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    Wave equation on a rectangular domain

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    Consider the following wave equation:

    utt = c2 uxx, 0<x<a, 0<y<b

    Subject to the following boundary conditions:

    u(0,y, t) = 0, u(a, y, t) = 0, 0<y<b, t>0
    u(x,0, t) = 0, u(x, b, t) = 0, 0<x<a, t>0

    Find an expression for the solution if the initial conditions are:

    a) u(x, y, 0) = xy(a-x)(b-y), ut(x,y) = 0

    b) u(x, y) = 0, ut(x,y) = sin(5x/a) sin(7y/b)

    c) u(x, y, 0) = xy(a-x)(b-y), ut(x,y) = sin(5x/a) sin(7y/b)

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    https://brainmass.com/math/fourier-analysis/wave-equation-rectangular-domain-626377

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    The wave equation on the rectangular membrane is:
    (1.1)
    The boundary values are Dirichlet's boundary conditions:
    (1.2)
    And
    (1.3)
    And general initial conditions:
    (1.4)
    We start wit writing the wave function as a product of univariate functions:
    (1.5)
    This makes the boundary conditions a sngle-function conditions:

    (1.6)
    And:
    (1.7)

    We them plug (1.5) into (1.1), and the partial derivatives turn into full derivatives:

    (1.8)
    The left hand side is a function of t while the right hand side is a function of , and since it must hold true for any combination of both sides must be equal to a constant:
    (1.9)
    The right hand side of (1.8) can be deconstructed further:
    (1.10)
    Now the left hand side is a function of x while the right hand side is a function of y, so both sides must be equal the same constant:
    (1.11)
    But we could also write:
    (1.12)
    Therefore, we can define a different constant :
    (1.13)
    Note that
    (1.14)
    So we can say we have three independent second order ordinary differential equations.
    One is t-dependent:
    (1.15)
    The second is x-dependent
    (1.16)
    With boundary conditions
    And the third equation is y-dependent
    (1.17)
    With initial boundary conditions
    Note that ...

    Solution Summary

    The 16-page solution shows how to find the solution for a wave equation on a rectangular domain with different initial conditions

    $2.19

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