This preview shows page 1. Sign up to view the full content.
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,...
View
Full
Document
This document was uploaded on 01/12/2014.
 Winter '14
 Equations

Click to edit the document details