Theory


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

wave equation in two dimensions:

[Graphics:chladweb.txtgr1.gif]

 

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:

          [Graphics:chladweb.txtgr3.gif]

        [Graphics:chladweb.txtgr4.gif]

        [Graphics:chladweb2.txtgr4.gif]

                                   where                [Graphics:chladweb.txtgr6.gif]

Thus   [Graphics:chladweb.txtgr7.gif]

These are equations for a simple harmonic oscillator.  After each is solved, the total solution in Cartesian coordinates is:        [Graphics:chladweb.txtgr8.gif]

Note:   [Graphics:chladweb.txtgr9.gif]

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

[Graphics:chladweb.txtgr2.gif][Graphics:chladweb.txtgr11.gif]

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,     [Graphics:chladweb.txtgr12.gif]

[Graphics:chladweb.txtgr2.gif][Graphics:chladweb.txtgr13.gif]

[Graphics:chladweb.txtgr2.gif][Graphics:chladweb.txtgr14.gif]

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.

[Graphics:squarechlad.txtgr7.gif]

From the relationship   [Graphics:chladweb.txtgr7.gif]  , we see that

the modal frequencies will be         [Graphics:chladweb.txtgr16.gif]

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    [Graphics:chladweb.txtgr18.gif],

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

[Graphics:chladweb.txtgr2.gif][Graphics:chladweb.txtgr19.gif][Graphics:chladweb.txtgr17.gif]

Theory for Circular Plates:

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

 [Graphics:chladweb2.txtgr5.gif]

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

Here is   [Graphics:chladweb2.txtgr3.gif]:

[Graphics:chladweb.txtgr22.gif]

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

So, [Graphics:chladweb.txtgr23.gif]

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; = [Graphics:chladweb.txtgr24.gif], [Graphics:chladweb.txtgr25.gif], 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:

[Graphics:bessel3.txtgr4.gif][Graphics:bessel3.txtgr7.gif]

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.