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    Two-Dimensional Wave Equation

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    1. Find the solution to the two-dimensional wave equation
    [see the attachment for the full equation]

    2. Solve the two-dimensional wave equation for a quarter-circular membrane
    [see the attachment for the full equation]
    The boundary condition is such that u=0 on the entire boundary.

    3. Consider Laplace's equation
    [see the attachment for the full equation]
    a. Give a brief physical interpretation of this equation.
    b. Suppose that u(x,y,t)=f(x)g(y)h(t)
    What ordinary differential equations are satisfied by f, g, and h?

    © BrainMass Inc. brainmass.com June 4, 2020, 3:58 am ad1c9bdddf
    https://brainmass.com/math/numerical-analysis/two-dimensional-wave-equation-545036

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    1.

    The two dimensional wave equation on the square membrane is:
    (1.1)
    with boundary conditions:
    (1.2)
    And initial conditions:
    (1.3)
    We start with assuming we can write the solution as a product of three completely independent functions:
    (1.4)
    Therefore the partial derivative become full derivatives, for example:
    (1.5)
    Plugging this into (1.1) we get:

    (1.6)
    On the left hand side the expression depends only on t, while the right hand side is a function of x and y.
    Since this equation must hold for any (x, y, t), the only way this can be satisfied is if both sides equal the same (arbitrary) constant:
    (1.7)
    The time dependent equation is:
    (1.8)
    While the spatial equation is:
    (1.9)
    Here we see that the left hand side is a function of x while the right hand side depends on y. Again, this can occur for any (x,y) if and only if both sides equal the same constant:
    (1.10)
    The x dependent equation is:
    (1.11)
    While the y dependent equation is:
    (1.12)
    Where we define the constant:
    (1.13)
    From the boundary conditions we get conditions for the spatial equations namely:
    (1.14)
    And:
    (1.15)
    Note that equation (1.11) and equation (1.12) are identical (up to a constant) and have the same boundary conditions, hence they will have identical solutions.
    We begin with noting the three cases.

    Case 1:
    The equation becomes:
    (1.16)
    Its solution is:
    (1.17)
    Applying conditions (1.14):

    We obtain the trivial solution
    (1.18)
    Case 2:
    The equation becomes:
    (1.19)
    Its solution is:
    (1.20)

    Applying conditions (1.14):

    We obtain the trivial solution
    (1.21)

    Case 3:
    The equation becomes a simple harmonic:
    (1.22)
    Its solution is:
    (1.23)
    Applying conditions (1.14):

    We obtain the non-trivial solution
    (1.24)
    And the eigenvalue is:
    (1.25)
    By the same process we see that the solution for the y-dependent equation (1.12) is:

    (1.26)
    And its eigenvalues are:
    (1.27)
    Thus, from equation (1.13):
    (1.28)
    Plugging this in the temporal equation (1.8) we obtain:

    (1.29)

    The general solution of equation (1.1) is a linear combination of all possible solutions:
    ...

    Solution Summary

    The expert finds the solution to the two-dimensional wave equations. Laplace's equations are considered.

    $2.19

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