5.11 Comments and References 353
5.11 Comments and References
We have seen just a few problems in two or three dimensions, but they are
sufficient to illustrate the complications that may arise. A serious drawback
to the solution by separation of variables is that double and triple series tend
to converge slowly, if at all. Thus, if a numerical solution to a two- or three-
dimensional problem is needed, it may be advisable to sidestep the analytical
solution by using an approximate numerical technique from the beginning.
One advantage of using special coordinate systems is that some problems
that are two-dimensional in Cartesian coordinates may be one-dimensional
in another system. This is the case, for instance, when distance from a point
(rin polar orρin spherical coordinates) is the only significant space variable.
Of course, nonrectangular systems may arise naturally from the geometry of a
problem.
As Sections 5.3 and 5.4 point out, solving the two-dimensional heat or wave
equation in a regionRof the plane depends on being able to solve the eigen-
value problem∇^2 φ=−λ^2 φinRwithφ=0 on the boundary. The solution
of this problem in a region bounded by coordinate curves (that is, in a gen-
eralized rectangle) is known for many coordinate systems. We have discussed
the most common cases; others can be found inMethods of Theoretical Physics
by Morse and Feshbach. Information about the special functions involved is
available from theHandbook of Mathematical Functionsby Abramowitz and
Stegun and also fromSpecial Functions of Mathematics for Engineersby L.C.
Andrews. Eigenfunctions and eigenvalues are known for a few regions that are
not generalized rectangles. (See Miscellaneous Exercises 20 and 21 in the text
that follows.)
Eigenvalues of the Laplacian in a region can be estimated by a Rayleigh quo-
tient, much as in Section 3.5. Furthermore, we have theorems of the follow-
ing type. Letλ^21 be the lowest eigenvalue of∇φ=−λ^2 φinRwithφ= 0
on the boundary. Letλ ̄^21 have the same meaning for another region,R ̄.IfR ̄
fits insideR,thenλ ̄^21 ≥λ^21. [The smaller the region, the larger the first eigen-
value. For further information, seeMethods of Mathematical Physics, Vol. 1,
by Courant and Hilbert. In the famous article “Can one hear the shape of a
drum?,”American Mathematical Monthly, 73 (1966): 1–23], Mark Kac shows
that one can find the area, perimeter, and connectivity of a region from the
eigenvalues of the Laplacian for that region. However, Kac’s title question has
been answered negatively. In theBulletin of the American Mathematical Soci-
ety, 27 (1992): 134–138, authors C. Gordon, D.L. Webb, and S. Wolpert display
two plane regions, or “drums,” of different shapes, on which the Laplacian has
exactly the same eigenvalues.
The nodal curves of a membrane shown in Fig. 9 can be realized physically.
Photographs of such curves, along with an explanation of the physics of the