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Rays, waves, and scattering. Topics in classical mathematical physics. (English) Zbl 1386.00065

Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14837-3/hbk; 978-1-4008-8540-4/ebook). xv, 588 p. (2017).
The book provides a compressive general survey of rays, waves and scattering with the aim to connect the physical sciences with mathematical methods of physics and methods of mathematical physics (the author differenciates between the two methods), which is relevant for several types of undergraduate and senior/graduate courses in applied mathematics, theoretic physics, or engineering. Instructors could select topics of the wide range of contents and use it as a treasure because the volume has properties of an anthology, a monograph, and a textbook. The reader should have some knowledge in advanced calculus, ordinary and partial differential equations, functions of a complex variable, classical mechanics, electromagnetic theory, and elementary quantum mechanics. A special aim of the author consists in the mathematical and physical presentation of ‘dynamical phenomena taking place in the everyday world around us – in the sky, in clouds, rivers, lakes, oceans, puddles, faucets, sinks coffee cups, and bathtubs’. The author includes as many topics as possible and consequently the book is written for anyone interested in the union of ray, wave and scattering problems in different media rather than in a special research field.
The author emphasizes that the unifying nature of mathematics enables the reader to recognize the similarities between the different kinds of waves, such as electromagnetic, acoustic, gravity, seismic, water waves, and so forth.
The 588 + XXIV pages long book consists of a preface, acknowledgments, an introductory chapter, five parts with all in all 28 chapters, divided into sections and subsections, 6 appendices, 18 pages of references, and a 4 pages long index of relevant terms. The 274 references are partially provided with hints for the reader. Exercises are included in the sections.
In the Introduction the author gives an overview about the five parts of the book. The titles of the parts are (I) Rays, (II) Waves, (III) Classical Scattering, (IV) Semiclassical Scattering, and (V) Special Topics in Scattering Theory.
The author starts with the theory of the rainbow which has been formulated in different levels: the geometrical-optics theory of Descartes (based on rays), the wave-optics approach of Airy, or Mie’s scattering theory. The rainbow serves as an unifying template throughout the most parts of the book. The author describes the theories by already confronting the reader with the concepts of rays, waves, classical and semiclassical scattering, caustics and diffraction catastrophes, although the terms are exactly defined later.
Part 1 begins in Chapter 2 with the ‘Introduction of the ‘Physics’ of Rays’, outlining that a ray is only a “mathematical idealization of an infinitesimally narrow beam of light” that does not exist physically, but is a very useful artificial notion in geometric optics with the property to have a position, a direction, a speed, a density (power per unit area), to carry energy, to propagate in media, and to undergo reflection or refraction at interfaces.
Introduced are the refraction index, the geometric path length, geometric wavefronts, the eikonal equation, Fermat’s principle, the intensity law, and Snell’s laws. Snell’s laws describe the relationship between the angles of incidence and refraction referring to the bending of a path of a light wave as it passes across the boundary separating two different isotropic media. After the heuristic derivation of the laws a geometric, a wave-theoretic and an algebraic proof are presented.
The mathematics of rays is the subject of Chapter 3 with the main result that geometric optics can be formulated by the so-called ray equation, a set of ordinary differential equations, and by the Hamilton-Jacobi equations, which are first-order partial differential equations. It is shown that these formulations are equivalent by deriving that the ray equations are the characteristic equations of the Hamilton-Jacobi equations. Topics are the Helmholtz equation, the method of characteristics, and the Hamilton-Jacobi theory (dealing with the Euler-Lagrange equation, Hamilton’s principle, the optical path length, Fermat’s principle, and the eikonal equation) with the interpretation that the geometric light rays are the orthogonal trajectories to the geometrical wavefronts. Treated are also the mirage theorem, the D’Alembert solution of the one-dimensional wave equation, the dispersion relation based on the Fourier transform of the one-dimensional wave equation, the plan wave and the group velocity. The theory is applied to a choice of atmospheric waves. Starting from a governing wave equation for so-called acoustic-gravity waves their dispersion relations are discussed. Declared are lee waves, gravity waves in water, planetary waves, and Rossby waves, to name a few. Further subjects are the general solution of the linear wave equation, the asymptotics for oscillatory sources, based on the consideration of time-dependent three-dimensional partial differential operators, radiation conditions, slowly varying environments, conservation laws, wavepackets, and the group velocity.
Chapter 4 starts with a rainbow description using the ray theory of light. Based on the assumption, that a rainbow is formed by refractions and reflections from a immense number of geometrically equivalent raindrops, the path of a ray of light through a single spherical drop is considered to calculate the features of the ‘classical rainbow’ by Snell’s laws of reflection and refraction taking into account a primary rainbow (caused by a ray reflected from the outer surface of the drop) and high-order rainbows. The last are caused by the part of the ray which is refracted into the drop. This ray undergoes one or more internal reflections at the respective back of the drop and a part of it is there refracted and transmitted out. The variation in the deviation of the incident ray as a function of the incidence angel and the ray of minimum deviation are considered. The author demonstrates in detail how the refraction dependence of the wavelength of light causes the color of the rainbow, including the dispersion effect.
A number of other effects which influence the features of a rainbow are outlined, among others, outside the realm of geometric optics, the dependence of light reflection on its degree of polarization. Treated are further topics of meteorological optics, such as the glory (consisting of one or more color concentric rings), coronas, and the Rayleigh scattering taking into account the ratios between the wavelength and the droplet size.
The classical geometric-optics theory of the rainbow is improved by Airy’s theory (a linear description of the propagation of waves), that is described in detail in Chapter 5. Especially the existence of supernumerary bows, caused by self-interference in waves bunching out the angle of minimum deviation, cannot predict by the classical theory. Airy’s theory results in a shift of a few tenths of a degree in the angular position of the rainbow compared with the classical one. Topics are the Airy’s rainbow integral and the colors distribution in the Airy rainbow.
Optical caustics and diffraction catastrophes and their relevance for the rainbow theory are the subject of Chapter 6.
An introduction into the Wenzel-Kramers-Brillouin (WKB) approximation is given in Chapter 7. The approximation is a method to find approximate solutions of linear ordinary differential equations with spatially varying coefficients. It is used in semiclassical calculations in quantum mechanics for the solution of the one-dimensional time-independent stationary Schrödinger equation for a particle of mass \(m\) with the energy \(E\) moving in a potential \(F(x)\) in which the wavefunction is assumed to have a slowly varying amplitude and/or phase. Topics are the physical interpretation of the WKB approximation, Airy’s differential equation, the treatment of turning points, Airy functions and its expressions in terms of Bessel functions.
In contrast to the last three chapters which have a wavelike character Chapter 8 and 9 are based on the ray approximation only. In Chapter 9 islands rays are treated. Based on the eikonal equation surface water waves are considered. The reader is informed about gravity surface waves, plane waves incident on a ridge, shallow water waves, and tsunamis, to name a few.
Earthquakes and explosions generate seismic rays which are used to create three-dimensional images from the interior of the spherical earth. There are seismic and surface seismic waves. The seismic waves are considered in Chapter 9.
Two types of surface seismic waves occur within in the top of about 30 km of the earth’s crust, the Rayleigh waves having horizontal components in the direction of propagation, and Love waves having horizontal displacements perpendicular to their direction of propagation. Because these waves cannot treated in the ray theory only they will considered in Part 2, Chapter 10.
The seismic waves are decomposed in refracted \(P\) waves and reflected \(P\) waves because the earth is layered. In difference to sound, electromagnetic, or water waves the seismic waves are decomposed additionally into refracted \(S\) waves and reflected \(S\) waves because the rock boundaries are compressed and sheared (called wave conversion). The Wiechert-Herglotz inversion for the determination of the wavespeed quantities is derived.
Part 2 of the book begins with Chapter 10 “Elastic Waves”. The author starts with the field equations for a continuous elastic medium. These equations of motion are the so-called Navier equations, which are deduced from the conservation laws of mass, momentum, and energy. The molecular and crystalline structure of the medium is not taken into account (continuity assumption), i. e., a large differential volume element is used in order to average effects of individual molecules. Starting from basic notations and the relation between the stress and strain (Hooke’s law) the governing equations of motion are deduced for elastic media of constant density. Primary body and shear (or secondary) waves equations are derived from these equations. There are two kinds of body waves, which can propagate in elastic medium: (I) longitudinal waves of compression and dilation, (ii) sheared and twisted waves, which cannot propagate in gases or liquids, but in solids the particles move transversely to the direction of the waves. The Helmholtz representation of Navier equations is deduced, and the in Chapter 9 mentioned surface waves are treated.
In Chapter 11 an overview about the generation of surface gravity waves in fluid dynamics is presented. The author starts with some notions: These waves are generated at the interface between two media if the gravity or a buoyancy force restores the equilibrium which is changed by a dilation of a fluid element, for instance by wind. Gravity waves between the atmosphere and sea are called gravity or surface waves while gravity waves within the deep ocean are named internal waves. Two kinds of wind-generated waves are considered: the ocean tides and tsunamis, the last in Chapter 13. Another type of waves are the gravity-capillary waves and (neglecting the gravity) capillarity waves.
The treatment of surface waves is a part of hydrodynamics. The corresponding equations of motion are simpler than the general case because: (i) the fluid can be assumed to be inviscid , incompressible, and irrotational (ideal fluid dynamics), (ii) the water waves are two-dimensional, (iii) the waves are on the surface on which no external forces act.
Based on these prerequisites the Bernoulli equation is deduced. Linearizing this equation and considering dynamic and kinematic boundary conditions as well as relations between speed and wavelength different types of water waves are characterized: deep water waves, capillary waves, shallow water waves.
After these rather simple cases (linearization) nonlinear effects caused essentially by high amplitude waves come into consideration. Questions of hydrodynamical stability are treated. Solitary waves are only mentioned.
Some sections are devoted to an extensive analysis of ship waves (among others Whitham’s ship wave analysis, geometric approach to ship waves and wakes, ship waves in shallow water, short and long waves).
The transmission of sound waves in the ocean, which depends on the depth, the configural and physical properties of the bottom, and the shape of the surface, is treated in Chapter 12. Starting from the wave propagation in an infinite nonhomogeneous two-dimensional domain the governing equation is formulated and considering time-harmonic waves the corresponding Helmholtz equation is established. Some types of ocean waveguides are discussed on basic of this equation including the so-called radiation condition and boundary conditions. Treated are guide waves with low and higher velocity speed (leaky modes) in the central region, one-dimensional waves in an inhomogeneous medium, and acoustic waves in stratified medium, to name a few, including solution methods (method of Darboux).
A mathematical representation of a tsunami is given in Chapter 13. The author defines the tsunami as a series of ocean waves with very long wavelength caused by sudden displacements in the see floor. Tsunamis are characterized by wave heights of few inches in the deep ocean which increase to fast moving walls reaching the coastline.
Assuming a two-dimensional ocean of constant depth, a free surface, a rigid boundary at the sea floor only, an equation for the atmospheric pressure, and a displacement on the seafloor (e. a. due to an earthquake) a boundary value problem for the space and time development of a tsunami is formulated, by using notations from Chapter 11. Discussed are then the tsunami generation by a displacement of the free surface, surface waves on deep water, wave energy propagation, and the tsunami generation by a displacement of the seafloor.
Atmospheric waves are considered in Chapter 14. Looking back to fluids, the author starts with the general statement that ‘waves occur between stable layers of fluids of different density’. Various forces disturb the equilibrium and cause the waves. “Gravity or buoyancy forces try to restore the equilibrium”. The same principle induces the atmospheric waves. A number of forces which cause atmospheric waves are mentioned.
Taking into account the earth rotation, the continuity equation, the first law of thermodynamics for adiabatic motion, the equations of geostrophic and hydrostatic equilibrium, and introducing the buoyancy frequency, the density scale height, and the Eckart coefficient, with the aim to establish a dispersion relation from a single ordinary differential equation for one of the field variables the governing equation for the density-modified vertical velocity perturbation is formulated. Based on these general results acoustic, gravity, Rossby, Kelvin, Rossby-gravity, and Yanai waves are treated. Further topics are mathematical models of terrain-generated gravity waves, such as waves over an isolated mountain ridge and trapped lee waves.
The last three parts of the monograph are devoted to III Classical Scattering, IV Semiclassical Scattering, and V Special Topics in Scattering Theory. Generally one distinguishes three kinds of scattering. The size of the object is: (i) small in comparison to the wavelength, (ii) comparable with the wavelength, (iii) large compared with the wavelength, of the incident wave.
Part III starts with the alternative approach to derive equations of motion. The action of a physical system is a mathematical functional. The path which is followed by a physical system is that for which the action is stationary. The Euler-Lagrange equations are used to determine where the functional is stationary giving Newton’s second law for the motion of a particle. The Lagrangian, the Hamiltonian, the Hamilton-Jacobi equations, the eikonal equation of geometric optics, the classical wave equation, (restricting to time-harmonic waves) the Helmholtz equation, Hamilton’s equations, and the time-independent Schrödinger equation are formulated. The classical scattering, the deflection and scattering angles, and scattering cross section are introduced. Rainbow and glory scattering are considered.
The subject of Chapter 16 is gravitational scattering, the deflection of the path of particles near a massive body. Topics are planetary orbits, the Kepler problem, hard sphere scattering and Rutherford scattering.
The scattering of surface gravity waves by islands, reefs, and barriers is treated in Chapter 17.
Although the title of Chapter 18 is “Acoustic Scattering”, the author starts with an overview about acoustic and electromagnetic scattering and the statement ‘all what we see and much of what we hear, is respectively, a consequence of the scattering of electromagnetic and acoustic waves from various objects’. The historically by Lorenz, Mei, and Debye developed mathematical theory of electromagnetic scattering from spheres which do not conducts electric current (dielectric spheres) is also applied to plane acoustic waves by spheres, even though the last is a scalar problem in contrast to the vector problem of electromagnetic scattering. Generally one distinguishes three kinds of scattering. The size of the object is: (i) small in comparison to the wavelength, (ii) comparable with the wavelength, (iii) large compared with the wavelength, of the incident wave. The molecular scattering of light in the atmosphere, responsible for why the sky is blue (Rayleigh scattering), belongs to the region (i). The inclusion of scattering for improved models for a rainbow, a glory, and a corona is a part of region (iii). Scattering problems in quantum mechanics (wave-particle duality) go with region (ii).
Based on the scalar Helmholtz equation a number of problems associated with acoustic scattering are discussed, such as scattering by a cylinder, by a impenetrable sphere, by a rigid sphere, by a rigid pulsating sphere, to name a few. Further themes are the Sommerfeld radiation condition, the sound of mountain streams, and bubble collapse.
The author starts in Chapter 19, “Electromagnetic Scattering: The Mie Solution”, with phenomenological contemplations of the scattering of sunlight (why the sky is blue and not violet) taking into account the different wavelength of the sun colors, the non uniformly intense at all wavelength, the more sensitivity of humane eyes to blue, the influence of dust particles which are not small in comparison with the wavelength, and so on. To understand scattering from an analytical point of view Maxwell’s equations and the vector Helmholtz equation for electromagnetic waves are introduced. The represented Mie solution is based on Maxwell’s equations for a monochromatic plane wave from infinity incident on a homogeneous isotropic sphere of a given radius. The author uses also the notion Lorentz-Mie solution and communicates detailed historical dates about the first publications by Lorentz, Mie, Debye, and Clebsch.
Part IV, “Semiclassical Scattering”, is devoted to problems which exists between geometrical optics and wave optics. The author states that geometric optics, particle and particle/ray- like trajectories belong to the classical domain and physical optics, acoustic and electromagnetic waves, and quantum mechanics to the wave domain.
Chapter 20 contains a number of problems of semiclassical scattering, two are mentioned here. The classical wave connection is an example for the semiclassical domain. The author deduces the similarity between the inhomogeneous Helmholtz equation for acoustic wave propagation in radially inhomogeneous media and the time-independent Schrödinger equation with the result, that the wave function can be considered both as the spatial part of the acoustic pressure (classical) and as the Schrödinger wave function (quantum mechanically).
Expanding the electric field of the governing equations for the electromagnetic scattering from a radially inhomogeneous sphere in terms of vector spherical harmonics yields a solution in terms of transverse electric and transverse magnetic modes with the radial Debye potentials. Also in this case a connection to the canonical time-independent Schrödinger equation is established.
In Chapter 21, the WKB(J) approximation is characterized as a semiclassical calculation in quantum mechanics. The wave function is assumed to be an exponential function which slowly varies in comparison to the de Broglie wavelength. The function is semiclassical expanded.
Chapter 23 is devoted to the Sturm-Liouville equation and its connection with the time-independent one-dimensional Schrödinger equation.
The subject of Part V, “Special topics in Scattering Theory”, is the so-called \(S\)-matrix. The \(S\)-matrix expresses the state of a scattering system. For electromagnetic or acoustic waves the matrix relates the phase, polarization, and intensity of the outgoing waves in the far field to the direction and polarization of the incident wave toward an obstacle. In quantum mechanics the matrix plays a role in the description of particle scattering.
The subjects of Chapter 24 until 27 are a detailed analysis of the \(S\)-matrix, technical details of the Jost solution, one-dimensional Jost solutions, and morphology-dependent resonances.
In Chapter 28 the author emphasizes that the rainbow functions are a ‘sort of template for the topics of this book’ and he tells the genesis of his fascination with rainbows.
Some problems that have been only outlined in the chapters are represented at full length in the appendices
Although the rainbow plays a role in much chapters of the book the reader is in detail informed about the very comprehensive domain of waves.

MSC:

00A79 Physics
78-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A45 Diffraction, scattering
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B20 Ship waves
76B55 Internal waves for incompressible inviscid fluids
76B60 Atmospheric waves (MSC2010)
76B65 Rossby waves (MSC2010)
76E17 Interfacial stability and instability in hydrodynamic stability
76Q05 Hydro- and aero-acoustics
74Jxx Waves in solid mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
81U20 \(S\)-matrix theory, etc. in quantum theory
85A25 Radiative transfer in astronomy and astrophysics
86A05 Hydrology, hydrography, oceanography
86A15 Seismology (including tsunami modeling), earthquakes
35Q60 PDEs in connection with optics and electromagnetic theory
35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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