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bebween V and ï than between W and , just because they are more used to i%
that way. Thus, since Ï = đ@/đf# = iuậ, we can just substitute ƒ/26 for ậ and get
Ÿ = („L+ R+ 1/iu@)Ê = ÔÏ. (23.19)
--- Trang 417 ---
Another way is to rewrite Eq. (23.17), so that it looks more familiar; one often
sees it written this way:
Ld1/dt + RT + q/© ƒ T dt = V(t). (23.20)
At any rate, we ñnd the relation (23.19) between voltage Ÿ and current Í which
is Just the same as (23.18) except divided by 2œ, and that produces Ed. (23.19).
The quantity #8 + /œÙ + 1/2 is a complex number, and is used so mụch in
electrical engineering that it has a name: it is called the cornplez ứmpedance,
2. Thus we can write Ÿ = 2Ÿ. The reason that the engineers like to do this
1s that they learned something when they were young: V = ÏÏ for resistances,
when they only knew about resistances and DƠ. Now they have become more
educated and have AC circuits, so they want the equation to look the same. Thus
they write Ÿ =ÊŸ, the only diference being that the resistance is replaced by a
more complicated thing, a complex quantity. So they insist that they cannot use
what everyone else in the world uses for imaginary numbers, they have to use a 7
for that; it is a miracle that they did not insist also that the letter Z be an ?#l
(Then they get into trouble when they talk about current densities, for which
they also use 7. 'The difficulties of science are to a large extent the dificulties of
notations, the units, and all the other artificialities which are invented by man,
not by nature.)
23-4 Resonance in nature
Although we have discussed the electrical case in detail, we could also bring
up case after case in many fields, and show exactly how the resonance equation
is the same. 'Phere are many circumstances in nature in which something is
“oscillating” and in which the resonance phenomenon occurs. We said that in
an earlier chapter; let us now demonstrate it. If we walk around our study,
pulling books of the shelves and simply looking through them to ñnd an example
of a curve that corresponds to Eig. 23-2 and comes from the same equation,
what do we fnd? Just to demonstrate the wide range obtained by taking the
smallest possible sample, it takes only five or six books to produce quite a series
of phenomena which show resonances.
The first two are from mechanics, the frst on a large scale: the atmosphere oŸ
the whole earth. If the atmosphere, which we suppose surrounds the earth evenly
--- Trang 418 ---
Cycles per day
4a Ị a
1aha2‡ tahoo, 1ohạo
Fig. 23-6. Response of the atmosphere to external excitation. a Is
the required response if the atmospheric Sa-tide is of gravitational origin;
peak amplification ¡is 100 : 1. b ¡s derived from observed magnification
and phase of M;-tide. [Munk and MacDonald, “Rotation of the Earth,”
Cambridge University Press (1960)]
on all sides, is pulled to one side by the moon or, rather, squashed prolate into a
double tide, and if we could then let it go, it would go sloshing up and down; it is
an oscillator. Thịs oscillator is đrZuen by the moon, which is eÑfectively revolving
about the earth; any one component of the force, say in the z-direction, has a
cosine component, and so the response of the earth's atmosphere to the tidal pull
of the moon is that of an oscillator. The expected response of the atmosphere is
shown in Eig. 23-6, curve Ö (curve ø is another theoretical curve under discussion
in the book from which this is taken out of context). Now one might think
that we only have one point on this resonance curve, since we only have the one
frequency, corresponding to the rotation of the earth under the moon, which
Occurs at a period of 12.42 hours——12 hours for the earth (the tide is a double
bump), plus a little more because the moon is goïing around. But rom the size oŸ
the atmospheric tides, and from the phase, the amount of delay, we can get both
p and Ø. From those we can get œạ and +, and thus draw the entire curvel 'This is
an example of very poor seience. From two numbers we obtain two numbers, and
from those two numbers we draw a beautiful curve, which of course goes through
the very point that determined the curvel Ït is of no use nÏess te can Tncasure
sơmethzng else, and in the case of geophysics that is often very difcult. But in
this particular case there is another thing which we can show theoretically must
--- Trang 419 ---
have the same timing as the natural frequency œọ: that is, if someone disturbed
the atmosphere, it would oscillate with the frequency œg. Now there +0øs such a
sharp disturbance in 1883; the Krakatoa volcano exploded and half the island
blew of, and ¡it made such a terrific explosion in the atmosphere that the period
of oscillation of the atmosphere could be measured. It came out to 105 hours.
'The œọ obtained from Eig. 23-6 comes out 10 hours and 20 minutes, so there we
have at least one check on the reality of our understanding of the atmospheric
tides.
Next we go to the small scale of mechanical oscillation. This time we take a
sodium chỉloride crystal, which has sodium ions and chlorine iIons next to each
other, as we described in an early chapter. 'Phese ions are electrically charged,
alternately plus and minus. Now there is an interesting oscillation possible.
uppose that we could drive all the plus charges to the right and all the negative
charges to the left, and let go; they would then oscillate back and forth, the
sodium lattice against the chlorine lattice. How can we ever drive such a thing?
'That is easy, for If we apply an electric fñeld on the crystal, it will push the plus
charge one way and the minus charge the other wayl So, by having an external
electric field we can perhaps get the crystal to oscillate. The frequency of the
electric fñeld needed is so high, however, that it corresponds to ?nƒfrared radiatiom
So we try to fnd a resonance curve by measuring the absorption of infrared light
by sodium chloride. Such a curve is shown in Fig. 23-7. "The abscissa is not
frequenecy, but is given in terms of wavelength, but that is just a technical matter,
Of course, since for a wave there is a definite relation bebween Írequency and
wavelength; so it is really a frequency scale, and a certain Írequency corresponds
to the resonant frequency.
But what about the width? What determines the width? There are many
cases in which the width that is seen on the curve is not really the natural width +
that one would have theoretically. There are two reasons why there can be a
wider curve than the theoretical curve. If the objects do not all have the same
frequency, as might happen ïf the crystal were strained in certain reglons, so that
in those regions the oscillation frequency were slightly diferent than in other
regions, then what we have is many resonance curves on top oŸ each other; so we
apparently get a wider curve. The other kind of width is simply this: perhaps
we cannot measure the frequency precisely enough-——if we open the slit of the
spectrometer fairly wide, so although we thought we had only one Írequency,