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 Summary
The solution shows how to use separation of variables and Fourier expansion to solve the 1D wave equation.
Solution Preview
The equation is that of a string clamped on both ends
(1.1)
We guess a solution in the form
(1.2)
Plugging it in we obtain:
(1.3)
Dividing by it becomes
(1.4)
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
(1.5)
The spatial equation is
(1.6)
With boundary conditions
(1.7)
Case 1:
The equation becomes and its solution is
(1.8)
Applying boundary conditions
We get the trivial solution
Case 2 :
The equation becomes ...
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