The Vibrations of a Circular Membrane

The solution of the differential equation for the vibrations of a circular membrane, 

We see that the solution may be taken as the product of a function R depending on r only and a function depending on cp only i.e. in polar co-ordinates the variables are separable. The differential equation can then be split up into two equations with the single independent variables r, j respectively, by means of a separation parameter, which we shall call mh 

It is essential that the wave function shall be one-valued but this is not the case unless m is an integer, for otherwise an increase of 2tt in the value of f would give a different value for the wave function hence the separation parameter m must be an integer (m 0). 

The differential equation in r is the well-known equation defining the Bessel function J VXr) of the m-th order with argument VAr. Here, however, we must take the boundary conditions into account. As a particular case, we assume that the membrane is fixed at the circumference, so that for all points on the boundary R p) = 0, where p is the radius of the membrane. Now the Bessel function of any order has an infinite number of zeros (in fact if the value of the argument 

In the first case the wave function has no zeros other than those for r = p and (where m 0) for r = 0 in the second case it has one other zero, in the third case two other zeros hence in general for the proper value 

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As our first example of a special function, we consider a two-dimensional problem the vibration of a circular membrane such as a drumhead. The amplitude of vibration is determined by solution of the Helmholtz equation in two dimensions, most appropriately chosen as the polar coordinates r, 9. Using the scale factors 2r = 1 and Q = r for the two-dimensional Laplacian, the Helmholtz equation can be written as... 

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