# hwy - Chapter 17 Dynamics of Planar Media In this chapter...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern