4. Consider the wave equation for a vibrating rectangular membrane 0 < y < H. 0 < x < L, J²u J²u J²u მt2 მე2 + მყ2 subject to the initial condition ди u(x, y, 0) = 0, and (x, y, 0) = f(x, y). Ət Solve the initial value problem if ди ди Ju ди (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x,H,t) = 0. მე მე მყ მყ Note: You may assume without derivation that product solutions u(x, y, t) = (x,y)h(t) satisfy dh = -Ac²h dt and the two-dimensional eigenvalue problem V2 + X = 0, and you may use results of the two-dimensional eigenvalue problem.
4. Consider the wave equation for a vibrating rectangular membrane 0 < y < H. 0 < x < L, J²u J²u J²u მt2 მე2 + მყ2 subject to the initial condition ди u(x, y, 0) = 0, and (x, y, 0) = f(x, y). Ət Solve the initial value problem if ди ди Ju ди (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x,H,t) = 0. მე მე მყ მყ Note: You may assume without derivation that product solutions u(x, y, t) = (x,y)h(t) satisfy dh = -Ac²h dt and the two-dimensional eigenvalue problem V2 + X = 0, and you may use results of the two-dimensional eigenvalue problem.
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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Please solve the following by hand and without the use of AI. Please be thorough and use detailed mathematical formulas to solve. Thank you.
Transcribed Image Text:4. Consider the wave equation for a vibrating rectangular membrane
0 < y < H.
0 < x < L,
J²u
J²u
J²u
მt2
მე2
+
მყ2
subject to the initial condition
ди
u(x, y, 0) = 0, and
(x, y, 0) = f(x, y).
Ət
Solve the initial value problem if
ди
ди
Ju
ди
(0, y, t) = 0,
(L, y, t) = 0,
(x, 0,t) = 0,
(x,H,t) = 0.
მე
მე
მყ
მყ
Note:
You may assume without derivation that product solutions
u(x, y, t) = (x,y)h(t) satisfy
dh
=
-Ac²h
dt
and the two-dimensional eigenvalue problem V2 + X = 0, and you may use
results of the two-dimensional eigenvalue problem.
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