6. Let d(t) be the Dirac-delta function. The position u(x, t) of a vibrating string which satisfies U = 6(t – 1) + 8(t – 2), 0 < r < L,t >t, Utt %3D u(0, t) = u(L, t) = 0, t >0, u(r, 0) = u,(r, 0) = 0, 0 < r < L. (a) Find the series solution. (b) Does the solution decay in time? Explain the physical interpretation of your result.
6. Let d(t) be the Dirac-delta function. The position u(x, t) of a vibrating string which satisfies U = 6(t – 1) + 8(t – 2), 0 < r < L,t >t, Utt %3D u(0, t) = u(L, t) = 0, t >0, u(r, 0) = u,(r, 0) = 0, 0 < r < L. (a) Find the series solution. (b) Does the solution decay in time? Explain the physical interpretation of your result.
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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![6. Let \(\delta(t)\) be the Dirac-delta function. The position \(u(x, t)\) of a vibrating string which satisfies
\[
u_{tt} - u_{xx} = \delta(t - 1) + \delta(t - 2), \quad 0 < x < L, t \ge t_0,
\]
\[
u(0, t) = u(L, t) = 0, \quad t > 0,
\]
\[
u(x, 0) = u_t(x, 0) = 0, \quad 0 < x < L.
\]
(a) Find the series solution.
(b) Does the solution decay in time? Explain the physical interpretation of your result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc986032-179a-454d-ad20-2c76f1bd2607%2F555404dd-8d92-4d44-bd18-9e9e55d7611c%2Fix0muib_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. Let \(\delta(t)\) be the Dirac-delta function. The position \(u(x, t)\) of a vibrating string which satisfies
\[
u_{tt} - u_{xx} = \delta(t - 1) + \delta(t - 2), \quad 0 < x < L, t \ge t_0,
\]
\[
u(0, t) = u(L, t) = 0, \quad t > 0,
\]
\[
u(x, 0) = u_t(x, 0) = 0, \quad 0 < x < L.
\]
(a) Find the series solution.
(b) Does the solution decay in time? Explain the physical interpretation of your result.
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