The Mathematical Theory of Huygens' Principle

Abstract

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 polarization, 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. ?. ?. 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. ?. ?. 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 specialized 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.

The Mathematical Theory of Huygens' Principle

By Prof. Bevan B. Baker Prof. E. T. Copson. Pp. vii + 156. (Oxford: Clarendon Press; London: Oxford University Press, 1939.) 12s. 6d. net.