About Foundations of Wave Phenomena
This text provides an introduction to some of the foundations of wave phenomena. Wave phenomena appear in a wide variety of physical settings, for example, electrodynamics, quantum mechanics, fluids, plasmas, atmospheric physics, seismology, and so forth. Of course, there are already a number of fine texts on the general subject of "waves" and related physical phenomena, and some of these texts are very comprehensive. So it is natural to ask why you might want to work through this rather short, condensed treatment which is largely devoid of detailed applications. The answer is that this course has a slightly different — and perhaps more general — aim than found in the more conventional courses on waves. Indeed, an alternative title for this course might be something like "Introduction to Mathematical Physics with Applications to Wave Phenomena". So, while one of the principal goals here is to introduce you to many of the features of waves, an equally — if not more — important goal is to get you up to speed with the plethora of mathematical techniques that you will encounter as you continue your studies in the physical sciences.
It is often said that "mathematics is the language of physics". Unfortunately for you — the student — you are expected to learn the language as you learn the concepts. In physics courses any new mathematical tools are introduced only as needed and usually in the context of the current application. Compare this to the traditional progression of a course in mathematics (which you have surely encountered by now), where a branch of the subject is given its theoretical development from scratch, mostly in the abstract, with applications used to illustrate the key mathematical points. Both ways of introducing the mathematics have their advantages. The mathematical approach has the virtue of rigor and completeness. The physics approach – while usually less complete and less rigorous – is very efficient and helps to keep clear precisely why/how this or that mathematical idea is being developed. Moreover, the physics approach implements a style of instruction that many students in science and engineering find accessible: abstract concepts are taught in the context of concrete examples. Still, there are definite drawbacks to the usual physics approach to the introduction of mathematical ideas. The student cannot be taught all the math that is needed in a physics course. This is exacerbated by the fact that (prerequisites notwithstanding) the students in a given class will naturally have some variability in their mathematics background. Moreover, if mathematics tools are taught only as needed, the student is never fully armed with the needed arsenal of mathematical tools until very late in his/her studies, i.e., until enough courses have been taken to introduce and gain experience with the majority of the mathematical material that is needed. Of course, a curriculum in science and/or engineering includes prerequisite mathematical courses which serve to mitigate these difficulties. But you may have noticed already that these mathematics courses, which are designed not just for scientists and engineers but also for mathematicians, often involve a lot of material that simply is not needed by the typical scientist and the engineer. For example, the scientist may be interested in what the theorems are and how to apply them but not so interested in the details of the proofs of the theorems, which are of course the bread and butter of the mathematician. And, there is always the well-known but somewhat mysterious difficulty that science/engineering students almost always seem to have when translating what they have learned in a pure mathematics course into the context of the desired application.
The traditional answer to this dilemma is to o↵er some kind of course in "Mathematical Physics", designed for those who are interested more in applications and less in the underlying theory. The course you are about to take is, in effect, a Mathematical Physics course for undergraduates – but with a twist. Rather than just presenting a litany of important mathematical techniques, selected for their utility in the sciences, as is often done in the traditional Mathematical Physics course, this course tries to present a (slightly shorter) litany of techniques, always framed in the context of a single underlying theme: wave phenomena. This topic was chosen for its intrinsic importance in science and engineering, but also because it allows for a treatment of a wide variety of mathematical concepts. The hope is that this way of doing things combines some of the advantages of both the mathematician’s and the physicist’s ways of learning the language of physics. In addition, unlike many mathematical physics texts which try to give a more comprehensive "last word" on the subject, this text only aspires to give you an introduction to the key mathematical ideas. The hope is that when you encounter these ideas again at a more sophisticated level you will find them much more palatable and easy to work with, having already played with them in the context of wave phenomena.
This text is designed to accommodate a range of student backgrounds and needs. But, at the very least, it is necessary that a student has had an introductory (calculus-based) physics course, and hopefully a modern physics course. Mathematics prerequisites include: multivariable calculus and linear algebra. Typically, one can expect to cover most (if not all) of the material presented here in one semester.