Chapter 17 Dynamics of PlanarMedia

In this chapter, we continue our ascent of the dimensional ladder for linear systems. In Chapter 6, we embarked on our journey with equilibrium configurations of discrete systems — mass–spring chains, circuits, and structures — which are governed by certain linear algebraic systems. In Chapter 9, the dynamical behavior of such discrete systems was modeled by systems of linear ordinary differential equations. Chapter 11 began our treatment of continuous media with the boundary value problems that describe the equilibria of one-dimensional bars, strings and beams. Their dynamical motions formed the topic of Chapter 14, in the simplest case leading to two fundamental partial differential equations: the heat equation describing thermal diffusion, and the wave equation modeling vibrations. In Chapters 15 and 16, we focussed our attention on the boundary value problems describing equilibria of planar bodies — plates and membranes — with primary emphasis on solving the ubiquitous Laplace equation, both analytically or numerically. We now turn to the analysis of their dynamics, as governed by the two-dimensional forms of the heat and wave equations. The heat equation describes diffusion of, say, heat energy, or population, or pollutants in a homogeneous two-dimensional domain. The wave equation models small vibrations of two-dimensional membranes such as a drum. Although the increase in dimension does challenge our analytical prowess, we have, in fact, already mastered the key techniques: separation of variables and fundamental solutions. (Disappointingly, conformal mappings are not particularly helpful in the dynamical universe.) When applied to partial differential equation in higher dimensions, separation of variables often leads to new ordinary differential equations, whose solutions are no longer elementary functions. These so-called special functions , which include the Bessel functions appearing in the present chapter, and the Legendre functions, spherical harmonics, and spherical Bessel functions in three-dimensional problems, play a ubiquitous role in more advanced applications in physics, engineering and mathematics. Basic series solution techniques for ordinary differential equations, and key properties of the most important classes of special functions, can be found in Appendix C In Appendix C, we collect together the required results about the most important classes of special functions, including a short presentation of the series approach for solving non-elementary ordinary differential equations.