We treat the electromagnetic fields arising from the passage of a point source through an interface separating two dissimilar dispersive media. Since the media are dispersive, they are partially characterized by a characteristic frequency λ. Our expansions are valid for λa/c » 1, where a is a characteristic length and c is the speed of light.

Geometrical optics fails to account for the phenomenon of diffraction, i.e., the existence of nonzero fields in the geometrical shadow. Keller's geometrical theory of diffraction accounts for this phenomenon by providing correction terms to the geometrical optics field, in the form of a high-frequency asymptotic expansion. In problems involving screens with apertures, this asymptotic expansion fails at the edge of the screen and on shadow boundaries where the expansion has singularities. The uniform asymptotic theory presented here provides a new asymptotic solution of the diffraction problem which is uniformly valid near edges and shadow boundaries. Away from these regions the solution reduces to that of Keller's theory. However, singularities at any caustics other than the edge are not corrected.

The problem of Cerenkov radiation in infinite inhomogeneous media is considered. The mathematical description of this phenomenon is given by the integro-differential system of equations for the electromagnetic field in a dispersive medium. The leading term of the asymptotic expansion of the electromagnetic field is obtained by applying an expansion procedure called the ``ray method.'' In this method all the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. The source which gives rise to the radiation is taken to be quite general. In fact, it is shown that any multipole moving along an arbitrary trajectory is a special case of the general source considered. From the expansion of the fields an expression for the total energy of the radiation is determined. Then, as an example, the case of plane-stratified media is treated in detail.

A fundamental problem of statistical mechanics is to obtain simplified descriptions of complex systems. A general principle is presented for obtaining equations of motion for such descriptions. The principle involves maximizing an appropriate entropy functional. It also involves the particle dynamics through the Liouville equation. Various special cases are presented in which the principle yields the Vlasov equation, the Boltzmann equation, Euler's hydrodynamic equations, a generalization of Grad's ten-moment approximation, the Gibbs distribution (i.e., equilibrium statistical mechanics), and Onsager's equations of irreversible thermodynamics. The principle also yields, trivially, the Liouville equation and Hamilton's equations of classical mechanics. Some of these results have been derived elsewhere by very similar procedures, but apparently the generality of the principle has been unrecognized. In terms of the general principle, the origin of irreversibility in the various equations of motion is easily seen, and the relation between the numerous definitions of entropy is clarified. No a priori justification of the principle itself is given.

Recently developed methods of asymptotic analysis are applied to the problem of Cerenkov radiation. The mathematical description of this physical phenomenon is given by the integro-differential system of equations for the electromagnetic field in a dispersive medium. The parameter λ is introduced into these equations, where λ is a characteristic frequency of the medium. It is for large λ that the asymptotic expansion of the electromagnetic field is sought. Isotropic, uniaxial crystalline and gyrotropic media are treated in detailed. The source function which appears in the field equations is taken to be quite general, e.g., it may be used to represent the current associated with any moving ``multipole'' source. By applying the method of stationary phase to an integral representation of the solution, the leading term of the asymptotic expansion of the electromagnetic field is obtained. More precisely, a parametric representation of the expansion is found in which certain space-time curves called ``rays'' play a key role. An expression for the total energy of the radiation is then determined.

Periodic plane waves are considered in a plasma pervaded by a uniform magnetic field. The appropriate linearized Boltzmann equations for the distribution functions of the electrons and of each type of ion are solved exactly. Collisions are taken into account by the inclusion of a simplified collision term. The solutions are used to compute the electric current, from which the conductivity tensor of each ionic species is found. The total conductivity is then used in Maxwell's equations to determine the electromagnetic field, and this leads to the dispersion equation for plane waves. This equation is solved in various parameter ranges for the case of longitudinal waves when the conductivity of only one species is taken into account. The solutions of Landau, Gordeyev, Gross, Bernstein, and others are recovered when the collision frequency vanishes. In addition, various power series and asymptotic expansions for the conductivity tensor are given, and it is shown to reduce to that of the magneto-ionic theory under suitable conditions. The Appendix contains a systematic study of a function which plays a central role in the theory.

The $s$-particle distribution functions ($s=1, 2, \ensuremath{\cdots}$) of classical equilibrium statistical mechanics are determined for a crystal, as power series in the temperature. They are obtained by solving Bogolyubov's functional differential equation. From the distribution functions, the thermodynamic functions of a crystal are computed as power series in the temperature. The leading terms in these series are the usual classical results which are customarily derived by assuming that the potential energy is a quadratic function of the particle displacements. The further terms, which depend upon the nonquadratic or anharmonic terms in the potential, provide corrections to the usual results, which become more important as the temperature increases. If only a few terms in the series are used, the results will be valid at temperatures low compared to some characteristic temperature of the crystal, e.g., the melting temperature. Since they are based on classical mechanics, the results are valid only at temperatures high compared to the Debye temperature.The series expansions of the distribution functions and thermodynamic functions may be viewed as the low-temperature analogs of the virial expansions, which are low-density expansions. As in the case of the virial expansions, all the terms are determined explicitly in analytic form, but their actual evaluation is difficult.

Communications on Pure and Applied MathematicsVolume 9, Issue 2 p. 207-265 Article Asymptotic solution of some diffraction problems† J. B. Keller, J. B. KellerSearch for more papers by this authorR. M. Lewis, R. M. LewisSearch for more papers by this authorB. D. Seckler, B. D. SecklerSearch for more papers by this author J. B. Keller, J. B. KellerSearch for more papers by this authorR. M. Lewis, R. M. LewisSearch for more papers by this authorB. D. Seckler, B. D. SecklerSearch for more papers by this author First published: May 1956 https://doi.org/10.1002/cpa.3160090205Citations: 162 † The Table of Contents is on page 264. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume9, Issue2May 1956Pages 207-265 RelatedInformation