Theory

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

wave equation in two dimensions:

Solution for Rectangular Plates:

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.

Theory for Circular Plates:

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.

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Source: William C. Elmore and Mark A. Heald. Physics of Waves. New York: Dover Publications.