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.
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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![5° Consider the equation
def
Lu Au(x) + k²u(x) = 0, x≤R³
called Helmoltz or reduced wave equation.
(a) Show that the radial solutions u = u(r), r = |x|, satisfying the outgoing Sommerfeld
condition
are of the form
u₁ + iku = 0 (17/12)
0(
O(=
o(r, k) = c
e-ikr
Select the constant c such that
r
as r → ∞,
‚c € C.
"
(b) For f smooth and compactly supported in R³ define the potential
U(x) = c [ f (y)² dy.
e-ik|x-y|
|x − y|
-
R3
LU(x) = -f(x)
[Answer: c = ¹].
4T
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 o defined on R³, A6 = ²6 +²aro. Then the problem can
be reduced to solving a simple ODE for the quantity ro.
Tr
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8a9802b-ded9-4a7b-93c8-71218002814a%2Fb7916e77-9304-45f0-90ec-4e737fefb724%2F3k1uwis_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5° Consider the equation
def
Lu Au(x) + k²u(x) = 0, x≤R³
called Helmoltz or reduced wave equation.
(a) Show that the radial solutions u = u(r), r = |x|, satisfying the outgoing Sommerfeld
condition
are of the form
u₁ + iku = 0 (17/12)
0(
O(=
o(r, k) = c
e-ikr
Select the constant c such that
r
as r → ∞,
‚c € C.
"
(b) For f smooth and compactly supported in R³ define the potential
U(x) = c [ f (y)² dy.
e-ik|x-y|
|x − y|
-
R3
LU(x) = -f(x)
[Answer: c = ¹].
4T
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 o defined on R³, A6 = ²6 +²aro. Then the problem can
be reduced to solving a simple ODE for the quantity ro.
Tr
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.
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