For us, the material presented in Chapters 1 through 6 of this book is a preamble: the material allows us to solve problems that arise in the analysis of physical problems, which so often end in partial differential equations. It is true that the analysis of Fourier transforms, for example, or eigenfunction expansions are, in themselves, interesting mathematical pursuits. However, our reason for studying what has come before this chapter, and the motivation for most applied mathematicians, is pragmatic: We are thereby enabled to approach solutions to those very physical problems we wish to solve.
With the advent and now exploding use of tools, such as RANS, DNS, and LES for solving the Navier–Stokes equations numerically, one might presume that the methodologies of this book are out of date. However, it has been our experience in our years of fluid dynamics research that the cross–fertilization of numerics and analysis, functioning synergistically alongside each other, provides insights into physical problems that are not available from either one standing alone. So, this chapter and Chapter 8 present some relatively simple, but real–world problems, that use more than one method from the previous chapters.
We now turn to a problem that is important in atmospheric flows, namely the standing gravity waves downstream of mountains, known as Lee waves. Early work on Lee waves may be found in (Janowitz) and (Miles), for example; an excellent, recent summary of the topic is in (Wurtele, et al).