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    Wave Equations : Comparison Between Circular and Square Vibrating Membranes

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    Consider a circular membrane of radius a and a square membrane
    Assume the two membranes
    (i) have the same area.
    ....
    (ii) obey the same wave equation...
    (iii) Have the same boundary conditions phi = 0 at their boundaries.
    A) TABULATE (i) the 3 lowest frequencies for each of the two membranes
    (ii) all the concomitant normal modes.
    B) For each of the normal modes of the circular membrane DRAW a picture of the nodes, i.e. the locus of points where = 0. LABEL each of the normal mode pictures by
    (a) its frequency and
    (b) its formula for the amplitude.
    C) DO the same for the other membrane. (Caution: Watch out for degeneracies!)
    D) CAN ONE HEAR THE SHAPE of a DRUM?
    Roots: is the jth root of the Bessel Functions
    n=0 n=1 n=2 n=3 n=4
    j = 1
    2.405
    3.832
    5.136
    6.380
    7.586
    j = 2
    5.520
    7.016
    8.417
    9.760
    11.064
    j = 3
    8.654
    10.173
    11.620
    13.017
    14.373
    j = 4
    11.792
    13.324
    14.796
    16.223
    16.223

    © BrainMass Inc. brainmass.com June 3, 2020, 5:16 pm ad1c9bdddf
    https://brainmass.com/math/trigonometry/wave-equations-comparison-between-circular-and-square-vibrating-membranes-20152

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    Solution Summary

    Bessel functions are employed to answer questions about a wave equation. The solution is detailed and well-presented.

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