Pde - Rectangular Drum

1. Sep 25, 2009

Void123

1. The problem statement, all variables and given/known data

Draw a picture to illustrate the two-dimensional drumhead in the x-y plane. Label the coordinates of the sides of the drumhead. Use this picture to illustrate the "modes" of vibration.

2. Relevant equations

$$\frac{\partial^{2}Z}{\partial x^{2}} + \frac{\partial^{2}Z}{\partial y^{2}} - \frac{1}{v^{2}}\frac{\partial^{2}Z}{\partial t^{2}} = 0$$ ,

where Z(x, y, t) is the amplitude of vibration.

Boundary conditions:

Z(0, y, t) = Z(x, 0, t) = Z(a, y, t) = Z(x, b, t) = 0

The sides of the drumhead are parallel to the x and y axes with the corners being located at (0, 0), (0, b), (a, b), and (a, 0).

3. The attempt at a solution

It sounds like a very simple illustration, without too many details, though I want to be sure I am on the right track (since this subject is vague to me). And I think by 'drumhead' they mean a vibrating membrane. Hence, would I just draw a simple rectangle in the xy-plane with nodal lines to indicate the zero amplitude? What would that look like, without too many complicated details? This is all I have so far.

2. Sep 26, 2009

No input?