# Calculating decay rates for modes of a circular membrane

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I am trying to solve for the theoretical relative 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.

Imagine a simple isolated drum head being struck by a stick. The membrane should be considered fixed with even tension 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:

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

http://www.math.ubc.ca/~nagata/sci1/drum.pdf
https://courses.physics.illinois.ed...P406POM_Lecture_Notes/P406POM_Lect4_Part2.pdf
http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_29.pdf

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 constant 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.

Thanks for any help!

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I was able to find a website summarizing a damped wave equation for a circular membrane here:

where u is the amplitude of vibration of the membrane, r and
are polar coordinates of membrane, a is the damping factor, c is the speed of a wave on the membrane.

http://www.math.ust.hk/~machas/drum/

However, the membrane simulation sounds terrible (audio clips at the end of that site) because they aren't employing the proper frequency/mode dependent per-partial damping that occurs in nature.

As far as I can tell, this is just damping all the modes equally over time, which is useless.

Am I understanding this correctly, and if so, is there any obvious way to improve this and get a more representative damped wave equation?

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