1. Consider the wave equation that describes the vibrating string with fixed ends J²u მ2 = J²u მ2 ди u(x, 0) = f(x); (x, 0) = g(x) Ət u(0,t) = u(L, t) = 0. Use its Fourier series solution u(x,t) u(x, t)=sin( to show that -Σsin ("7") [a, cos(x) + b, sin (TC)] n=1 u(x,t) = R(x-ct) + S(x + ct), where R and S are some functions. Hints: i. sin a cos b = 1 [sin(a + b) + sin(a − b)]. ii. sin a sin b [cos(a - b) - cos(a + b)]. L
1. Consider the wave equation that describes the vibrating string with fixed ends J²u მ2 = J²u მ2 ди u(x, 0) = f(x); (x, 0) = g(x) Ət u(0,t) = u(L, t) = 0. Use its Fourier series solution u(x,t) u(x, t)=sin( to show that -Σsin ("7") [a, cos(x) + b, sin (TC)] n=1 u(x,t) = R(x-ct) + S(x + ct), where R and S are some functions. Hints: i. sin a cos b = 1 [sin(a + b) + sin(a − b)]. ii. sin a sin b [cos(a - b) - cos(a + b)]. L
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
Question
Please solve the following by hand and without the use of AI. Please be thorough and use detailed mathematical formulas to solve. Thank you.
![1. Consider the wave equation that describes the vibrating string with fixed ends
J²u
მ2
=
J²u
მ2
ди
u(x, 0) = f(x);
(x, 0) = g(x)
Ət
u(0,t) = u(L, t) = 0.
Use its Fourier series solution
u(x,t)
u(x, t)=sin(
to show that
-Σsin ("7") [a, cos(x) + b, sin (TC)]
n=1
u(x,t) = R(x-ct) + S(x + ct),
where R and S are some functions.
Hints:
i. sin a cos b =
1
[sin(a + b) + sin(a − b)].
ii. sin a sin b
[cos(a - b) - cos(a + b)].
L](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf4d6898-2dc3-4334-a7a9-00be2cdf120e%2F15d73d79-4d53-415e-adaa-d3feb978ae74%2F7ma851_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider the wave equation that describes the vibrating string with fixed ends
J²u
მ2
=
J²u
მ2
ди
u(x, 0) = f(x);
(x, 0) = g(x)
Ət
u(0,t) = u(L, t) = 0.
Use its Fourier series solution
u(x,t)
u(x, t)=sin(
to show that
-Σsin ("7") [a, cos(x) + b, sin (TC)]
n=1
u(x,t) = R(x-ct) + S(x + ct),
where R and S are some functions.
Hints:
i. sin a cos b =
1
[sin(a + b) + sin(a − b)].
ii. sin a sin b
[cos(a - b) - cos(a + b)].
L
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