Vibration of a Circular Membrane Equation

[Moss Pauly]

Hello,
I am currently in the process of writing an Extended essay on the the "Effect of Tension on the fundamental Vibration of a circular Membrane" I have already completed the practical side of this by making 5 diaphragms with a 3 micron thick mylar membrane and have written up the methods and the results. I am currently at the stage where i have to do a comparison of experimental data to theoretical values. As i am only a Grade 12 maths student (Australia) my Knowledge of P.D is limited and the only equations that i can find seem to be quite complex P.D equations I am happy to put time into trying to understand the partial differential equation but i need help. My post here is to basically ask of the best way of going about this. Is there any other equation that i could use to plug in variables with tension being one. or is anyone able to go through how to operate the P.D equation with me?

Thanks,
Moss

 

[Moss Pauly]

Anyone?

 

[Nyxie]

You're advanced at your age!
You know the boundary conditions, right? So a circular membrane (or 'drum') has radially symmetric initial conditions, which means that they only depend on radius r.

The equation you need to use here is the 'two-dimensional wave equation' in polar form - i.e. you need to use the Laplacian in there written in polar form: http://en.wikipedia.org/wiki/Laplace_operator
(Look for the *polar* form in this article. Don't worry about where it came from right away; just focus on how to solve the wave equation in general. After that it's just changing coordinate systems; here we use polar coordinates.)

Can you solve the one-dimensional wave equation? If not you should ask your teacher for help. You need to know Bessel functions; they're weird at first but pretty simple to use. What book(s) are you using?

The tension part fits right into the wave equation: http://en.wikipedia.org/wiki/Wave_equation

Here is a decent source on how to solve the 2-D wave eq.: http://personal.rhul.ac.uk/UHAP/027/PH2130/PH2130_files/membrane.pdf
In the link above scroll down to the polar form of the wave eq. and read from there, as long as it's helping. If not, hopefully your teacher can show you or someone here who has LaTeX might be able to outline the steps.

I think it might be a bit ambitious to learn on your own in a short time, but then I don't know how much you know about it.

Any more specific details about your experiment would be great, like how you're measuring your data?