First arriving signals in layered waveguides. An approach based on dispersion diagrams

The first arriving signal (FAS) in a layered waveguide is investigated. It is well known that the velocity of such a signal is close to the velocity of the fastest medium in the waveguide, and it may be bigger than the fastest group velocity given by the dispersion diagram of the waveguide. Usually the FAS pulse decays with the propagation distance. A model layered waveguide is studied in the paper. It is shown that the FAS is associated with the pseudo-branch structure of the dispersion diagram. The velocity is determined by the slope of the pseudo-branch. The decay is exponential and it depends on the structure of the pseudo-branch. A new type of phase integral is introduced for FAS.Comment: 27 pages, 12 figure

An ODE--based approach to some Riemann--Hilbert problems motivated by wave diffraction

A novel approach to Riemann--Hilbert problems of particular class is introduced. The approach is applicable to problems in which the multiplicative jump is set on a half-line. Such problems are linked to some Wiener--Hopf problems motivated by diffraction theory. The new approach is based on ordinary differential equations: the Riemann--Hilbert problem is reduced to finding a coefficient of an ordinary differential equation and solving this equation. The new method leads to an efficient numerical algorithm and opens a road to new asymptotical and analytical advances.Comment: 17 pages, 7 figure

Wiener-Hopf matrix factorization using ordinary differential equations in the commutative case

A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a Riemann-Hilbert problem using the Hurd's method. Secondly, the Riemann-Hilbert problem is embedded into a family of Riemann-Hilbert problems indexed by a variable b taking values on a half-line. A linear ordinary differential equation (ODE1) with respect to b is derived. The coefficient of this equation remains unknown at this step. Finally, the coefficient of the ODE1 is computed. For this, it is proven that it obeys a non-linear ordinary differential equation (ODE2) on a half-line. Thus, the numerical procedure of matrix factorization becomes reduced to two runs of solving of ordinary differential equations on a half-line: first ODE2 for the coefficient of ODE1, and then ODE1 for the unknown function. The efficiency of the new method is demonstrated on some examples.Comment: 26 pages, 4 figure

Criteria for commutative factorization of a class of algebraic matrices

The problem of matrix factorization motivated by diffraction or elasticity is studied. A powerful tool for analyzing its solutions is introduced, namely analytical continuation formulae are derived. Necessary condition for commutative factorization is found for a class of "balanced" matrices. Together with Moiseyev's method and Hurd's idea, this gives a description of the class of commutatively solvable matrices. As a result, a simple analytical procedure is described, providing an answer, whether a given matrix is commutatively factorizable or not

Transient phenomena in a three-layer waveguide and the analytical structure of the dispersion diagram

Excitation of waves in a three-layer acoustic wavegide is studied. The wave field is presented as a sum of integrals. The summation is held over all waveguide modes. The integration is performed over the temporal frequency axis. The dispersion diagram of the waveguide is analytically continued, and the integral is transformed by deformation of the integration contour into the domain of complex frequencies. As the result, the expression for the fast components of the signal (i.e. for the transient fields) is simplified. The structure of the Riemann surface of the dispersion diagram of the waveguide is studied. For this, a family of auxiliary problems indexed by the parameters describing the links between layers is introduced. The family depends on the linking parameters analytically, and the limiting case of weak links can be solved analytically.Comment: 41 pages, 19 figure

Diffraction by an elongated body of revolution. A boundary integral equation based on the parabolic equation

A problem of diffraction by an elongated body of revolution is studied. The incident wave falls along the axis. The wavelength is small comparatively to the dimensions of the body. The parabolic equation of the diffraction theory is used to describe the diffraction process. A boundary integral equation is derived. The integral equation is solved analytically and by iterations for diffraction by a cone.Comment: 23 pages, 5 figure

Diffraction by a quarter-plane. Analytical continuation of spectral functions

The problem of diffraction by a Dirichlet quarter-plane (a flat cone) in a 3D space is studied. The Wiener-Hopf equation for this case is derived and involves two unknown (spectral) functions depending on two complex variables. The aim of the present work is to build an analytical continuation of these functions onto a well-described Riemann manifold and to study their behaviour and singularities on this manifold. In order to do so, integral formulae for analytical continuation of the spectral functions are derived and used. It is shown that the Wiener-Hopf problem can be reformulated using the concept of additive crossing of branch lines introduced in the paper. Both the integral formulae and the additive crossing reformulation are novel and represent the main results of this work

Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide

The problem of diffraction of a waveguide mode by a thin Neumann screen is considered. The incident mode is assumed to have frequency close to the cut-off. The problem is reduced to a propagation problem on a branched surface and then is considered in the parabolic approximation. Using the embedding formula approach, the reflection and transmission coefficients are expressed through the directivities of the edge Green's function of the propagation problem. The asymptotics of the directivities of the edge Green's functions are constructed for the case of small gaps between the screen and the walls of the waveguide. As the result, the reflection and transmission coefficients are found. The validity of known asymptotics of these coefficients is studied.Comment: 42 pages, 10 figure

Diffraction by an impedance strip II. Solving Riemann-Hilbert problems by OE-equation method

The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the problem was reduced to two matrix Riemann-Hilbert problem. Here the Riemann-Hilbert problems are solved with the help of a novel method of OE-equations. Each Riemann-Hilbert problem is embedded into a family of similar problems with the same coefficient and growth condition, but with some other cuts. The family is indexed by an artificial parameter. It is proven that the dependence of the solution on this parameter can be described by a simple ordinary differential equation (ODE1). The boundary conditions for this equation are known and the inverse problem of reconstruction of the coefficient of ODE1 from the boundary conditions is formulated. This problem is called the OE-equation. The OE-equation is solved by a simple numerical algorithm.Comment: 21 pages, 8 figure

Diffraction by a Dirichlet right angle on a discrete planar lattice

A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of [1]