3 exercises 571 show that among all rectangular

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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 ω11 c = 2π 2π π a 2 + π b 2 = 1 2 T ρ 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 ∂2u ∂2u ∂2u = + 2, ∂t2 ∂x2 ∂y u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0, 135 ∂u (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.

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