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1D wave equation

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Solve the following string equation problem

utt = 1/4* uxx, 0<x<1,t>0,
u(0, t) = 0, u(1,t)=0, t>0

1/2 * x, 0< x<1/2
u(x,0) =
1−x, 1/2<x<1.

ut (x,0) = 0

Solve using separation of variables and D'Alembert. Show solutions in detail.
Graph both solution u(x, t) for t = 0, 1, 2, 4.

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Solution Preview

The equation is that of a string clamped on both ends
We guess a solution in the form
Plugging it in we obtain:
Dividing by it becomes
Since the left side is a function of t only while the right side is a function of x only, for this equation to be true at all both sides must be equal a constant
The spatial equation is
With boundary conditions

Case 1:
The equation becomes and its solution is
Applying boundary conditions

We get the trivial solution

Case 2 :
The equation becomes ...

Solution Summary

The solution shows how to use separation of variables and Fourier expansion to solve the 1D wave equation.