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    2D wave equation on a wedge

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    7-4

    Consider the displacement of ,u(r,,t) , a "pie-shaped" membrane of radius a and angle /3 that satisfies:

    utt = c22u

    Assume that >0. Determine the natural frequencies of oscillation if the boundary conditions are:

    Problem a.

    a) u(r, 0, t) = 0, u(r, /3, t) = 0, ur(a, , t) = 0

    problem b.
    b) u(r, 0, t) = 0, u(r, /3, t) = 0, u(a, , t) = 0

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    https://brainmass.com/math/fourier-analysis/wave-equation-wedge-625952

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    The wave equation on circular domain is given (in polar coordinates):
    (1.1)
    With the boundary conditions
    (1.2)
    And
    (1.3)
    We want to find the modes (frequencies) of the oscillation.
    If we set:
    (1.4)
    The boundary conditions become
    (1.5)
    (1.6)
    (1.7)

    Plugging (1.4) into (1.1) we get:

    (1.8)
    The left hand side depends on t while the right hand side is a function of the spatial variables , hence both sides must be equal the same constant:
    (1.9)
    The negative sign is right now meaningless, since is an arbitrary constant, but it will help us later to recognize the radial equation.
    Furthermore:

    (1.10)

    Now ...

    Solution Summary

    The solution shows how to get the modes of the waves on a circular wedge with different radial boundary conditions.

    $2.49

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