(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?
(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
Problem 1RQ
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part C
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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