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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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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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