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Therefore, to the “forced” solution we can add any “free” solution, and we still |
have a solution. 'Phe free solution is called a frøns¿en‡ solution. |
'When we have no force acting, and suddenly turn one on, we do not imme- |
diately get the steady solution that we solved for with the sine wave solution, |
but for a while there is a transient which sooner or later dies out, IÝ we wait long |
enough. 'Phe “forced” solution does not die out, since it keeps on being driven |
by the force. Ultimately, for long periods of time, the solution 1s unique, but |
initially the motions are diferent for different circumstances, depending on how |
the system was started. |
25-2 Superposition of solutions |
Now we come to another interesting proposition. Suppose that we have a |
certain particular driving force #4 (let us say an oscillatory one with a cerbain |
œ = œ„, but our conclusions will be true for any functional form oŸ F2) and we |
have solved for the forced motion (with or without the transients; it makes no |
difference). NÑow suppose some other force is acting, let us say #}ÿ, and we solve |
* Solutions which cannot be expressed as linear combinations of each other are called |
independent. |
--- Trang 439 --- |
the same problem, but for this different force. hen suppose someone comes |
along and says, “[ have a new problem for you to solve; I have the force „ + Fỳ.” |
Can we do it? Of course we can do it, because the solution is the sum of the |
two solutions #„ and zø; for the forces taken separately——a most remarkable |
circumstance indeed. IÝ we use (25.3), we see that |
L(#¿ + #ụ) = L(z4) + L(œy) = †„) + tịÚ). (25.8) |
This is an example of what is called the prznciple oƒ superposition for linear |
systems, and it is very important. It means the following: if we have a complicated |
force which can be broken up in any convenient manner into a sum of separate |
pieces, each of which is in some way simple, in the sense that for each special |
piece into which we have divided the force we can solve the equation, then the |
answer is available for the +0hole force, because we may simply add the pieces of |
the solufion back together, in the same manner as the total ƒorce is compounded |
out of pieces (Eig. 25-1). |
Fạ + Fp |
Xa -F Xpb |
Fig. 25-1. An example of the principle of superposition for linear |
Systems. |
--- Trang 440 --- |
Let us give another example of the prineiple of superposition. In Chapter 12 |
we said that it was one of the great facts of the laws of electricity that if we have |
a certain distribution of charges ga and calculate the electric fñield #Z„ arising from |
these charges at a certain place , and ïf, on the other hand, we have another |
set of charges q; and we calculate the feld #; due to these at the corresponding |
place, then if both charge distributions are present at the same time, the field |
at P is the sưm of E„ due to one set plus #; due to the other. In other words, |
1ƒ we know the fñeld due to a certain charge, then the feld due to many charges |
is merely the vector sum of the ñelds of these charges taken individually. 'This |
is exactly analogous to the above proposition that if we know the result of two |
given forces taken at one time, then if the force is considered as a sum of them, |
the response is a sum of the corresponding individual responses. |
` dụ ` F |
Fig. 25-2. The principle of superposition in electrostatics. |
'The reason why this is true in electricity is that the great laws of electricity, |
Maxwell's equations, which determine the electric field, turn out to be diferential |
cquations which are ineør, ¡.e., which have the property (25.3). What corresponds |
to the force is the chørøe generating the electric fñeld, and the equation which |
determines the electric ñeld in terms of the charge is linear. |
As another interesting example of this proposition, let us ask how it is possible |
to “tune in” to a particular radio station at the same time as all the radio stations |
are broadcasting. 'Phe radio station transmits, fundamentally, an oscillating |
electric fñield of very high frequency which acts on our radio antenna. Ït is true |
that the amplitude of the oscillation of the field ¡is changed, modulated, to carry |
the signal of the voice, but that is very slow, and we are not going to worry about |
it. When one hears “'Phis station is broadcasting at a frequency of 780 kilocycles,” |
this indicates that 780,000 oscillations per second is the frequency of the electric |
field of the station antenna, and this drives the electrons up and down at that |
frequency in our antenna. Now at the same time we may have another radio |
station in the same town radiating at a diferent frequency, say 550 kilocycles per |
second; then the electrons in our antenna are also being driven by that frequency. |
Now the question is, how is it that we can separate the signals coming into the |
--- Trang 441 --- |
one radio at 780 kilocycles from those coming in at 550 kilocycles? We certainly |
do not hear both stations at the same tỉme. |
By the principle of superposition, the response of the electric circuit in the |
radio, the first part of which is a linear circuit, to the forces that are acting due |
to the electric field q + Fỳ, is z„ + øạ. It therefore looks as though we will never |
disentangle them. In fact, the very proposition of superposition seems to insist |
that we cannot øø0oø?d having both of them In our system. But remember, for a |
resonanf circuit, the response curve, the amount oŸ z per unit ?, as a function of |
the requency, looks like Fig. 25-3. If it were a very high @ circuit, the response |
would show a very sharp maximum. Now suppose that the two stations are |
comparable in strength, that is, the two ƒorces are of the same magnitude. 'Phe |
response that we get 1s the sum of ø„ and ø;. But, in Eig. 25-3, #ø„ is tremendous, |
while #; is small. 5o, in spite of the fact that the two signals are equal in strength, |
when they go through the sharp resonant circuit of the radio tuned for œ¿, the |
frequency of the transmission of one station, then the response to this station |
1s mụuch greater than to the other. Therefore the complete response, with both |
signals acting, is almost all made up oŸ œ„, and we have selected the station we |
œp_ (Úc ta ø |
Fig. 25-3. A sharply tuned resonance curve. |
Now what about the tuning? How do we tune it? We change œọ by changing |
the Ù or the Œ of the circuit, because the frequency of the circuit has to do with |
the combination of Ù and Œ. In particular, most radios are built so that one can |
change the capacitance. When we retune the radio, we can make a new setting |
of the dial, so that the natural frequenecy of the circuit is shifted, say, to œe. In |
those circumstances we hear neither one station nor the other; we get silence, |
provided there is no other station at frequency œ„. lf we keep on changing the |
--- Trang 442 --- |
capacitance until the resonance curve 1s at œụ, then of course we hear the other |
station. 'That is how radio tuning works; it is again the prineciple of superposition, |
combined with a resonant response.* |
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