Standing Waves in Finite Continuous Medium
We saw earlier, in Sections 4.3 and 5.2,
that a smallamplitude transverse wave on a uniform string, and a smallamplitude longitudinal
wave on a thin elastic rod, are both governed by the
wave equation, which (in one dimension) takes the general form

(6.1) 
where represents the wave disturbance, and is a constant, with the dimensions
of velocity, that is a property of the particular medium that supports the wave. Up to now, we have only considered media of finite length. That is, media that extend from (say)
to . Generally speaking, we have encountered two distinct types of physical
constraint that hold at the boundaries of such media. First, if a given boundary is
fixed then the wave disturbance is constrained to be zero there. For instance, if the left boundary is fixed then
. Second, if a given boundary is free then the spatial derivative
of the disturbance (which usually corresponds to some sort of force) is constrained to be
zero there. For instance, if the right boundary is free then
. It follows that a fixed boundary corresponds to a node; that is,
a point where the amplitude of the wave disturbance is always zero. On the other hand, a
free boundary corresponds to an antinode; that is, a point where the
amplitude of the wave disturbance is always locally maximal. Consequently, the
nodes and the antinodes of a wave, of definite wavelength, supported in a medium of finite length that has stationary boundaries (which can be either fixed or free) are constrained to be stationary. The only simple solution of the wave equation,
(6.1), that has stationary nodes and antinodes is a standing wave of the general form

(6.2) 
The associated nodes are located at the values of that satisfy

(6.3) 
which implies that they are stationary, and also evenly spaced a distance
apart, where
is the wavelength. Moreover, the antinodes
are situated halfway between the nodes. For example, suppose that both boundaries
of the medium
are fixed. It follows that the points and must each correspond to a node. This is
only possible if the length of the medium, , is a halfinteger number of wavelengths; that is,
, where is a positive integer. We conclude that, in this case, the possible
wavenumbers of standing wave solutions to the wave equation are quantized such that

(6.4) 
Moreover, the same is true if both boundaries are free. Finally, if one boundary is free, and the
other fixed, then the quantization of wavenumbers takes the slightly different form

(6.5) 
Those standing wave solutions that satisfy the appropriate quantization criterion are known as the
normal modes of the system. (See Chapter 3.) Moreover, substitution of Equation (6.2)
into the wave equation, (6.1), yields the standing wave dispersion relation

(6.6) 
The fact that normal mode wavenumbers are quantized immediately implies that the associated
oscillation frequencies are also quantized. Finally, because the
wave equation is linear, the most general solution that satisfies the
boundary conditions is a linear superposition of all of the normal modes. Such a
solution has the appropriate node or antinode at each of the boundaries, but
does not necessarily have any stationary nodes or antinodes in the interior of the medium.