Consider the wave equation: utt = uxx, −∞ < x < ∞, t > 0 with ut(x, 0) = 0 and u(x, 0) = ( 0, x < 0 and x > π) u(x, 0) = cos2 x, 0 ≤ x ≤ π a)Find a solution to this problem using the D’Alembert’s formula. Then, plot the solution at t = 0, π /4 , π /2 , π b)Find a solution to this problem using the Fourier transform and show that your solution is the same as part (a).
Consider the wave equation: utt = uxx, −∞ < x < ∞, t > 0 with ut(x, 0) = 0 and u(x, 0) = ( 0, x < 0 and x > π) u(x, 0) = cos2 x, 0 ≤ x ≤ π a)Find a solution to this problem using the D’Alembert’s formula. Then, plot the solution at t = 0, π /4 , π /2 , π b)Find a solution to this problem using the Fourier transform and show that your solution is the same as part (a).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the wave equation:
utt = uxx, −∞ < x < ∞, t > 0
with ut(x, 0) = 0 and
u(x, 0) = ( 0, x < 0 and x > π)
u(x, 0) = cos2 x, 0 ≤ x ≤ π
a)Find a solution to this problem using the D’Alembert’s formula.
Then, plot the solution at t = 0, π /4 , π /2 , π
b)Find a solution to this problem using the Fourier transform and show that your solution is the same as part (a).
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