Chapter 18 Partial Differential Equations in Three – Dimensional Space

At last we have ascended the dimensional ladder to its ultimate rung (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 basic spatial diffusion processes. Fortunately, almost everything of importance has already appeared in the oneand two-dimensional situations, and appending a third dimension is, for the most part, simply a matter of appropriately adapting the constructions. We have already seen the basic underlying solution techniques: separation of variables and Green’s functions or fundamental solutions. (Unfortunately, the most powerful 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. The first two do not produce anything fundamentally new, and are therefore relegated to the exercises. Separation in spherical coordinates leads to spherical harmonics 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 Green’s function for the three-dimensional Poisson equation in space can be identified as the classic Newtonian (and Coulomb) 1/r potential. The fundamental solution for the three-dimensional heat equation can be easily guessed from its oneand two-dimensional versions. The three-dimensional wave equation, surprisingly, has an explicit, although more intricate, solution formula of d’Alembert form, due to Poisson. Paradoxically, the best way to treat the two-dimensional version is by “descending” from the simpler three-dimensional formula. This result highlights a remarkable difference between waves in planar and spacial media. In three-dimensions, Huygens’ principle states that waves emanating from a localized initial disturbance remain localized as they propagate through space. In contrast, in two dimensions, initially concentrated pulses leave a slowly decaying remnant that never entirely disappears.