# Reviews of B B Baker and E T Copson, *The Mathematical Theory of Huygens' Principle*.

We present below some extracts from reviews of B B Baker and E T Copson,

*The Mathematical Theory of Huygens' Principle*. This book was first published in 1939 with a second edition in 1950 and a third edition in 1987. This third edition was reprinted by the American Mathematical Society in 2001, 2003, 2009, 2014.**1. The Mathematical Theory of Huygens' Principle (1939), by B B Baker and E T Copson.**

**1.1. From the Preface:**

Stimulated by a course of post-graduate lectures on the Partial Differential Equations of Mathematical Physics which Professor E T Whittaker gave sixteen years ago in the Mathematical Institute of Edinburgh University, one of the authors of the present work (B. B. B.) planned a comprehensive treatise covering the whole of this field. Unfortunately, ill health and pressure of other duties have, so far, prevented the completion of this scheme. In the meantime the subject has been treated from different points of view by Bateman (1932), Courant and Hilbert (1924 and 1938), and Webster (1927).

During the same period there have been great advances in mathematical physics, especially in the various developments of quantum mechanics. As these new theories are still developing rapidly, it would perhaps be unwise to attempt at the present juncture another general treatise on the mathematics of physics: and, after much consideration, we have decided to abandon the original plan and to replace it by the publication of a number of monographs, each complete in itself, on various special topics not adequately treated in existing books.

The present monograph deals with the mathematical theory of Huygens' principle in optics and its application to the theory of diffraction. No attempt has been made to give a complete account of the various methods of solving special diffraction problems. We are concerned only with the general theory of the solution of the partial differential equations governing the propagation of light and we discuss some of the simpler diffraction problems merely as illustrative examples. ...

The standard of knowledge of pure and applied mathematics expected of the reader is roughly that of the undergraduate who has completed the compulsory part of an honours course and is about to take up some specialised study.

We wish to express here our great indebtedness to Professor Whittaker for the original stimulus which led us to this work and for his continued interest, encouragement, and advice. We also desire to thank the Delegates of the Clarendon Press for undertaking this book and their Staff for their unfailing skill in printing it.

B. B. B., E. T. C.

**1.2. Review by: S Chapman.**

*The Mathematical Gazette*

**24**(259) (1940), 131-132.

This learned and difficult book, inspired by the lectures and influence of Professor E T Whittaker, is intended by its authors to be the first of a series of monographs, each complete in itself, on various special topics in the theory of the partial differential equations of mathematical physics. The subjects treated in this volume (and to be treated in later volumes) are chosen as being among those not yet adequately dealt with in existing treatises on the general theory.

Elementary students of physics first come across Huygens' principle in their course on light, and probably few, even among advanced students, learn very much about it. The principle was formulated at a time when light was regarded as a wave-disturbance similar to that of sound in air, although the phenomenon of polarisation, which shows that light propagation is essentially more complicated than that of sound, was discovered by Huygens himself. Even in the simplest geometrical form of the principle, however, and as applied to sound, it is not satisfactory without additional assumptions: Huygens had to assume that his secondary waves had effect only where they touched their envelope: further, since there may be an envelope to the rear of an advancing system of waves, as well as in front, it was necessary to assume that only one sheet of the envelope is to be considered. To avoid making this last assumption, it is necessary to give up the purely geometrical theory and resort to analysis. ...

The book is thus as far removed as possible from most of the mathematical treatises seen by school pupils and even by university students, in which all is well worn, cut and dried. Its rather severe character will limit its appeal, but it will give to its readers a sense of the untiring and continually renewed attacks, by a restricted mathematical elite, on the unsolved problems met with almost at the outset of the electromagnetic theory of light: it may be hoped and expected that it will also stimulate some among its readers to carry on this attack.

**1.3. Review by: A C Banerji.**

*Current Science*

**9**(5) (1940), 237-238.

The book is a monograph which deals with the Mathematical Theory of Huygens' Principle and discusses the general theory of the solution of the partial differential equations which govern the propagation of light. Some simple diffraction problems which serve as illustrations have also been discussed. The book is divided into four chapters. In the first chapter, the scalar form of Huygens' principle, as applied to the propagation of sound waves has been considered. ... In the second chapter, Kirchoff's and Kottler's theories of diffraction by a black screen have been discussed. ... In the third chapter, analytical formulation of Huygens' principle for electromagnetic waves has been developed ... In the fourth chapter, Sommerfeld's theory of diffraction has been fully discussed. ... The printing and get-up of the book are excellent and it may serve as quite a good text-book for those who want to make a specialised study of Huygens' principle. The number of diffraction problems that have been discussed is rather restricted. The usefulness of the book would have been greater and it would have served better as a standard book of reference if a more complete account of different methods of solving special diffraction problems were given. A judicious choice of additional problems illustrative of technical developments of the theory of diffraction would make the monograph complete in itself without unduly increasing its size.

**1.4. Review by: W E Bleick.**

*Bull. Amer. Math. Soc.*

**46**(5) (1940), 386-388.

This work is the first of a series of monographs planned by the authors on the mathematics of physics. Each monograph is to be complete in itself and deal with some special topic in the theory of the partial differential equations of mathematical physics not adequately treated in existing books.

The aims of the authors are admirably achieved in the first monograph which deals with the mathematical theory of Huygens' principle in the propagation of sound and light waves. The theme of the work is the general theory of the solution by Green's method of the partial differential equations governing these phenomena.

Huygens' principle is a well known elementary method for treating the propagation of waves. The method assumes that a spherical wavelet starts out with velocity

*c*from each point of a given wave front at time

*t*= 0. Each wavelet will have a radius

*ct*at time

*t*, and the envelope of these wavelets is taken to be the resulting wave front at this later time

*t*. A difficulty arises in that this geometrical construction would give a wave traveling backward, as well as one traveling forward. To avoid this difficulty it is necessary to generalise Huygens' principle by having recourse to an analysis of the partial differential equation governing the wave motion and of the boundary conditions to be satisfied....

The standard of knowledge expected of the reader of this work is that of a graduate student who has completed the usual courses in analysis and electromagnetic field theory. Twenty-three exercises are provided in the first three chapters. The text is replete with footnote references to papers that have appeared in the literature up to 1939. By rigour of logical treatment and careful attention to detail the authors have produced a critical treatise which will undoubtedly become a standard reference work.

**1.5. Review by: T H Piaggio.**

*Nature*

**145**(1940), 531-532.

There are several parts of the science of physics which appear very simple when expounded briefly in elementary text-books, but nevertheless present great difficulties when examined more carefully. A good example of this is furnished by Huygens' principle, which in its original form discusses the propagation of light by asserting that the wave front is the envelope of secondary waves whose centres are themselves on a previous wave front. The first difficulty in this elegant geometrical construction is that it gives not only the actual wave propagated forwards, but also another propagated backwards, which does not really exist. To the simple principle Huygens therefore added the special assumption that only one sheet of the envelope, namely that propagated forwards, was to be considered. The next difficulty is to explain diffraction. To do so, Fresnel replaced Huygens' isolated spherical waves by purely periodic trains of spherical waves, and made use of the principle of interference. He had to restrict his treatment to the case of small wave-lengths, and also had to make two additional assumptions concerning the relations of the amplitudes and phases of the secondary waves to those of the primary. The necessity for these two additional assumptions, which appear to be of an arbitrary character, has led some to consider Fresnel's theory as merely a convenient device for calculation without any sound physical basis. In any case, the principle so far takes no account of the phenomenon of polarisation, although this was discovered by Huygens himself. In fact we may say that what was put forward as a theory of optics cannot, in anything like its original form, be legitimately applied to that branch of physics, though it may apply to acoustics. The necessity for a careful re-examination of the subject is now apparent. Unfortunately, this seems possible only on an analytical basis, with rather complicated mathematics; the elementary geometrical treatment appears to fail to give the results required, unless it is supplemented by special assumptions. This re-examination of Huygens' principle was part of a much larger programme, covering the whole field of the partial differential equations of mathematical physics, which was the subject of a course of lectures by Prof E T Whittaker sixteen years ago, and was to have been treated in a comprehensive text-book by his pupil, now Prof B B Baker. Unfortunately, ill-health and pressure of other duties intervened, and Prof Baker, in collaboration with Prof E T Copson, intends to replace the projected treatise by a series of monographs, of which this is the first. The reader is assumed to have a knowledge of pure and applied mathematics roughly equal to what is possessed by an honours student of mathematics who has completed the compulsory parts of his degree course, and is about to enter upon some specialised study. The book deals with the subject in connexion with the general theory of the solutions of the partial differential equations involved, with some of the simpler diffraction problems as examples illustrating that theory.

**1.6. Review by: Harry Bateman.**

*Mathematical Reviews*

**1**(October) (1940), 315-316.

Huygens' geometrical construction, with its restriction that only one sheet of the envelope of the spherical wavelets is to be considered, is first justified in Chapter I by Poisson's analytical solution of the equation of wave-motions. A discussion is then given of the ideas of Fresnel and of the formula of Helmholtz which expresses these ideas in analytical form and gives the principle of Huygens for periodic processes. The diffraction formulae of Fresnel and Stokes are then obtained.

Kirchhoff's famous formula is first derived from the formula of Helmholtz and then proved directly. The formula is interpreted physically and the question of the uniqueness of the solution discussed. It is pointed out that to extend the theorem of Kirchhoff to the space outside a closed surface it is necessary to prescribe the asymptotic behaviour of the wave-functions under consideration. The peculiarities of wave-propagation in two dimensions are next indicated and Weber's analogue of Helmholtz's theorem is given. The analogue of Kirchhoff's formula, due to Volterra, is derived and an interesting account is given of a method, devised by Marcel Riesz and based on the theory of fractional integration, which provides a powerful method of solving initial value problems for equations like the wave equation.

The rest of the book is devoted chiefly to the problem of diffraction. Part of Chapter II includes a useful discussion of an important type of definite integral which occurs in the analysis of diffraction problems. Diffraction by a black screen is discussed in some detail. In Chapter III Huygens' principle is formulated for electromagnetic waves and Tedone's proof is given for some formulae which are associated with the names of Larmor and Tedone. Chapter IV contains a good account of Sommerfeld's theory of diffraction.

**2. The Mathematical Theory of Huygens' Principle 2nd Edition (1950), by B B Baker and E T Copson.**

**2.1. Review by: Ian N Sneddon.**

*The Mathematical Gazette*

**35**(311) (1951), 67.

Baker and Copson's monograph on the mathematical theory of Huygens' principle in optics and its application to the theory of diffraction is well known to those who conduct advanced lecture courses or seminars on the partial differential equations of mathematical physics. In this second edition the four chapters of the original are virtually unchanged, except for the addition of references to more recent work and the correction of minor misprints and errors. The main change is the addition of a chapter on the application of the theory of integral equations to problems of diffraction by a plane screen. In addition to a discussion of the classic papers of Rayleigh and Schwarzschild, there is given an account of recent work in the subject by Professor Copson, E N Fox and others. The chapter ends with a lucid exposition of the variational principle of Levine and Schwinger which has been of value in the solution of problems in short wave radio in which no rigorous solutions are known. There have been many interesting developments in the theory of diffraction since 1939, both in the theory of sound and in radar; the account given here is an excellent introduction to these studies, and will be of great value to the student beginning a study of the recent literature. This second edition, like the first, is characterised by the lucidity of the authors' style and the excellence of production traditionally associated with the Clarendon Press.

**2.2. Review by: W Franz.**

*Zentralblatt MATH,*

**Zbl 0040.12702**.

The book can only be described as an indispensable reference for its thorough and transparent account of the historical development of Huygens' principle. ... The text of the book is clear and easy to read, the computations are usually elegant and transparent, methodical but not quite uniform, and the physical considerations are generally referenced, rather than fully worked out. ... The great value of the book lies in the excellent historical account, which leaves hardly a single wish unfulfilled, and recounts a whole series of mistakes that have become customary in the literature. Familiarity with "Baker-Copson" is therefore essential for anyone who works in this field or who seeks a convenient and reliable introduction.

**3. The Mathematical Theory of Huygens' Principle 3rd Edition (1987), by B B Baker and E T Copson.**

**3.1. From the Publisher:**

Baker and Copson originally set themselves the task of writing a definitive text on partial differential equations in mathematical physics. However, at the time, the subject was changing rapidly and greatly, particularly via the developments coming from quantum mechanics. Instead, the authors chose to focus on a particular area of the broad theory, producing a monograph complete in itself. The resulting book deals with Huygens' principle in optics and its application to the theory of diffraction. Baker and Copson concern themselves with the general theory of the solution of the PDEs governing the propagation of light. Extensive use is made of Green's method. A chapter is dedicated to Sommerfeld's theory of diffraction, including diffraction of polarised light by a perfectly reflecting half-plane and by a black half-plane. New material was added for subsequent editions, notably Rayleigh's method of integral equations to the problem of diffraction by a planar screen. Some of the simpler diffraction problems are discussed as examples. Baker and Copson's book quickly became the standard reference on the subject of Huygens' principle. It remains so today.