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(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?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 97E
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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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