= Problem 9.12 In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (kr - wt + 8) and g (r, t) : B cos (k · r — wt + 8). Show that (ƒg) = (1/2)Re(ƒĝ*), where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and w, but they need not have the same amplitude or phase.] For example, and (S) 1 (u) = ½¼Re (€„Ẽ · Ēª + (EDE 4 1 E* -B. B* мо 1 -Re (Ex B"). 2μο
= Problem 9.12 In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (kr - wt + 8) and g (r, t) : B cos (k · r — wt + 8). Show that (ƒg) = (1/2)Re(ƒĝ*), where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and w, but they need not have the same amplitude or phase.] For example, and (S) 1 (u) = ½¼Re (€„Ẽ · Ēª + (EDE 4 1 E* -B. B* мо 1 -Re (Ex B"). 2μο
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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![=
Problem 9.12 In the complex notation there is a clever device for finding the
time average of a product. Suppose f (r, t) = A cos (kr - wt + 8) and g (r, t) :
B cos (k · r — wt + 8). Show that (ƒg) = (1/2)Re(ƒĝ*), where the star denotes
complex conjugation. [Note that this only works if the two waves have the same k
and w, but they need not have the same amplitude or phase.] For example,
and (S)
1
(u) = ½¼Re (€₁Ẽ · Ēª +
(EDE
4
1
E* -B. B*
мо
1
2μο
B*).
Re (Ex B*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0685f8f4-4062-4d5b-8aa1-43fd6f7d1cfb%2F5f909932-f368-4b1b-b9c0-d0d9a7bba6ca%2Fizrp1lm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:=
Problem 9.12 In the complex notation there is a clever device for finding the
time average of a product. Suppose f (r, t) = A cos (kr - wt + 8) and g (r, t) :
B cos (k · r — wt + 8). Show that (ƒg) = (1/2)Re(ƒĝ*), where the star denotes
complex conjugation. [Note that this only works if the two waves have the same k
and w, but they need not have the same amplitude or phase.] For example,
and (S)
1
(u) = ½¼Re (€₁Ẽ · Ēª +
(EDE
4
1
E* -B. B*
мо
1
2μο
B*).
Re (Ex B*
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