To understand the relation between a picture of ripples and a description in terms of tones, we consider the generic behavior of a string stretched between to endpoints. It will lead us to one of the most useful techniques in mathematics, a way of using superposition to characterize waves.
The resting position of this string is just a straight line. Any wave
we put on it can be described by the displacement sideways of each bit
of string from that bit's particular resting place. The string is
pinned at the ends, so the displacement always takes the value zero
there. Consider a completely general wave, such as that shown in
fig. 7.5.
Suppose somehow we get the string into that shape and let it go, and ask how it moves as time goes by. This looks like a very complicated problem. There is, however, a way to use the superposition principle to make it easily solvable.
Suppose, for a moment, that the wave had the much simpler form shown
in fig's 7.5a, 7.5b,
7.5c, etc.
These very symmetrical waves are familiar,
because they are the waves that are commonly seen on vibrating
strings. They are special because each of them maintains its relative
proportions and high degree of symmetry as it vibrates. Further, each
of them vibrates with a particular frequency. On a string, this
frequency determines the note heard from the vibration.
Constant-frequency vibration is a very simple motion to describe,
because it is the same regular back-and-forth motion as the
oscillation of a mass on a spring. Thus, we know very well how each
of these waves evolves in time. These special waves are called the
fundamental modes
of vibrations on the string.
There is one additional special property of the fundamental modes of
vibration on a string that we can note. Because the material of the
string is assumed to be the same everywhere, the shape of the
fundamental modes is the same on one hump as on all of the others.
Thus, each of the fundamental modes is a completely regular sequence
of the same type of hump, repeated over and over. Because the endpoints
are pinned, this requires that, for a fundamental mode to be able to
exist on a string, the length of each of the humps, called the
wavelength
of the mode, must fit exactly within the length of the string some
(integer) number of times. Thus, even though there are many more
regular waves than those in the sequence indicated in
fig. 7.5, they are not fundamental modes of this
string because their wavelength does not exactly divide the length of
the string, so they are not compatible with the requirement that the
endpoints of the string be pinned.
Now return to the example in fig. 7.5.
Suppose we took a wave that looked
like that in fig. 7.5a,
and added to it one that looked like that in fig. 7.5b.
Suppose then we subtracted a smaller wave of the form in
fig. 7.5c, and kept
on adding and subtracting waves from each of the fundamental modes. The
resulting wave is shown, after each step in this addition, in
fig. 7.5.
It is clear that, by choosing the right "amount" of each of the fundamental modes, we have produced the wave in fig. 7.5.
The wave in fig. 7.5 is not special in this
respect. It is a
mathematical theorem, called Fourier's Theorem,
that any wave
that can be placed on a string can be described as a superposition of
some combination of the fundamental modes. The fundamental modes
themselves are special because, while each fundamental mode can be
described as itself (a tautology), it can not be described as any
combination of the other fundamental modes, hence the name
fundamental.
This theorem, together with the experimental fact that small waves on strings (like that shown in this example) obey the superposition principle, allows us to easily describe the evolution of the wave of fig. 7.5 over time. We first simply determine how much of each of the fundamental modes is required to produce the wave at the starting instant. Then, to describe the position of the string at any later time, we find the positions of the fundamental modes at that later time (as noted, this is straightforward, because each evolves with a constant shape and with a definite frequency), and we add the resulting shapes. That will be the shape of the real string at the later time.
This result is actually familiar to most people. When a piano string is struck, it assumes a triangular form, because the hammer hits it at a narrow point. Such a shape is not one of the fundamental modes, but rather a combination of them. As a result, the struck piano string produces a sound which is a combination of pure tones, which gives it its distinctive character. Compare this to the sound from an electronic device like a pager, which typically produces a single tone. (Actually, piano notes are even more complicated, because of the non-identical tuning of the strings and the presence of the piano body, but that is just an elaboration on the same effect.) The piano with the thumbtack sounds different because the tack strikes the string more ``sharply'' than the rounded and yielding felt of the hammer, and in producing a ``sharper triangle'', introduces more of the higher-pitch tones which compose that shape.
Fourier's theorem has a far more general range of application than just to waves on strings. Any wave can be decomposed as a sum of some given collection of other waves. The theorem is particularly useful when you know how to describe a complete set of fundamental modes, as we did for the string, and when the system obeys the superposition principle. Then, if you know the amount of each fundamental mode in the initial wave, you know that each such mode evolves independently, and by treating each mode separately and adding the result, you can describe the wave at any later time.