# Damped 1D wave equation on a clamped string

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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#### Solution Preview

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.