4. Consider the vibrations of a homogeneous circular membrane modeled as follows: c?v²u_in D = {(n,8),n E [0, R], 6 E [-7, 1]} at2 au U(R, 6,t) = - (R,0, t) an U(R,0,0) = 0 ne (R,0,0) = 0 at Obtain an expression that determine the natural frequencies of vibrations and solve the problem
4. Consider the vibrations of a homogeneous circular membrane modeled as follows: c?v²u_in D = {(n,8),n E [0, R], 6 E [-7, 1]} at2 au U(R, 6,t) = - (R,0, t) an U(R,0,0) = 0 ne (R,0,0) = 0 at Obtain an expression that determine the natural frequencies of vibrations and solve the 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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![4. Consider the vibrations of a homogeneous circular membrane modeled as follows:
\[
\begin{align*}
\frac{\partial^2 U}{\partial t^2} &= C^2 \nabla^2 U \quad \text{in } D = \{ (r, \theta), r \in [0, R], \theta \in [-\pi, \pi] \} \\
U(R, \theta, t) &= -\frac{\partial U}{\partial n} (R, \theta, t) \\
U(R, \theta, 0) &= 0 \\
\frac{\partial U}{\partial t} (R, \theta, 0) &= 0
\end{align*}
\]
Obtain an expression that determines the natural frequencies of vibrations and solve the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40d18442-55cc-442d-893b-ae5bbbc2b444%2F12bc0a10-7bf1-49af-834e-516e2006286f%2Fzk0fxj_processed.png&w=3840&q=75)
Transcribed Image Text:4. Consider the vibrations of a homogeneous circular membrane modeled as follows:
\[
\begin{align*}
\frac{\partial^2 U}{\partial t^2} &= C^2 \nabla^2 U \quad \text{in } D = \{ (r, \theta), r \in [0, R], \theta \in [-\pi, \pi] \} \\
U(R, \theta, t) &= -\frac{\partial U}{\partial n} (R, \theta, t) \\
U(R, \theta, 0) &= 0 \\
\frac{\partial U}{\partial t} (R, \theta, 0) &= 0
\end{align*}
\]
Obtain an expression that determines the natural frequencies of vibrations and solve the problem.
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