As we observe the world around us, we encounter waves everywhere. Many of them are easily seen, such as ripples on the surface of a pond after one throws a small rock into it. Others are easily detected by our other senses such as hearing the sound waves produced during conversation with friends or feeling the warmth of electromagnetic waves from the sun absorbed on our skin on a clear summer day. Some waves are too large in scale to be easily seen by a person standing in one place, such as the sloshing motion of certain long and narrow lakes produced by passing winds. Others, such as seismic waves, originate deep within the earth, and yet others, such as quantum mechanical waves, are too small to be directly observed and require elaborate instrumentation for their detection. When a wave encounters an obstacle in its path, it is scattered by it. This enables us to hear a person talking in the next room and to see objects in front of us when they are illuminated by a lamp in an otherwise dark room at night. When the wavelength of the wave is small compared to the size of the obstacle, the scattered waves organize themselves into localized regions of constructive interference that we call rays, which travel in paths reminiscent of the trajectories of classical particles. This behavior is familiar to anyone who has seen a shaft of light coming through a high window in a dimly lit church.
In addition to the beauty of naturally occurring and man-made waves that is perceived by our senses, the richness and depth of the mathematics used to quantitatively describe them is another beauty that can be perceived by our mind. Rays, Waves, and Scattering—Topics in Classical Mathematical Physics by John A. Adam, a professor of mathematics at Old Dominion University, is a tour de force of the mathematical description of waves. The book encompasses a large variety of waves, including waves inside or on the surface of an elastic solid, waves in fluids, sound waves, collective wave motions within the atmosphere, electromagnetic waves, and quantum mechanical waves. The book frequently treats the physical mechanisms responsible for producing the waves rather briefly. Readers desiring longer and more qualitative descriptions would do well to consult Adam's 2003 book, Mathematics in Nature. However, the mathematical description of wave phenomena is covered in Rays, Waves, and Scattering in great depth and completeness. From a mathematical point of view, almost nothing is left to the reader's imagination, since the calculations and derivations explicitly give virtually every step. When teaching some of this material to junior-senior physics majors, on occasion one can tend to gloss over some of the more sophisticated mathematical fine points of various calculations. These fine points are always described in complete detail in Rays, Waves, and Scattering.
Following the large catalog of different types of waves, the book describes a number of specific wave scattering situations such as acoustic waves by a long cylinder, gravity waves in the ocean by an island, the exact Lorenz–Mie theory of a plane electromagnetic wave incident on a homogeneous sphere, and Coulomb scattering of quantum mechanical waves. The book also considers a number of general properties of wave scattering such as S-matrix theory and the pole structure of the scattering amplitude in the complex plane. Adam uses the continuing story of the mathematical description of the rainbow seen in the sky after a rain shower, interspersed at various places throughout the book, to motivate his choice of topics. This is a rich unifying example since the mathematical description of the rainbow can be told at many different levels of understanding: (i) Descartes' description using Snel's law and ray theory, (ii) Airy's physical optics model that incorporates some, but not all, of the features of wave theory, (iii) Mie and Debye's exact electromagnetic wave theory, the mathematical complexity of which obscures the physical description of the rainbow's most visible features, (iv) teasing out these features using the Watson transformation applied to Mie theory, and (v) the place of the rainbow in the hierarchy of structurally stable optical caustics.
I found this book to have a number of strong points about which I became very excited. For example, the motivation for and description of the Watson transformation and the Poisson summation formula are the best and clearest that I have ever seen. In addition, the study of fluid dynamics is today frequently not included in the “standard curriculum” of undergraduate physics majors, and is instead studied by mechanical engineering students. I must confess that after teaching physics for 40 years, my own knowledge of fluid mechanics remains quite rudimentary. Thus, I was amazed to learn the mathematical details of no less than ten types of wave phenomena in the ocean or atmosphere.
At the same time, I also found some instances where I would have organized the book in a somewhat different way. For example, starting the book with ray theory at the level of reflection and refraction at a flat interface and the Descartes theory of the rainbow is quite appropriate. On the other hand, I would have deferred the connection between light rays and the eikonal equation and Hamilton–Jacobi theory to somewhat later in the book, when the reader has already become familiar with sophisticated mathematical analyses of physical situations. Also, given the ubiquity and great importance of wave diffraction, I would have placed the chapter on diffraction earlier in the book and included a description of the evolution of diffraction by an aperture or obstacle from geometrical shadow casting in the near-zone, to Fresnel diffraction, to Fraunhofer diffraction in the far-zone. The diffraction chapter occurs much later in the book, and begins with a very nice calculation of scattering of an electromagnetic plane wave by an impenetrable cylinder using the Watson transformation. However, the calculation focuses on the directly reflected wave in the illuminated region and electromagnetic surface waves in the shadow region, rather than on the identification of the diffracted field and its evolution as the observer moves progressively farther from the cylinder. The chapter then gives a derivation of the Kirchhoff integral for diffraction by an aperture, and concludes with far-zone Fraunhofer diffraction for scattering by a sphere.
I thoroughly enjoyed reading Rays, Waves, and Scattering. I sincerely wish I had encountered such a book early in my teaching career. The material presented in it would have provided a very useful enhancement to a number of courses I have taught to undergraduate physics majors over the years. For example, in a two-semester junior-senior level course in “classical mechanics, oscillations, and waves,” the standard material in the first semester typically includes the Lagrangian and Hamiltonian formulations of mechanics, rotations in three dimensions, and collective motions of coupled systems. Material from Rays, Waves, and Scattering on waves in elastic solids and in fluids would have given an interesting direction to the material of the second semester. Similarly, in a two-semester junior-senior level course in electromagnetism, the standard material in the first semester typically includes electrostatics, magnetostatics, electric and magnetic properties of materials, induced emfs, and Maxwell's equations. The second semester frequently begins with electromagnetic plane waves in rectangular coordinates, reflection and refraction at a flat interface, and radiation by simple antennas. Material from the book on TE-polarized and TM-polarized spherical multipole waves, the partial wave version of scattering by a sphere, and diffraction would provide an interesting and practically useful conclusion to the course. While I do not think this book is organized in such a way to serve as a stand-alone textbook for a single course on wave phenomena, it is certainly an invaluable resource as a mathematical encyclopedia from which to draw material for many different undergraduate physics, engineering, and mathematics courses.
Last, I would like to provide two cautions for the reader of Rays, Waves, and Scattering. Every once in a while the author makes a sudden change in notation. Very fortunately, in every case he alerts the reader that this is happening. The reader should also give careful consideration to Sec. 1.2 in the Introduction, “A Mathematical Taste of Things to Come.” Depending on the reader's orientation and background, this section could be a little like drinking from a fire hose. It could easily elicit either excitement or fear. More mathematically inclined and experienced readers might think, “Look at all the wonderful jewels of knowledge and understanding that I am going to learn from this book.” A less mathematically inclined reader might instead think, “Look at all of this. How am I ever going to be able to understand it all?” I give encouragement to potential readers tending toward the latter point of view, that patience and perseverance are the two keys here. If one is willing to put in the time and effort as he or she progresses from section to section through Rays, Waves, and Scattering, the jewels contained within will be yours.
James A. Lock is Professor Emeritus of Physics at Cleveland State University. He does research on various topics in electromagnetic scattering of plane waves and shaped beams by particles having a relatively high degree of symmetry.
