side AF. " For, if we consider AB as the radius of a circle, the sine of the angle BAC is BQ and the sine of the angle ABN is AN. But the angles BAC and DAE are equal, for each is the complement of CAE. And the angle ABN is equal to NAF, since each is the complement of BAN. Hence the sine of the angle DAE is to the sine NAF as BO is to AN. But the
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ratio of BCto AN is the same as that of the speeds of light in the media on the side toward AE and the side inward AF, respectively; hence, also, the sine of the angle DAE bears to the sine of the angle NAF the same ratio as these two speeds of light." Since these speeds are properties of the media and not of the direction of the propagation of the waves, we have at once the law that the ratio of these sines is independ- ent of the angle of incidence. It is evidently different for different media, and will be shown to be different for waves of different wave length in the case of ether waves in a material medium. This law of refraction was first discovered experimentally by Snell (1591-1626). I! --fraction is a much more common phenomenon with ether waves than with air waves, and it. will be disenss.-d more fully in the section devoted t<> Light. CHAPTER XXI INTERFERENCE AND DIFFRACTION Interference Young's Experiments. — A most important phenomenon of wave motion, and one of particular interest historically because by means of its discovery Thomas Young in 1801 proved that light was due to waves, is wli.it is called inter- ference. In Young's own words: "When two undulations, from different origins, coincide either perfectly or very nearly in direction, their joint effect is a combination of the motions belonging to each. " Since every particle of the medium is affected by each undulation, wherever the directions coincide, the undulations can proceed no otherwise than by uniting their mo- tions, so that the joint motion may be the sum or difference of the separate motions, accordingly as similar or dissimilar parts of the undulations are ;. 16S. — Interference of waves on the surface of a liquid, which are sent out by two point sources. coincident." (This prin- ciple is illustrated in the cut, which represents the " interference " of two trains of water waves.) Two experiments may be described ; both are due to Young, and both may be performed easily with home-made apparatus. We shall describe them as if the waves to be 374 7JV TK /,•/•*/•:/.' i:\rf-:.i.v/> inrn;.ii IION 375 studied were light waves; but the same apparatus, suitably enlarged, would do equally well for air waves. Let there be trains
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of waves sent out by having some source placed near a long Marrow slit in an opaque screen. If the slit is sufficiently narrow, the disturbances will proceed out from the slit in all directions, making a train of waves with a cylindrical wave front. A second opaque screen with two X' if row slits, which are close together and both of which are parallel to the slit in the lirst screen, and at equal distances from it, is placed parallel to the latter. As the cylindrical waves reach these two slits, two cylindrical trains of waves are produced beyond the second screen. The importance of this arrangement lies in the fact that the two trains of waves thus produced are identical; that is, they have the same amplitude, the same wave length, and the same phase, because they arc produced by disturbances in the same wave front at the same distance from tin- lirst slit; so, if the original source of the waves changes its character in any way, the two cylindrical waves from the two slits both change in the same manner at the saint- instant. Then, if we consider the effect at any point in the space which is traversed l>y tin- two trains of waves, it is receiving disturb- ances from both waves, and the effect produced is the sum of two, one due to each train. Another mode of producing this result is to remove the screen with the two slits, and to place parallel to the slit in the first screen a narrow opaque object like a line wire or small needle. As will be shown in speaking of diffraetion (see page 880). disturbances are produced in the low. \ l( tly as if there were a source of waves along the edge of the obstacle ; so, in the case of the wire or needle. the points in the shadow are receiving distnrhanees from two parallel line sources,,f waves along the two ed: These two disturban then, in this case also due to 1 \\ o '{•'iff trains of Vfl 376 VIBRATIONS AND WAVES If these waves are light waves of a definite color, and if, from a point in the medium traversed by the two trains of waves, one looks in the direction of the two slits (or of the wire), or, better still, if a magnifying glass is used in front of the eye, a series of black and colored lines parallel to the slits (or wire)
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are seen. Similarly, if short sound waves are used, it is not difficult to prove by means of a sensitive flame (see page 192) that there are corresponding " bands of silence and sound," meaning that at points along a line parallel to the slits there are disturbances in the air, while along a neighboring line there are not. B C A C FIG. 169. — Diagram of Young's Interference experiment. 0t and Ot are two sources of waves and ACi& a screen on which the two trains of waves are received. U The explanation is not difficult. Let us consider the effect at various points on a screen parallel to the plane of the two slits. Let Ol and 02 be the traces of the slits on the paper, and A C that of the last screen. Let B be a point halfway between the slits ; draw a line BA perpendicular to the screens, and let 0 be any other point on the screen. This point receives disturbances due to two trains of waves ; but the lengths of the paths from C to the two sources 01 and 02 are not the same. This is shown on a large scale in part of the diagram. i.\TKi:ri-:nENCE AND DIFFRACTK>\ 877 FKU n 0.2 draw O.J* perpendicular to the line 0-^C. Since OL a jid 0.2 are iii reality extremely close together compared with the other distances in the apparatus, O^P is the differ- ence in path from 01 and 02 to C. If it amounts to a wave length exactly, or to any integral number of wave lengths, tin- disturbances reach O in the same phase, and so the effect is great ; but, if this difference in path is exactly half a wave length, or any odd number of half wave lengths, the disturb- ances arrive at O in exactly opposite phases, and so there is no effect. At the point A the two paths are of equal length, so the effect is great ; and as points near it are considered, constantly receding from A in either direction, the effect decreases, becomes zero when the difference in path is half a wave length, increases to a maximum when this difference is a wave length, decreases to zero again; etc. The effect at all points on a line through C parallel to the two sources is evidently the same ; and so the screen is covered with a pat- tern of bands, or. as they are
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the screens, and that of the sources are known, the wave length may be determined. This matter will be referred to later. It should be noted that in the formation of these interfer- ence fringes there is no destruction of energy ; it is simply distributed differently from what it would be if the screen t: A.\h Dll-'Fl; ACTION 379 were receiving waves from* two sources which had no perma- nent phase relation. < rfertno* frtagM irtrttlntil by Young's method. 380 VIBRATIONS AND WAVES Other cases of interference will be described in the section devoted to Light, but all interference phenomena dealing with light can be reproduced with waves in the air. Diffraction FresnePs Principle. — It is a well-known fact that, if an opaque obstacle is interposed in a beam of light, a shadow will be cast on any suitably placed screen, which is more or less sharply defined, depending upon the smallness of the source of the light. This is sometimes expressed by saying that "light travels in straight lines." In the case of sound, however, such an obstacle would not prevent a noise being heard behind it ; in other words, there is no sound shadow with such an obstacle. The explanation of the difference in the two cases was given by Fresnel, making use of Huygens's principle. It may be well to quote Fresnel's own words in Crew's translation. Fresnel's great memoir on Diffraction, from which these quotations are made, appeared in 1810. "I shall now show how by the aid of these interference formulae and by the principle of Huygens alone it is possible to explain, and even to compute, all the phenomena of dif- fraction. This principle, which I consider as a rigorous deduction from the basal hypothesis, may be expressed thus: The vibrations at each point in the wave front may be considered as the sum of the elementary motions which at any one instant are sent to that point from all parts of this wave in any one of its previous * positions, each of these parts acting inde- pendently the one of the other. It follows from the principle of the superposition of small motions that the vibrations pro- duced at any point in an elastic fluid by several disturbances * I am here discussing only an infinite train of waves, or the most general vibration of a fluid. It is only in this sense that one can speak of
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two light waves annulling one another when they are half a wave length apart. The formulae of interference just given do not apply to the case of a single wave, not to mention the fact that such waves do not occur in nature. /A'/-/-;/,- /••/•;/,• /-;.v< B.i.v/> inrni ACTION 381 are equal to the resultant of all the disturbances reaching this point at the same instant from different centres of vibra- tion, whatever be their number, their respective positions, their nature, or the epoch of the different disturbances. This general principle must apply to all particular cases. I shall suppose that all of these disturbances, infinite in num- ber, are of the same kind, that they take place simulta- neously, that they are contiguous, and occur in the single plane or on a single spherieal surface.... I have thus reconstructed a primary wave out of partial [secondary] dis- turbances. We may, therefore, say that the vibrations at each point in the wave front can be looked upon as the resultant of all the secondary displacements which reach it at the same instant from all parts of this same wave in some previous position, each of these parts acting independently one of the other. "If the intensity of the primary wave is uniform, it follows from theoretical as well as from all other considerations that this uniformity will be maintained throughout its path, pro- vided only that no part of the wave is intercepted or retarded with respect to its neighboring parts, because the resultant of the secondary displacements mentioned above will be the same at every point. But if a portion of the wave be stopped by the interposition of an opaque body, then the intensity of each point varies with its distance from the edge of tin- shadow, and these variations will be especially marked near the edge of the geometrical shadow." Rectilinear Propagation. — As a simple case, consider a train of plane waves advancing from left to right; let the paper be at ri.rht angles to them and let the trace on the the wave front at any instant be given in part by AB. The etl'eet at any later time at a point P in advance of the waves is determined, as already Stated, by deducing the effects there owing to the secondary waves from ea< -h point of the wave fn.nt. and adding these geometr
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ically VIBRATIONS AND WAVES It is evident that the effect at P of the secondary waves from any point Q on the wave front depends upon the length of the line QP for two rea- sons : the decrease in amplitude of spherical waves varies inversely as the distance, and the phase of the disturbance as it reaches P varies with it. There- fore, if 0 is the foot of the perpendicular let fall upon the FIG. 171. — AB is a section of a plane wave advancing toward P. B wave front from P, i.e. its "pole," as it is called, and if a circle with a radius equal to OQ be drawn on the wave front around 0, the secondary waves from each of the points on this circle will reach P with the same amplitude and in the same phase, because they start out with the same amplitude and in the same phase, and travel the same distance to reach P. But the directions of the displacements due to the sepa- rate secondary waves are not the same, and they must be added geometrically. (Since the phases are all the same, we have simply a case of vector addition.) Let us assume that the waves are longitudinal (the proof is similar, if they are transverse). Let Q1 and Q2 be two points at the end of a diameter of the circle round 0\ and let the displacement at P due to the secondary waves from Q1 be represented by FIG. 172. — Diagram to represent the resultant action at P of all the secondary waves from points <?,, Qt, etc., on a circle around 0. i\TKi;rEi;i-:\f'E A.\D DIFFRACTION 383 then that due to the secondary waves from Q2 is repre- sented by PA2. Their resultant is PA^ a displacement perpendicular to the plane wave front. Similarly, the resultant displacement due to all the secondary waves coming from points in the circle around 0, through Qr is proportional to PA. But calling the angle (OPQ^) the " inclination " of Qv and representing it by jY", it is evident tliat PA = 2 PAl cos N; and therefore, as the inclination increases, the resultant displacement decreases. The task of compounding the effects at P, due to the secondary waves from all the points in the wave front, is not at first sight a simple one ; but Fresnel invented a most brilliant method
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at d n. iliime. Tim area of this 386 VIBRATIONS AND II zone is wla very approximately, if - is ;i small quantity. a (For visible ether waves I is not far from 0.00005 cm., and so if a = 10 cm., the area Trla equals 0.0016 sq. cm.) Spherical Waves. — The case.of spherical or cylindrical waves may be treated in the same manner ; and in some cases other modes of describing " Fresnel zones " are prefer- able. The result in them all is, however, the same. If from a point in advance of the wave front a perpendicular line is let fall upon it, the effect at this point, due to the whole wave front, is one half of that due to the first central zone around the point where this perpendicular meets the wave front, i.e. the "pole." (It is assumed, of course, that the velocity of a disturbance in the medium is independent of the direction of propagation ; otherwise, referring to the previous cuts, the effects at P from points in a circle around 0 would not reach it in the same time, and so they would be in different phases. This case of non-isotropic media will be discussed later in speaking of Double Refraction.) Let us, then, consider the propagation of a train of waves having a spherical wave front, spreading out from the point S. Let AB be a portion of the wave front at any insta n t : the effect at P at a later time will be one half that pro- duced there by the central zone around 0, where 0 is the pole of P. If the wave length is ex- tremely small so F». 174. -Case of spherical waves advancing toward P. ^ ^ ^ Q£ ^ central zone is small, we may say that the effect at P is due to the disturbance at the point 0; and so in turn, when the wave front reaches P, we may say that the effect at a point AM* birn;A< T/o.v 887 I\. farther cut on the line SOPV is due to the effect at the point 1\ etc. In this sense, the disturbance due to a train of waves having a small wave length is propagated in a ight line. 1 his line, SOPPr is called a " ray." (In an •tly similar sense, the disturbance of an exceedingly thin ••
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sufficiently large, e.g. a mountain. But we see that, if the waves are long and the obstacle is of ordinary size, the former will penetrate a great distance in the shadow. This phenomenon of the peculiarities of a shadow produced by trains of waves incident upon an obstacle is called " dif- fraction." It was first described in the case of light by the Italian priest Grimaldi in 1666 ; but its explanation was given by Fresnel. Diffraction through a Small Opening. — An important illus- tration of diffraction is afforded when a train of waves falls upon an opaque obstacle which has in it a single small open- ing. The general features of the phenomena may be deduced easily. Let the waves come from such a direction that their wave front is tangent to the plane of the opening. Draw the line OB perpendicular to this plane, and OP oblique to it ; we shall consider the effect at various points on a screen perpendicular to OB. The effect at B depends upon the number of zones that can be drawn in the small opening at 0; if it is even, the effect is zero; if it is odd, the effect is practically as great as if the whole obstacle were removed, viz., £ mv The effect at P in a similar manner depends upon the number of zones that can be drawn for it in the opening. The centre of the zones is the pole of P on the wave front, Fio. 178. —Diffraction through a small opening 0. / \ / /• /; /••/;/; /•; \ i /•;. i \ i> DIFFRA GT/OJV* 391 and is therefore far from the opening ; and, since the edges of the zones get closer and closer as one recedes farther from the centre of the zones, P will have more zones — or rather portions of them — included by the opening than does B. So, if B has three zones in the opening, it will receive a maxi- mum effect; and, if P is sufficiently far away from B, it will have portions of four zones included, and so will receive a minimum effect ; farther out still, there will be a point for which there will be portions of five zones in the opening, and which accordingly receives a maximum effect; this is, however, much less than that at BI owing to the intimation of the zones and t<> the fact that only n:irr.-\v -
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