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Unformatted text preview: 2,
2t + 1 ∂t
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
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
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.
- Winter '14