The dispersion relation (6.3) can also be obtained by using separation of variables.

__Example 6. .1__Consider the vibrations of a square membrane inside the region
and . The wave equation is

describing the displacement, , of the membrane. Suppose the boundary conditions are taken as on the sides of the square , and ,. Try a solution of the form

so that the wave equation may be expressed in the separable form as

Since the left hand side of the equation is only a function of time and the right hand side depends only on spatial variations, they must both be equal to a constant. This separation process can be continued to show that both terms on the right hand side are constants. Hence,

where the separation constant is chsoen as

in order to satisfy the boundary conditions in . Here is an integer. Similarly,

where

and is an integer so that the boundary conditions in are satisfied. Combining these results we obtain the equation for as

Thus, we have

where the frequency satisfies the dispersion relation

Thus, the general solution is obtained by adding together all the possible solution to give

The constants and are determined by initial conditions on and . Each value of and corresponds to a different possible mode of oscillations. The fundamental mode is given by and with

and has

It corresponds to the whole membrane oscillating up and down. The mode and has a spatial form given by

and has

These two normal modes are illustrated in Figure 6.1

Note that we could have obtained exactly the same result by assuming
the solution could be expressed, in complex form, as

This is, of course, a separable solution since

Expressing this in terms of real and imaginary parts gives

and so one solution is

where

is a constant and and in order to satisfy the boundary conditions in and .