Standing waves on arbitrary membrane

  1. 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.

    Thanks

    Andy

    Additional:

    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?
     
    Last edited: Nov 29, 2012
  2. jcsd
  3. Pythagorean

    Pythagorean 4,622
    Gold Member

    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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