# Calculating the decay rates for modes of an ideal circular membrane (ie. drum head) using wave equations?

I am attempting to solve for the theoretical decay rates of the various (m,n) modes of an ideal circular membrane, if that membrane is excited momentarily by an impulse or deformation.

I would ideally like the decays of the (m,n) modes in dB/s.

The membrane should be considered fixed entirely around its perimeter. The excitation impulse/deformation should be at its center or x*radius from its center.

Someone on another site said of this problem:

If the air damps it linearly enough, you can probably solve it analytically. Use plate theory to generate a PDE, then work out all the eigenmodes. The decay rate will be determined by the real components of the eigenvalues, and can be converted into dBs-1 using a few logs.

The wave equation for modes of an ideal circular membrane is given by: https://pasteboard.co/HqpEmjv.png

The full wave equations are described/explained further in these documents:

I can use the Bessel zeros to calculate the frequencies of the various (m,n) modes and have done so already. However, I am unsure how to get the decay rates for these modes as he describes.

Does the method he suggests make sense? If so, can anyone elaborate further on how I would go about doing this? Or is there a better way?

Ideally I'd like an equation I can put in (m,n) for, plus perhaps an arbitrary damping coefficient, and get the decay of that mode in dB/s. If the decay rate of any mode might vary depending on the point of excitation, some way to specify for excitation position might be useful also.

Thanks for any help.

## 1 Answer

To take the losses into account, you need to add (at least) one term in the wave equation. The simplest one would be a first derivative of the function (deformation of the membrane) multiplied by a coefficient. However, the value of this coefficient will depend on the material of your membrane, and the properties of the surrounding medium, it could even be frequency dependent. I'm not sure if you can easily find a value to predict the decay rate of your modes...

Take a look at https://ccrma.stanford.edu/~jos/pasp/Lossy_1D_Wave_Equation.html#sec:lwe for more info on the losses on a string.

The new term to add is analogous to the term added in the harmonic oscillator eq. that is proportional to the speed of the oscillator https://en.wikipedia.org/wiki/Harmonic_oscillator#Damped_harmonic_oscillator.

If the decay rate of any mode might vary depending on the point of excitation, some way to specify for excitation position might be useful also

The modes are independent of any excitation (that is part of their definition), so their decay rate will not depend on the point of excitation. However, the modes you will be able to excite will indeed depend on the excitation point

• Thanks David! That's very helpful. I found this site which summarizes a damped wave equation for a circular membrane (drum head): math.ust.hk/~machas/drum However, their damped equation does not seem to have appropriate frequency or mode dependent damping and as a result it sounds terrible: math.ust.hk/~machas/drum/Image12.gif You can hear samples at the end of that site. This is what happens when you don't damp the modes correctly, which is what I'm trying to fix/avoid. Any suggestions to improve this model and damp it more properly?
– mike
Commented Jun 19, 2018 at 1:10
• "However, their damped equation does not seem to have appropriate frequency or mode dependent damping". It seems indeed that since their coeff "a" is constant, all modes will have the same damping rate, leading in part to the poor sound quality. What you can do as a first approximation is take the modes of the undamped case, and assign a different damping rate to each one of them (as an exponential decay factor). However, synthesizing sounds like that is no trivial matter and I wouldn't expect to obtain something "realist" without a lot of work. Commented Jun 19, 2018 at 14:32