Matrix Wiener-Hopf factorisation II

A direct method is described for effecting the explicit Wiener-Hopf factorisation of a class of (2 x 2}—matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner. The factorisation technique is applied to the matrices that arise from two problems in diffraction theory, thus permitting these diffraction problems to be solved in closed form (at least in principle)

Spherical wave diffraction by a rational wedge

In this paper we derive a new expression for the point source Green's function for the reduced wave equation, valid in an angular sector, whoseangle is equal to a rational multiple of . This Green's function is used to find new expressions for the field produced by the diffraction of a spherical wave source by a wedge, whose angle can be expressed as a rational multiple of . The expressions obtained are in the form of source terms and real integrals which represent the diffracted field. The general resultobtained includes as special cases Macdonald’s solution for diffraction by ahalf plane; a solution for the problem of diffraction by a wedge of open angle 3/2, i.e. a corner; a new representation for the solution of the problemof diffraction by a mixed soft/hard half plane; and a new representation for the point source Green's function for Laplace's equation, valid in an angularsector whose angle is equal to a rational multiple of

A trifurcated waveguide problem

We consider the diffraction of the dominant wave mode which propagates out of the mouth of a semi-infinite waveguide made of a soft and hard half plane. This semi-infinite waveguide is symmetrically located inside an infinite waveguide whose infinite plates are soft and hard. The whole system constitutes a trifurcated waveguide. A closed form solution of the resulting matrix Wiener-Hopf equation is obtained

A trifurcated waveguide problem II

We consider the diffraction of the dominant plane wave mode which propagates out of the end of a semi-infinite waveguide. This waveguide is made up of a half plane with a Dirichlet boundary condition and a half plane with a Neumann boundary condition. This semi-infinite waveguide is symmetrically located inside another infinite waveguide one of whose infinite plates has a Dirichlet and the other a Neumann boundary condition. A closed form solution of the resulting matrix Weiner-Hopf equation is obtained

A note on Wiener Hopf matrix factorisation

In this paper the most general class of (2x2) - matrices is determined, which permit a Wiener-Hopf factorisation by the procedure of Rawlins and Williams [1]. According to this procedure, the factorisation problem is reduced to a matrix Hilbert problem on a half-line, where the matrix involved in the Hilbert problem is required to have zero diagonal elements

Plane wave diffraction by a rational wedge

In this paper new expressions for the acoustic field produced when a plane wave source of sound is diffracted by a soft, hard or mixed soft/hard wedge whose angle can be expressed as a rational multiple of are given. The solution is expressed in terms of geometrical acoustic source terms and real integrals which represent the diffracted field. The expressions are in a form which allows easy calculation of the acoustic field. Uniformly valid ex-pressions for the far field are also given for all values of the angular variable. The general result obtained includes as special cases, Sommerfeld' s solution for diffraction by a half plane, Reiche’s result for the diffraction by a right angled wedge, and a new representation for the solution of the problem of diffraction by a mixed soft/hard half plane

Matrix Wiener-Hopf factorisation

A direct method is described for effecting the explicit Wiener-Hopf factorisation of a class of (2 x 2)-matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner. The factorisation technique is applied to the matrices that arise from two problems in diffraction theory, thus permitting these diffraction problems to be solved in closed form (at least in principle)

A note on the Euler-Maclaurin Sum formula

In this note we give a real variable approach for calculating the constant term that arises in the application of the Euler-Maclaurin expansion for a special class of series of the form Σ =n1rf(r).asn∞→ In particular the method is used to derive the approximate summation of the expression where ℓ is a non negative integer. Σ=n1rnr,rl