The next example of the same sort concerns waves in a box. Here, we will suppose that the wavelike medium is three dimensional, rather than one dimensional, as it was for the string. An example could be our previous description of sound waves in a room, where the medium is the air, and the waves are pressure and velocity waves. Another, as mentioned, is the classical electromagnetic description of light in a hot cavity. Here the medium is the electromagnetic field, and the wave is the smooth displacement of that field from its equilibrium value.
The first task in using Fourier's theorem to describe general waves is identifying the fundamental modes. In both cases, all the waves we consider are assumed to have zero displacement at the walls of the box, just as the waves on the string had no displacement at the endpoints of the string, where it is held in place. For sound, this is the assumption that the stiff walls of the box stop motion of the air perpendicular to their surfaces. In the other case, since we are trying to identify the light with waves in the electromagnetic field, the opacity of the walls means that the light, and hence the waves, are contained in the interior of the box, and must stop propagating at its walls.
The fundamental modes for both of these waves look very much like
those for the string, but in higher dimensions. A cross-section
through a graph of the displacement would show each of the
fundamentals as a smooth wave of exactly the form shown in
fig. 7.5.
Each of the fundamental modes has zero displacement at the walls of
the box, just as those of the string did. This also requires that the
wavelengths of the modes exactly divide the lengths of the sides of
the box. Now, however, there can be a different wavelength for the
oscillation in each direction, so the set of the fundamental modes is
more complicated than it was for the string. An example of this for
the more easily visualized case of waves on a two-dimensional membrane
is shown in fig. 7.5.
Each of the fundamentals for the box can be pictured as a product of one of the fundamentals for the string along each direction. For now we will only worry about waves that can be described by single numbers, like pressure waves (avoiding the more complicated cases that require arrows, which we will encounter in a later section). This way, we can not only add, but also multiply their values at each point of the medium.
For waves on the string, each of the fundamental modes was described by a single number, either its wavelength, or the number of times its wavelength fit into the length of the string. For waves in the box, each mode is described by three numbers, the three wavelengths in each of the three directions, or equivalently, the three factors that tell how many times each of the wavelengths divides the length of the respective side of the box.
Fourier's theorem also holds for this example, as does the superposition principle, so any wave in the box can be described as a sum of these fundamental modes and no others.