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Unformatted text preview: nπy/b)[A cos(ωmn t) + B sin(ωmn t)].
The general solution to to the wave equation with Dirichlet boundary conditions
is a superposition of these product solutions,
∞ u(x, y, t) = sin(mπx/a) sin(nπy/b)[Amn cos(ωmn t) + Bmn sin(ωmn t)].
m,n=1 The constants Amn and Bmn are determined from the initial conditions.
The initial value problem considered in this section could represent the motion of a vibrating membrane. Just like in the case of the vibrating string, the
motion of the membrane is a superposition of inﬁnitely many modes, the mode
corresponding to the pair (m, n) oscillating with frequency ωmn /2π . The lowest
frequency of vibration or fundamental frequency is
a 2 + π
b 2 = 1
a 2 + 1
b 2 . Exercises:
5.5.1. Solve the following initial value problem for a vibrating square membrane:
Find u(x, y, t), 0 ≤ x ≤ π, 0 ≤ y ≤ π , such that
u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0,
(x, y, 0) = 0.
∂t u(x, y, 0) = 3 sin x sin y + 7 sin...
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This document was uploaded on 01/12/2014.
- Winter '14