# Theory

To understand the modal patterns of the
circular and rectangular plates, we must first
investigate the solutions to

wave equation in two dimensions:

By assuming a product solution u(x,y,t) = X(x)Y(y)T(t), we separate variables and
obtain three distinct equations:

###

###

where

Thus* *

These are equations for a simple harmonic oscillator. After each is solved, the
total solution in Cartesian coordinates is: * *

Note:* *

We can also write the real part of the equation as:

This equation essentially describes two wavefronts.
One travelling in the x direction and one travelling in the y direction. For
rectangular plate with length "a" and width "b" and the edges fixed,
the amplitude must go to zero at the boundary.

So,* *

There will be (n-1) nodes running in the y-direction and (m-1) running in the
x-direction. Here is a Mathematica representation of the n=4, m=4 state.

From the relationship , we see that

the modal frequencies will be * *

Notice that the modal frequencies are not integral
multiples of each other, as is the case with a vibrating string.

If we graph on a log-log scale the modal frequencies w versus* **,*

we should get a straight line of slope 1/2.

For the circular plate, the wave equation in polar coordinates solves out to be:

For large values of r, these Bessel functions look sinusoidal.

Here is** :**

For a fixed plate with radius "a", the function goes to zero at r = a.

So,

A zero of the Bessel Function must occur at the boundary. Zeros occuring before
the mth zero form (m-1) concentric circular nodes.

* *Notice that for values of* n*theta; = , , *etc. there
will be a diametric mode through the center of the plate.

With the help of Mathematica, we can see a representation of two different modes:

In the first case, n=1,m=2. In the second case, n=2, m=3.

#### Source: William C. Elmore and Mark A. Heald. __Physics
of Waves__. New York: Dover Publications.