From the Fourier theorem, we see that, even for general waves, we can describe them by telling how much of each fundamental mode is required to build them. However, it is often not practically possible to measure such things as how much of a wave is due to a particular fundamental mode. For example, the ear is very good at discerning pitch in sound, but much worse at identifying direction. Thus most of what we notice about sounds concerns only their frequency. Whether the different wavelengths that create that frequency are oriented one way or another (their directional content) is largely missed. In such cases. we measure only how much of a wave has a given frequency. (This is also what the eye does, crudely, when it recognizes how bright the contribution from a certain color is to a light of mixed color.)
For systems like the string, each fundamental mode has a unique
frequency and wavelength, so the two questions are the same. But for
the box, because each fundamental mode is described by a wavelength in
three independent directions, different fundamental modes can have the
same overall frequency. If the box is square, an obvious example is
that given in fig's 7.5a and
7.5b,
again drawn for the simpler case of the membrane. The two modes differ only in their orientation, so their overall frequency must be the same. However, they are different fundamental modes, because one cannot be built as a sum containing the other.
Thus, when we ask how much of a wave has a given overall frequency, we are asking how much of the wave is described by the collection of all the fundamental modes that have that frequency. To find out, we need to know not only how much of the wave comes from each fundamental mode, but also how many such modes there are at each frequency.
The answer to this question involves a little calculation, so has been relegated to an appendix. For the case of waves in a box, which will concern us most, the number of modes in the box in each frequency range is proportional to the volume of the box, divided by the fourth power of the frequency around which the range is taken.
For classical light waves in a box, this has the following consequences. Suppose we were to ask how many fundamental modes there are at two colors. Like Young, we do not really need to understand anything about the Electromagnetic Field, or even about light, except that it is a wave, and what frequency corresponds to what color. For light, it is often convenient to speak in terms of an ``effective'' wavelength, which is formed as one would expect on dimensional grounds, by defining
For example, we could choose light of wavelength 4500 Å, which is
just slightly bluer than the eye can see, and at 9000 Å, which is
just at the limit of the red range we can see. The result is that,
because the blue wavelength is one half the red wavelength, there are
, or 16 times as many fundamental modes at the blue color as at the
red. Further, if we were to compare the number of fundamental modes
in two boxes, one with twice the volume of the other, there are twice
as many fundamental modes in the larger box as in the smaller.
This can be visualized easily from a simple example. Suppose that somehow, we were to construct a classical light wave in such a way that it was built from exactly the same amount of each fundamental mode, in the two color ranges used above. What would such a wave look like? Because we only see how much of the light has a given frequency, we would see all of the fundamental modes at the blue color as the same blue light, and all of those at the red color as the same red light. The brightness of each of the fundamental modes is the same, but because there are sixteen times as many of them at the blue color, the blue contribution to the light would seem sixteen times as bright as the red contribution.
A graph of these values is given in fig. 7.5.
We leave Fourier's theorem and the counting of modes at this point, because we have enough to understand the problems with radiation emanating from a cavity, when we encounter them. The next topic to address is the very interesting subject of currents, which we will need more immediately when we try to understand the behaviors, even at the classical level, of the Electromagnetic Field. Our sample case will be a different aspect of the behavior of water.