Diffraction of light by superposed parallel supersonic waves,one being then-th harmonic of the other. Solution of the system of difference-differential equations for the amplitudes by a series expansion method

In the problem of the diffraction of light by two parallel supersonic waves, consisting of a fundamental tone and itsn-th harmonic, the solution of the system of difference-differential equations for the amplitudes has been reduced to the integration of a partial differential equation. The expressions for the amplitudes of the diffracted light waves are obtained as the coefficients of the Laurent expansion of the solution of this partial differential equation. The latter has been integrated for two approximations:

1. Forρ = 0, the results of Murty’s elementary theory are reestablished.
2. Forρ ≤ 1, a power series inρ, the terms of which are calculated as far as the third one, leads to a new expression for the intensities of the diffracted light waves, verifying the general symmetry properties obtained by Mertens.

     

Diffraction of light bv two parallel superposed supersonic waves,one being the second harmonic,the other the third harmonic of the same fundamental. Description of two methods leading to approximate solutions in finite form

The diffraction of light by superposed parallel ultrasonic waves, with frequency ratio 2:3, is solved by the NOA-method (N?th order approximation method). Explicit calculations of the intensities are given forN = 2. The same problem is also treated with the SA-method (method of successive approximations). The latter results are improved by iteration. In both methods the general symmetry property of the diffraction pattern is verified, namely, that symmetry with respect to the zero order occurs when the phase-difference of the supersonic waves δ = (2k + 1) π/4.     

Diffraction of light by supersonic waves: The solution of the raman-nath equations

The partial differential equation associated with the system of difference-differential equations of Raman-Nath for the amplitudes of the diffracted light-waves is solved exactly by the method of the separation of the variables. The solution is presented as a double infinite series containing the Fourier coefficients of the even periodic Mathieu functions with periodπ and the corresponding eigenvalues. Considering this solution as a Laurent series in one of the variables, the Laurent coefficients immediately give the exact expressions for the amplitudes of the diffracted light-waves, from which the formulae for the intensities are calculated. The connection between the Raman-Nath method and Brillouin’s Mathieu function method has thus been achieved.     

Diffraction of light by supersonic waves: The solution of the raman-nath equations—I

In the problem of the diffraction of light by a supersonic wave, at normal incidence of the light, the solution of the system of difference-differential equations of Raman and Nath, for the amplitudes of the diffracted light beams, is reduced to the integration of a partial differential equation. The coefficients of the Laurent expansion of the solution of the latter equation yield the expressions for the amplitudes of the diffracted light waves. The partial differential equation has been integrated for two approximations. (1) Forρ=0, the well-known results of Raman and Nath’s preliminary theory are re-established. (2) Forρ?1 a power series inρ, the terms of which are calculated as far as the third one, leads to the solution of Mertens and Berry obtained by a perturbation method.     

Diffraction of light by two superposed parallel ultrasonic waves,having arbitrary frequencies. General symmetry properties; method of solution by series expansion

In this study about the diffraction of light by superposed parallel ultrasonics, with frequency ration ₁:n ₂, we deduce a general symmetry property for the intensities of the diffraction pattern: if the intensities of the ordersn and ?n are equal the phase-difference must be of the form:

$$\delta = \frac{{n_1 - n_2 }}{{n_1 }} \begin{array}{*{20}c} \pi \\ 2 \\ \end{array} + p \frac{\pi }{{n_1 }}$$     

Wiener-hopf equations for waves scattered by a system of parallel sommerfeld half-planes

A scalar time-harmonic wave (governed by Helmholtz's equation) impinges on N semi-infinite half-planes. The scattered field is sought when first, second, and third-kind boundary conditions or even general linear transmission conditions on the plates ∑_m and their complementary parts ∑ are prescribed. Making use of the Fourier transform a representation formula for H¹ (Ω) solutions is presented. The boundary/transmission problem is shown to be equivalent to a (2N × 2N)-Wiener–Hopf (WH) system for jumps of the Dirichlet–and Neumann–Cauchy data across the semi-infinite screens ∑_m. The (2N × 2N)-Fourier symbol matrix ??_?? contains N block matrices on the diagonal corresponding to Sommerfeld boundary/transmission problems for a single plate. These (2 × 2)-symbol matrices are factorizable and thus the full WH system is invertible by a perturbation argument for not too small spacings of neighbouring screens ∑_m.     

A geometrical proof of the maximum principle for systems represented by difference-differential equations

This note derives a maximum principle for a linear class of difference-differential systems. The control functions are assumed to be piecewise continuous, and the derivation closely follows the method of Halkin for the case of ordinary differential systems.     

Preconditioned CG-type methods for solving the coupled system of fundamental semiconductor equations

This paper presents some of the authors' experimental results in applying Preconditioned CG-type methods to nonsymmetric systems of linear equations arising in the numerical solution of the coupled system of fundamental stationary semiconductor equations. For this type of problem it is shown that these iterative methods are efficient both in computation times and in storage requirements. All results have been obtained on an HP 350 computer.     

Solution of the Cauchy problem for a system of integrodifferential equations of viscoelasticity

Existence and uniqueness of the solution of the Cauchy problem is proved for a system of integrodifferential equations of the hereditary theory of viscoelasticity. A method of constructing an approximate solution is proposed. An estimate of the error of the approximate solution is presented.     

The fundamental matrix of the system of linear elastodynamics in hexagonal media. Solution to the problem of conical refraction

An explicit integral representation by single definite integralsof the fundamental matrix (Green's tensor) of the time-dependentsystem of hexagonal elastic media is derived. Thereby the problemof internal conical refraction in such media is solved.