13.8 Higher-Dimensional Problems
INTRODUCTION
In the preceding sections we solved one-dimensional forms of the heat and wave equations. In this section we extend the method of separation of variables to certain problems involving the two-dimensional heat and wave equations.
Heat and Wave Equations in Two Dimensions
Suppose the rectangular region in FIGURE 13.8.1(a) is a thin plate in which the temperature u is a function of time t and position (x, y). Then, under suitable conditions, u(x, y, t) can be shown to satisfy the two-dimensional heat equation
FIGURE 13.8.1 (a) Rectangular plate (b) Rectangular membrane
.(1)
On the other hand, ...
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