Daljit S. Ahluwalia

Diffraction by a Curved Wire

A geometrical theory of diffraction by a curved piece of wire, with or without ends, is formulated. Diffracted rays and diffraction coefficients are introduced, as in the geometrical theory of diffraction by edges. The diffraction coefficients are determined by means of canonical problems for both scalar and electromagnetic fields. Then the theory is applied to diffraction of a normally incident plane wave by a circular loop of wire of circular cross section, i.e., a torus. The results are compared with the variational results of Kouyoumjian, the toroidal coordinate analysis of Weston and the experimental results of Keys and Primich. Good agreement is found at short wavelengths, and even for wavelengths as large as three times the loop radius.

Application of Ludwig’s uniform progressing wave ansatz to a smooth caustic

It is well known that the usual harmonic ansatz of geometrical acoustics fails at a caustic and that uniform expansions can be found which remain valid in the neighborhood of the caustic and reduce asymptotically into the usual geometrical acoustic ansatz far from the caustic. A similar uniform ansatz can be constructed for transient acoustic fields using the so‐called progressing wave ansatz. In this paper some of the details of this construction are considered for a point source in a time independent, refracting medium in which a smooth caustic is formed. The time dependence at the source is taken to be a rather general function of time. Particular attention is given to the nature of the leading term of the uniform progressing wave expansion and its construction from the nonuniform harmonic ansatz. Explicit expressions are obtained for times near the direct arrival time and the caustic arrival time as well as for large time. An analytic example is examined when the squared refractive index is linear and...

Ordinary differential equations with applications

First Reviewer. First, I would like to give some information about each book, then I will compare and discuss them and give my personal appreciation of them. Balser (B) is intended as a reference book in the special area of the analytic theory of linear meromorphic ordinary differential equations (ODEs) as well as an introduction to this topic for students who want to work in this or adjacent fields. After an introduction containing several examples motivating the study of divergent formal solutions of ODEs, the basic properties of solutions and singularities of the first kind are briefly studied. Then the following topics are discussed in depth: formal solutions, asymptotic expansions, Boreland Laplace-transforms, Gevrey asymptotics, summability, the Stokes’ phenomenon, and multisummability as well as tools required such as the Cauchy–Heine transform and Ecalle’s acceleration operators. Finally, related topics as the Riemann–Hilbert and reduction problems are briefly mentioned, as are applications of the methods to adjacent areas (in particular nonlinear ODEs, difference equations, and singular perturbations). Chicone (C) is intended as a text for the graduate level. In the first chapter, the basic theory of ODEs is covered—in contrast to the “logical” order of the theory, some results are introduced that are treated in depth in later chapters. The following chapters treat linear systems and stability theory, applications (real-life applications from physics!), invariant manifolds, persistence of periodic solutions under perturbations, Melnikov’s method and homoclinic orbits, and averaging and bifurcation theory. Applications are given throughout the book, often as motivation for the theory. Hsieh and Sibuya (HS) is also intended as a textbook for the graduate level. The first chapters cover the basic theory of ODEs—here an in-depth study of nonuniqueness is exceptional. Then, singularities of the first kind are treated (using S–N decompositions), as are boundary value problems, Lyapunov-type numbers, stability

Uniform Asymptotic Solution of Eigenvalue Problems for Convex Plane Domains

The ray method of Kelley and Rubinow [1] determines asymptotically certain of the large eigenvalues and corresponding eigenfunctions of the Laplace operator in convex domains, but the expression for each eigenfunction is singular at a certain caustic of the rays. We shall obtain uniform asymptotic expansions of these eigenfunctions in the two-dimensional case, and improved expressions for the eigenvalues, by using the uniform expansion of Kravtsov [2], [3] and Ludwig [4]. For caustics near the boundary the results agree with these of Buldyrev [5]. We also treat the case of caustics near the boundary by using the uniform expansion of Lewis, Bleistein and Ludwig [6] and the boundary layer method used by Matkowsky [7], both of which enable us to treat the impedance boundary condition and to obtain further terms more easily.

Elastic Waves Produced by Surface Displacements

Waves produced in a homogeneous elastic body by time-dependent displacements of all or part of the body surface are determined by the geometrical theory of diffraction. Waves associated with the rays of geometrical acoustics, and with certain diffracted rays, are treated. The latter arise from curves of discontinuity of the applied displacement and from curves separating the displaced part of the surface from the free part. The expression for each type of diffracted wave contains a diffraction coefficient. Certain diffraction coefficients are determined from appropriate canonical problems which are solved exactly by the conical field method. The theory is applied to the impulsive and time-harmonic angular displacements of a circular region on the otherwise stress free surface of a half-space. The results are shown to agree with and extend those found previously in these cases by other methods.

Scattering of low‐frequency acoustic waves by baffled membranes and plates

The method of matched asymptotic expansions is used to study the scattering of plane, monochromatic, acoustic waves from baffled flexible surfaces in the limit as L/λ→0. Here, λ is the wavelength of the incident acoustic wave and L is a characteristic size of the flexible surface, such as its maximum diameter. The baffled surface is either a membrane or a thin plate. Uniform asymptotic expansions of the scattered fields are obtained for both surfaces as L/λ→0. Thus, for example, it is valid in the nearfields and farfields of the flexible surface. The method is applied to obtain the fields scattered by rectangular membranes and plates. It is found that plates are more efficient scatterers than ‘‘similar’’ membranes if the plates are sufficiently thin.

Advection–diffusion around a curved obstacle

Advection and diffusion of a substance around a curved obstacle is analyzed when the advection velocity is large compared to the diffusion velocity, i.e., when the Peclet number is large. Asymptotic expressions for the concentration are obtained by the use of boundary layer theory, matched asymptotic expansions, etc. The results supplement and extend previous ones for straight obstacles. They apply to electrophoresis, the flow of ground water, chromatography, sedimentation, etc.

SOLUTIONS OF A CLASS OF MIXED FREE BOUNDARY PROBLEMS.

Analytical solutions to a class of mixed free boundary problems are constructed. The methods used are the hodograph method and matched asymptotics. Problems with equality and/or inequality boundary conditions are treated. The solutions exhibit transition from one steady solution to another. The physical problems treated are the formation of ice in a channel and the peeling of a membrane glued to a flat surface.

Scattering by a slender body

Scattering of a time harmonic scalar field by a hard, soft, or transparent slender body is calculated. The wavelength is assumed to be long compared to the radius of the body cross section, but of any size compared to the body length. The scattered field is represented as an integral of fields from monopoles and dipoles distributed along the centerline of the body. The strengths of these are sources are determined by solving integral equations. Then the field is evaluated for bodies either long or short compared to the wavelength. For long bodies, the results of the geometrical theory of diffraction are obtained and the end diffraction coefficients are determined. For and the end diffraction coefficients are determined. For short bodies, the low‐frequency results are found.

Direct and inverse scattering of acoustic waves by low‐speed free shear layers

In this paper, a direct method and a simple inverse method, which can be used to determine the velocity profile of a shear layer, are presented. Specifically, an infinite acoustic medium, with constant density and sound speed, containing a free shear layer of infinite extent, is considered. The free shear layer is probed with a two‐dimensional plane wave, incident on the layer from outside, and it is assumed that the maximum Mach numbers for the flows in the shear layers are small, e.g., M=10−3 to 10−4 are typical values for oceanic shear layers. Then, the method of matched asymptotic expansions is employed to obtain an asymptotic expansion of the solution of the direct scattering problem as M→0 that is uniformly valid in the dimensionless wave number k and the angle α of the incident plane wave. It is found that the reflection coefficient of the scattered wave is proportional to the Fourier transform of the velocity profile of the free shear layer. This transform is inverted to obtain an asymptotic appro...