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Unformatted text preview: le functions T (x, y ) and ρ(x, y ) respectively, then the motion of the string is governed by the equation ∂2u 1 = ∇ · (T (x, y )∇u) . ∂t2 ρ(x, y ) 133 5.5 Initial value problems for wave equations The most natural initial value problem for the wave equation is the following: Let D be a bounded region in the (x, y )-plane which is bounded by a piecewise smooth curve ∂D, and let h1 , h2 : D → R be continuous functions. Find a function u(x, y, t) such that 1. u satisfies the wave equation ∂2u = c2 ∂t2 ∂2u ∂2u +2 ∂x2 ∂y . (5.21) 2. u satisfies the “Dirichlet boundary condition” u(x, y, t) = 0, for (x, y ) ∈ ∂D. 3. u satisfies the initial condition ∂u (x, y, 0) = h2 (x, y ). ∂t u(x, y, 0) = h1 (x, y ), Solution of this initial value problem via separation of variables is very similar to the solution of the initial value problem for the heat equation which was presented in Section 5.3. As before, let us suppose that D = {(x, y ) ∈ R 2 : 0 ≤ x ≤ a, 0 ≤ y ≤ b}, so that the Dirichlet boundary condition becomes u(0, y, t) = u(a, y,...
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This document was uploaded on 01/12/2014.

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