[37] Consider the vibrations of a membrane of the shape of a quarter disk: x² + y² 0, y > 0. Assume that the boundary displacement is fixed at zero for all times. (a) Write down the initial-boundary value problem in polar coordinates. (b) Find all product solutions u = T(t)R(r)Q(0) of the PDE and the homogeneous boundary conditions you gave in (a). (c) Find the solution formula for the initial-boundary value problem you gave in (a).
[37] Consider the vibrations of a membrane of the shape of a quarter disk: x² + y² 0, y > 0. Assume that the boundary displacement is fixed at zero for all times. (a) Write down the initial-boundary value problem in polar coordinates. (b) Find all product solutions u = T(t)R(r)Q(0) of the PDE and the homogeneous boundary conditions you gave in (a). (c) Find the solution formula for the initial-boundary value problem you gave in (a).
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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![[37] Consider the vibrations of a membrane of the shape of a quarter disk: x² + y² <a², x>
0, y > 0. Assume that the boundary displacement is fixed at zero for all times.
(a) Write down the initial-boundary value problem in polar coordinates.
(b) Find all product solutions u
=
T(t)R(r)Q(0) of the PDE and the homogeneous
boundary conditions you gave in (a).
(c) Find the solution formula for the initial-boundary value problem you gave in (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2F9ba6877a-f741-43d3-be4d-3c35c90f8a3a%2Fwbdk156_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[37] Consider the vibrations of a membrane of the shape of a quarter disk: x² + y² <a², x>
0, y > 0. Assume that the boundary displacement is fixed at zero for all times.
(a) Write down the initial-boundary value problem in polar coordinates.
(b) Find all product solutions u
=
T(t)R(r)Q(0) of the PDE and the homogeneous
boundary conditions you gave in (a).
(c) Find the solution formula for the initial-boundary value problem you gave in (a).
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