The general solution to to the wave equation with

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Unformatted text preview: 2, 2t + 1 ∂t ∂x2 ∂y u(x, 0, t) = u(x, π, t) = u(0, y, t) = u(π, y, t) = 0, u(x, y, 0) = 2 sin x sin y + 3 sin 2x sin y. 5.3.4. Find the general solution to the heat equation ∂u ∂2u ∂2u = +2 ∂t ∂x2 ∂y subject to the boundary conditions u(x, 0, t) = u(0, y, t) = u(π, y, t) = 0, u(x, π, t) = sin x − 2 sin 2x + 3 sin 3x. 128 5.4 Two derivations of the wave equation We have already seen how the one-dimensional wave equation describes the motion of a vibrating string. In this section we will show that the motion of a vibrating membrane is described by the two-dimensional wave equation, while sound waves are described by a three-dimensional wave equation. In fact, we will see that sound waves arise from “linearization” of the nonlinear equations of fluid mechanics. The vibrating membrane. Suppose that a homogeneous membrane is fastened down along the boundary of a region D in the (x, y )-plane. Suppose, moreover, that a point on the membrane can move only in the vertical direction, and let u(x, y, t) denote the height of the point with coordinates (x, y ) at time t. If ρ denotes the density of the membrane (assumed to be constant), then by Newton’s second law, the force F acting on a small rectangular piece...
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This document was uploaded on 01/12/2014.

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