(a) Alembert's formula for the solution of the wave equation on R is given by 1 U(x, t) = (x + ct) + f(x – ct)) + , 8(9)ds. 20 Provide a brief discussion of the meaning of the two terms in the right-hand side of the above formula. (b) Show by direct computation that D'Alembert's formula is a solution to the problem U-cUrx 0, U(x,0) = f(x), U,(x,0) = g(x). xE R, t> 0, %3D %3! %3D (c) Find the solution to the problem Un - cUx = 0, U(x,0) = 0, U:(x,0) = cos x. x E R, t > 0, (d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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part C

(a) D'Alembert's formula for the solution of the wave equation on R is given by
1
rx+ct
U(x,t) -(* + ct) + f(x - ct) + 8()ds.
%3D
Provide a brief discussion of the meaning of the two terms in the right-hand
side of the above formula.
(b) Show by direct computation that D'Alembert's formula is a solution to the
problem
Un - eux = 0,
U(x,0) = f(x),
U(x,0) = g(x).
xE R,
t> 0,
(c) Find the solution to the problem
Un - cU = 0,
U(x,0) = 0,
U;(x,0)
x E R,
t > 0,
= COS X.
(d) What is the main difference between the wave equation and the heat equation
in terms of the speed of propagation of information?
Transcribed Image Text:(a) D'Alembert's formula for the solution of the wave equation on R is given by 1 rx+ct U(x,t) -(* + ct) + f(x - ct) + 8()ds. %3D Provide a brief discussion of the meaning of the two terms in the right-hand side of the above formula. (b) Show by direct computation that D'Alembert's formula is a solution to the problem Un - eux = 0, U(x,0) = f(x), U(x,0) = g(x). xE R, t> 0, (c) Find the solution to the problem Un - cU = 0, U(x,0) = 0, U;(x,0) x E R, t > 0, = COS X. (d) What is the main difference between the wave equation and the heat equation in terms of the speed of propagation of information?
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