Let u(x, t) be the solution of the one-dimensional wave equation with speed c > 0Utt = c²uxxu(0, t) = 0 = u(L, t),u(x, 0) = f(x)0 < x < L,t> 0t> 0u₁(x, 0) = 0.Show thatu(x,t)=[F(x-[F(x − ct) + F(x + ct)]where F(x) is the odd periodic extension of f(x).

Given wave equation is the following :we solve it using Separation of Variables method,Let, By…

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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 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).