Contour integral solutions of the parabolic wave equation

We present a simple systematic construction and analysis of solutions of the two-dimensional parabolic wave equation that exhibit far-field localisation near a given algebraic plane curve. Our solutions are complex contour integral superpositions of elementary plane wave solutions with polynomial phase, the desired localisation being associated with the coalescence of saddle points. Our solutions provide a unified framework in which to describe some classical phenomena in two-dimensional high frequency wave propagation, including smooth and cusped caustics, whispering gallery and creeping waves, and tangent ray diffraction by a smooth boundary. We also study a subclass of solutions exhibiting localisation near a cubic parabola, and discuss their possible relevance to the study of the canonical inflection point problem governing the transition from whispering gallery waves to creeping waves.

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title: The boundary-layer method in diffraction problems
abstract: Translator's Introduction.- 1 Introduction.- 2 The Ray Method.- 2.1 The Starting Point: Formulas for the Scalar Case.- 2.2 The Eikonal Equation Rays Wave Fronts.- 2.3 Ray Coordinates.- 2.4 Fundamental Recurrence Formulas.- 2.5 Reflection of a Wave Given by a Ray  3 The Caustic Problem.- 3.1 Ray Expansion in the Neighborhood of a Caustic.- 3.2 The Analytic Nature of the Eikonal for Incoming and Outgoing Waves Near a Caustic.- 3.3 Ray Series in (s,n) and (s,?) Coordinates.- 3.4 The Field in a Boundary Layer Surrounding the Caustic.- 3.5 Fundamental Formulas.- 4 Whispering Gallery and Creeping Waves.- 4.1 Whispering Gallery Waves.- 4.2 Whispering Gallery Quasimodes.- 4.3 Creeping Waves.- 4.4 The Friedlander-Keller Solution (Diffraction Rays).- 4.5 Matching of Creeping Waves and Diffraction Rays.- 5 Oscillations Concentrated in the Neighborhood of a Ray (Gaussian Beams).- 5.1 Rays in the First Approximation.- 5.2 Derivation of the Boundary-Layer Equation.- 5.3 Solution of the System of Recurrence Equations for Vj.- 5.4 Stability of an Extremal Diameter of a Region.- 5.5 Quasimodes of the "Bouncing-Ball" Type in the First Approximation.- 5.6 Construction of Higher Approximations.- 6 Shortwave Diffraction from a Smooth Convex Body.- 6.1 The Parabolic Equation Method.- 6.2 The Analytic Nature of the Functions $$ {{\rm{\bar V}}_{\rm{j}}} $$ and Vj..- 6.3 The Boundary Layer in the Deep Shadow Zone.- 6.4 Continuation of the Solution from the Vicinity of the Point C into the Transition Region.- 6.5 Analytic Representation of the Incident Wave in the Neighborhood of the Limiting Ray.- 6.6 System of Recurrence Equations for the Neighborhood of the Limiting Ray.- 6.7 Extension of the Transition Region Formulas into the Neighborhood of the Limiting Ray.- 6.8 Formulas for the Field in the Shadow and in the Penumbra.- 7 The Problem of an Oscillating Point Source.- 7.1 The Ray Method for a Central Field of Rays.- 7.2 Expansion in the Transition Region.- 7.3 Expansions in the Neighborhood of the Origin.- 8 Survey of Literature.- References.

title: Thin-Layer Solutions of the Helmholtz and Related Equations
abstract: This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions.

title: Numerical solution of the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary
abstract: A method of numerically solving the problem on whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary is described, and results are presented of calculating the wave field in the case of the first and second whispering gallery waves.

title: Wave field in the caustic shadow in a neighborhood of an inflection point of the boundary
abstract: The problem of the propagation of whispering gallery waves in a neighborhood of an inflection point of the boundary is considered. It is shown that a caustic shadow zone occurs away from the boundary along the normal. The asymptotics of the wave field in the caustic shadow are obtained, and their geometric interpretation in terms of complex rays is given.

title: Direction diagram of radiation in the problem of an inflection point of the boundary
abstract: The coefficient of the first term of the ray asymptotics in this problem is computed for the first three whispering-gallery waves incident on a point of rectification of the boundary.

title: Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone
abstract: The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦→+∞ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t→−∞ The expansion constructed loses asymptotic character near the curve x=1/6 t3 (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.

title: Special function related to the concave–convex boundary problem of the diffraction theory
abstract: The concave–convex boundary problem of the diffraction theory is studied. It corresponds to the scattering of a whispering gallery mode on the point of inflection of the boundary. A new special function related to this boundary problem is introduced and its particular properties are discussed. This special function is defined as a contour integral on the complex plane and its behaviour in different domains of parameters is considered.

title: Whispering gallery waves in a neighborhood of a flat point of a concave boundary
abstract: The propagation of whispering gallery waves near a concave (from the side of the wave field) boundary having a flat point is studied. As in , the problem that arises for an equation of Schrodinger type is solved by the method of grids. The results of the computations are presented as shadow figures.

title: Whispering gallery waves in a neighborhood of a higher order aero of the curvature of the boundary
abstract: Let Ω⊂R 2 be a domain with a smooth boundary Γ. Suppose Ω is a media where some wave propagates with a speed, say 1 for simplicity. When Ω is strictly convex, there is a high frequency wave well known by the name of whispering gallery waves which propagates along Γ

title: Special Function Related to the Scattering of the Whispering Gallery Mode at a Point of Local Straightening
abstract: We consider the scattering of the whispering gallery mode at a point of local straightening of the boundary. A special function describing the main features of the wave field in the vicinity of the point of local straightening is constructed and examined.  12 titles.

title: Sound radiation from a cylindrical duct. Part 1. Ray structure of the duct modes and of the external field
abstract: This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debye's approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Keller's geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargill's meridional rays, is summarized.

title: Tangent ray diffraction and the Pekeris caret function
abstract: We study the classical problem of high frequency scattering of an incident plane wave by a smooth convex two-dimensional body. We present a new integral representation of the leading order solution in the "Fock region", i.e. the neighbourhood of a point of tangency between the incident rays and the scatterer boundary, from which the penumbra (light-shadow boundary) effects originate. The new representation, which is equivalent to the classical Fourier integral representation and its well-studied "forked contour" regularisation, reveals that the Pekeris caret function (sometimes referred to as a "Fock-type integral" or a "Fock scattering function"), a special function already known to describe the field in the penumbra, is also an intrinsic part of the solution in the inner Fock region. We also provide the correct interpretation of a divergent integral arising in the analysis of Tew et al. (Wave Motion 32, 2000), enabling the results of that paper to be used for quantitative calculations.

title: An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives
abstract: We present a code to compute the oscillatory cuspoid canonical integrals and their first order partial derivatives. The algorithm is based on the method of Connor and Curtis [ (1982) 1179–1190], in which the integration path along the real axis is replaced by a more convenient contour in the complex plane, rendering the oscillatory integrand more amenable to numerical quadrature. Our code is a modern implementation of this method, presented in a modular fashion as a Fortran 90 module containing the relevant subroutines. The code is robust, with the novel feature that the algorithm implements an adaptive contour procedure, choosing contours that avoid the violent oscillatory and exponential natures of the integrand and modifying its choice as necessary. In many cases, the subroutines are efficient and provide results of high accuracy. Plots and results for the Pearcey and swallowtail cuspoid integrals given in previous articles are reproduced and verified. Our subroutines significantly extend the range of application of this earlier work.

title: Scalar wave diffraction by tangent rays
abstract: Modern asymptotic methods are used to provide as complete as possible a description of the scattering of a high-frequency time-harmonic acoustic plane wave by a two-dimensional convex obstacle. The cases of infinite and finite scatterers are considered separately, and descriptions of the diffracted fields and the transition solutions valid across shadow boundaries are presented. For a finite scatterer, expressions are given for the directivity function (or angular variation) of the scattered field in the very far-field encompassing all angular directions; in particular this results in the prediction of a narrow range of angles for which the directivity is an order of magnitude lower than in general.

title: The Stokes set of the cusp diffraction catastrophe
abstract: Stokes and anti-Stokes lines are familiar in the asymptotic approximation of functions of a complex variable. The author generalises this notion and define the Stokes and anti-Stokes sets of a complex function of many (possibly complex) variables, defined by a diffraction-type integral. They are subsets of the Maxwell set of catastrophe theory, extended to complex variables. On the Stokes set the number of complex stationary points contributing to the integral changes by one, whereas on the caustic the number of real stationary points changes by two. Knowledge of the location of the Stokes set is essential to perform a full stationary phase analysis of a diffraction integral, and it imposes a constraint upon the positions of wavefront dislocations. For the canonical cusp diffraction catastrophe the author finds the explicit equation of the Stokes set, which is a broadened mirror image of the cusp caustic.

title: On the Theory of Complex Rays
abstract: The article surveys the application of complex-ray theory to the scalar Helmholtz equation in two dimensions. 
The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Sp{} in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterizations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex Wentzel--Kramers--Brilbuin expansion of these wavefields. 
Examples are given for several cases of physical importance.

title: A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: application to Pearcey's integral and its derivatives
abstract: A numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described. The method exploits Cauchy's theorem and Jordan's lemma to write the infinite integration path along different contours in the complex plane. The method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques. Application is made to Pearcey's integral P(x,y) and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem. Tables of P(x,y), delta P(x,y)/ delta x, delta P(x,y)/ delta y and the real zeros of P(x,y) are presented for the grid -8.0<or=x<or=8.0 and 0<or=y<or=8.0.