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The dispersion relation (6.3) can also be obtained by using
separation of variables.
Example 6. .1Consider 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
Figure 6.1:
The fundamental mode and the harmonics in the
-direction.
|
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
.
Next: Sound Waves
Up: Waves
Previous: Waves
Prof. Alan Hood
2000-11-06