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    Damped 1D wave equation on a clamped string

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    The displacement u(x, t) from the vertical at distance x from its left endpoint, at time t, of a string of length L, fastened at both endpoints, satisfies the PDE
    utt + aut = c2uxx, where a is a positive constant, with initial conditions
    u(x, 0) = f(x), ut(x, 0) = g(x).
    1. Solve the equation by separation of variables. The solution could depend on where a is, compared to the eigenvalues of the corresponding Sturm Liouville problem. That is, a could be less than all the eigenvalues, or between two of the eigenvalues.

    2. Let λ be the first eigenvalue of the corresponding Sturm-Liouville problem. There is a qualitative difference in the behavior of the string if a > 2 λ and the behavior for 0 < a < λ. Describe it.

    3. What can be said about limt→∞ u(x, t)?

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    https://brainmass.com/math/fourier-analysis/damped-wave-equation-clamped-string-626003

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    The damped wave equation is
    (1.1)
    Where is the damping coefficient.
    The string is of length L and is clamped on both ends, hence the boundary conditions are:
    (1.2)
    And the initial conditions are:
    (1.3)
    We start with defining:
    (1.4)
    Which makes the boundary conditions:
    (1.5)
    And
    (1.6)
    Plugging (1.4) into the wave equation we get:

    Finally:
    (1.7)
    The left hand side is a function of t exclusively, and the right hand side is f function of x exclusively. Since this equation must hold for any the two sides must be equal a constant:
    (1.8)
    The x-dependent equation is
    (1.9)
    And the t-dependent equation is
    (1.10)
    To solve equation (1.9) we distinguish between three cases.
    Case 1:
    The equation is and its solution is
    (1.11)
    Applying boundary conditions:

    We obtain the trivial solution
    . Case 2:
    The equation is and its solution is
    (1.12)
    Applying boundary conditions:

    We obtain the trivial solution
    Case 2:
    The equation is and its solution is
    (1.13)
    Applying boundary conditions:

    The eigenfunctions are
    (1.14)

    And the eigenvalues are
    ...

    Solution Summary

    The 13-pages long solution shows, in detail, how to solve the damped wave equation on a clamped string.

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