Let u(x, t) be the solution of the one-dimensional wave equation with speed c> 0 Show that Utt = c²uxx u(0, t) = 0 = u(L, t), 0 < x 0 t>0 u(x, 0) = f(x) u₁(x, 0) = 0. u(x, t) = ½ [F(x − ct) + F(x + ct)] where F(x) is the odd periodic extension of f(x).
Let u(x, t) be the solution of the one-dimensional wave equation with speed c> 0 Show that Utt = c²uxx u(0, t) = 0 = u(L, t), 0 < x 0 t>0 u(x, 0) = f(x) u₁(x, 0) = 0. u(x, t) = ½ [F(x − ct) + F(x + ct)] where F(x) is the odd periodic extension of f(x).
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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![Let u(x, t) be the solution of the one-dimensional wave equation with speed c > 0
Utt = c²uxx
u(0, t) = 0 = u(L, t),
u(x, 0) = f(x)
0 < x < L,
t> 0
t> 0
u₁(x, 0) = 0.
Show that
u(x,t)
=
[F(x-
[F(x − ct) + F(x + ct)]
where F(x) is the odd periodic extension of f(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6cd132a-964f-4a31-af3d-1b1c2cf60729%2F7ab79c86-8f50-4cca-a87f-31948d7f9f16%2F6wyg02p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let u(x, t) be the solution of the one-dimensional wave equation with speed c > 0
Utt = c²uxx
u(0, t) = 0 = u(L, t),
u(x, 0) = f(x)
0 < x < L,
t> 0
t> 0
u₁(x, 0) = 0.
Show that
u(x,t)
=
[F(x-
[F(x − ct) + F(x + ct)]
where F(x) is the odd periodic extension of f(x).
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