standing waves on arbitrary membrane
My (probably misguided) intuition says the following :
'Take a closed loop of wire and bend it into any arbitrary shape so that it lies flat on a table. stretch a membrane over it (i.e. a soap membrane say). Then, I should be able to vibrate it at just the right frequency to generate (at least) a fundamental mode of vibration.'
In other words I think my intuition is telling me that there are solutions to the 2D wave equation with a zero-displacement condition on an arbitrary closed boundary.
Is my intuition right or wrong? If wrong, why?
Also, my intuition is telling me that for a complicated irregular boundary that there would be fewer modes of vibration or that they would be spaced more widely apart in terms of frequency.
If the intuition is incorrect, then is this something to do with the fact that a real world membrane is elastic and can stretch in ways that dont satisfy the wave equation?
Re: standing waves on arbitrary membrane
I would think you'd generate several modes for different spatial scales that, in a real membrane, would quickly attenuate the whole membrane to the steady state as tey compete with each other.
A circle only has one spatial scale (the radius or diameter if you like), the arbitrary shape could have several.
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