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