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Unformatted text preview: Chapter 12 Partial Differential Equations in Space At last we have reached the ultimate rung of the dimensional ladder (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. As in the one- and two-dimensional settings developed in the preceding chapters, the three key examples are the three-dimensional Laplace equation, modeling equilibrium configurations of solid bodies, the three-dimensional wave equation, governing vibrations of solids, liquids, gasses, and electromagnetic waves, and the three-dimensional heat equation, modeling spatial diffusion processes. Fortunately, almost everything of importance has already appeared, and appending a third dimension is, for the most part, simply a matter of appropriately adapting the constructions. We have already developed the basic solution techniques: separation of variables and Greens functions or fundamental solutions. (Unfortunately, the most pow- erful of our planar tools, conformal mapping, does not carry over to higher dimensions.) In three-dimensional problems, separation of variables is applicable in rectangular, cylindrical and spherical coordinates (and eight additional exotic coordinate systems that we do not treat, [ 94 ]). The first two do not produce anything fundamentally new, and are therefore relegated to the exercises. Separation in spherical coordinates leads to spherical harmon- ics and spherical Bessel functions, whose properties are investigated in some detail. These new special functions play important roles in a number of physical systems, including the quantum theory of atomic structure that underlies the spectral and chemical properties of atoms. The Greens function for the three-dimensional Poisson equation in space can be iden- tified as the classic Newtonian (or Coulomb) 1 /r gravitational (electrostatic) potential. The fundamental solution for the three-dimensional heat equation can be easily guessed from its one- and two-dimensional versions. The three-dimensional wave equation, sur- prisingly, has an explicit, although more intricate, solution formula of dAlembert form, originally due to Poisson. Counter-intuitively, the best way to handle the two-dimensional wave equation is by descending from the simpler (!) three-dimensional formula. This strategy highlights a remarkable difference between waves in planar and spatial media. Huygens Principle states that three-dimensional waves emanating from a localized initial disturbance remain localized as they propagate through space. In contrast, in two di- mensions, initially concentrated disturbances leave a slowly decaying remnant that never entirely disappears....
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