5° Consider the equation Lu are of the form def - called Helmoltz or reduced wave equation. (a) Show that the radial solutions u = u(r), r = |x|, satisfying the outgoing Sommerfeld condition Au(x) +k²u(x) = 0, x=R³ Ur + iku =0 (1/2) e-ikr o(r, k) = C -,CE C. r as r∞, (b) For f smooth and compactly supported in R³ define the potential e-ik|x-yl |x − y| U(x) = c [ f(y)² R3 Select the constant c such that LU(x) = -f(x) -dy. [Answer: c= 11. Hint: This is Problem 3.21 on pg. 155 of Salsa's book. For part a), use the fact that for a spherically symmetric function & defined on R³, A6 = 8² +0,. Then the problem can be reduced to solving a simple ODE for the quantity ro. For part b), revisit the proof of the theorem from Class notes 7 in which we solved Poisson's equation in R". Remark 0.0.1. In part a), you are deriving a fundamental solution for the Helmholtz equation, up to a constant factor c. In part b), you are finding the correct constant c.
5° Consider the equation Lu are of the form def - called Helmoltz or reduced wave equation. (a) Show that the radial solutions u = u(r), r = |x|, satisfying the outgoing Sommerfeld condition Au(x) +k²u(x) = 0, x=R³ Ur + iku =0 (1/2) e-ikr o(r, k) = C -,CE C. r as r∞, (b) For f smooth and compactly supported in R³ define the potential e-ik|x-yl |x − y| U(x) = c [ f(y)² R3 Select the constant c such that LU(x) = -f(x) -dy. [Answer: c= 11. Hint: This is Problem 3.21 on pg. 155 of Salsa's book. For part a), use the fact that for a spherically symmetric function & defined on R³, A6 = 8² +0,. Then the problem can be reduced to solving a simple ODE for the quantity ro. For part b), revisit the proof of the theorem from Class notes 7 in which we solved Poisson's equation in R". Remark 0.0.1. In part a), you are deriving a fundamental solution for the Helmholtz equation, up to a constant factor c. In part b), you are finding the correct constant c.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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