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To conclude this discussion, let us describe qualitatively what happens if we
proceed further in analyzing a linear problem with a given force, when the force is
quite complicated. Out of the many possible procedures, there are two especially
useful general ways that we can solve the problem. Ône is this: suppose that we
can solve it for special known forces, such as sine waves of different frequencies.
W©e know it is child”s play to solve it for sine waves. So we have the so-called
“child°s play” cases. Now the question is whether our very complicated force
can be represented as the sum oŸ two or more “child”s play” forces. In Fig. 25-1
we already had a fairly complicated curve, and of course we can make i% more
complicated still if we add in more sine waves. So 1t is certainly possible to
obtain very complicated curves. And, in fact, the reverse is also true: practically
every curve can be obtained by adding together ?mfinite muwmbers oŸ sỉine waves of
điferent wavelengths (or frequencies) for each one of which we know the answer.
W© just have to know how mụch of each sine wave to put in to make the given #',
and then our answer, zø, is the corresponding sum of the # sine waves, each
multipled by its effective ratio of z to #'Ô This method of solution is called
the method of Fourier transƒforms or Fourier œnalusis. YNe are not going to
actually carry out such an analysis just now; we only wish to describe the idea
involved.
Another way in which our complicated problem can be solved is the following
very interesting one. Suppose that, by some tremendous mental efort, it were
possible to solve our problem for a special force, namely an Zmpuise. 'The Íorce is
quickly turned on and then of; ït ¡is all over. Actually we need only solve for an
impulse of some unit strength, any other strength can be gotten by multiplication
by an appropriate factor. We know that the response ø for an impulse is a
damped oscillation. NÑow what can we say about some other Íorce, for instance a
force like that of Fig. 25-4?
Such a force can be likened to a succession of blows with a hammer. First there
1s no force, and all of a sudden there is a steady force—impulse, impulse, impulse,
* In modern superheterodyne receivers the actual operation is more complex. The amplifers
are all tuned to a fixed frequency (called IEF frequency) and an oscillator of variable tunable
frequency is combined with the input signal in a n„onl¿near circuit to produce a new frequency
(the diference of signal and oscillator frequency) equal to the IE frequency, which is then
amplifed. 'Phis will be discussed in Chapter 50.
--- Trang 443 ---
Fig. 25-4. A complicated force may be treated as a succession of
sharp Iimpulses.
impulse,... and then it stops. In other words, we imagine the continuous force
to be a series of impulses, very close together. Now, we know the result for an
Impulse, so the result for a whole series of impulses will be a whole series of
damped oscillations: it will be the curve for the first impulse, and then (slightly
later) we add to that the curve for the second impulse, and the curve for the
third impulse, and so on. 'Phus we can represent, mathematically, the complete
solution for arbitrary functions if we know the answer for an impulse. We get
the answer for any other force simply by integrating. This method is called the
Greens ƒunction rmnethod. A Green?s function is a response to an impulse, and
the method of analyzing any force by putting together the response of impulses
1s called the Green's function method.
The physical prineiples involved in both of these schemes are so simple,
involving just the linear equation, that they can be readily understood, but the
mathemaftical problems that are involved, the complicated integrations and so on,
are a little too advanced for us to attack right now. You will most likely return
to this some day when you have had more practice in mathematics. But the ?dea
1s very simple indeed.
Finally, we make some remarks on why Ï?near systems are so important. The
answer is simple: because we can solve theml So most of the tỉme we solve linear
problems. Second (and most important), it turns out that the ƒundamental laus
0ƒ phụsics are oflen linear. The Maxwell equations for the laws of electricity are
linear, for example. "The great laws of quantum mechanics turn out, so far as
we know, to be linear equations. 7 hø‡ is why we spend so much time on linear
cequations: because if we understand linear equations, we are ready, in principle,
to understand a lot of things.
We mention another situation where linear equations are found. When
displacements are small, many functions can be øpprozzmaœtcd linearly. For
--- Trang 444 ---
example, if we have a simple pendulum, the correct equation for its motion is
d20/dt? = —(g/L) sin 0. (25.9)
This equation can be solved by elliptic funections, but the easiest way to solve 1W is
numerically, as was shown in Chapter 9 on Newton”s Laws of Motion. A nonlinear
equation cannot be solved, ordinarily, any other way 0u# numerically. Now for
small Ø, sinØ is practically equal to Ø, and we have a linear equation. It turns
out that there are many circumstances where small efects are linear: for the
example here the swing of a pendulum through small arcs. As another example,
1ƒ we pull a little bit on a spring, the force is proportional to the extension. lÝ we
pull hard, we break the spring, and the force is a completely diferent function of
the distancel Linear equations are important. In fact they are so important that
perhaps fifty percent of the time we are solving linear equations in physics and
1n engineering.
25-3 Oscillations ỉn linear systems
Let us now review the things we have been talking about in the past few
chapters. lt is very easy for the physics of oscillators to become obscured by
the mathematics. The physics is actually very simple, and if we may forget the
mathematics for a moment we shall see that we can understand almost everything
that happens in an oscillating system. First, ifƒ we have only the spring and the
weight, it is easy to understand why the system oscillates—it is a consequence
of inertia. We pull the mass down and the force pulls it back up; as 1È passes
zoro, which is the place it likes to be, it cannot Just suddenly stop; because of its
qmomentum it keeps on goïing and swings to the other side, and back and forth.
So, if there were no fÍriction, we would surely expect an oscillatory motion, and
indeed we get one. But if there is even a little bit of friction, then on the return
cycle, the swing will not be quite as high as it was the first time.
Now what happens, cycle by cycle? 'Phat depends on the kind and amount
of friction. Suppose that we could concoct a kind of friction force that always
remains in the same proportion to the other forces, of inertia and in the spring,
as the amplitude of oscillation varies. In other words, for smaller oscillations
the friction should be weaker than for big oscillations. Ordinary friction does
not have this property, so a special kind of friction must be carefully invented
for the very purpose of creating a friction that is directly proportional to the
--- Trang 445 ---
velocity——so that for big oscillations iỀ is stronger and for small oscillations it