Chapter 17
Dynamics of Planar Media
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 equi-
libria 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 model-
ing 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 solu-
tions. (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 tech-
niques 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.
†
Throughout, “dimension” refers to the number of space variables. In Newtonian dynamics,
the time “dimension” is accorded a separate status, which distinguishes dynamics from equilib-
rium. Of course, in the more complicated relativistic universe, time and space must be regarded
on an equal footing, and the dimension count modified accordinagly.