The Sommerfeld half-plane problem revisited,IV: Variations on a theme of Carlson and Heins

A plane wave is incident upon an infinite set of equally spaced, semi-infinite parallel and staggered plates. The boundary conditions on the plates alternate between the Dirichlet and Neumann ones. This problem is formulated as a pair of coupled Wiener-Hopf integral equations and solved by a method proposed by A. E. Heins in 1950. For the case of specular reflection, that is, a single reflected plane wave, the magnitudes of the reflection coefficient and the transmission coefficients are determined.     

Wave scattering by soft–hard three spaced waveguide

In this work we consider the sound radiation of a fundamental plane wave mode from a semi-infinite soft–hard duct. This duct is symmetrically located inside an infinite duct. This infinite waveguide consist of soft and hard plates. The whole system constitutes a three spaced waveguide. A closed form solution is obtained by using eigenfunction expansion matching method. This particular problem has been solved previously by Rawlins in closed form but without numerical work. Here the numerical results for reflection coefficient are given when the lowest mode propagates out from the semi-infinite duct. At the end we give comparison to both methods.     

The sommerfeld half-plane problem revisited III: Parallel plate media with mixed boundary conditions

A plane wave is normally incident upon an infinite stack of equally spaced parallel plates which are semi-infinite and not staggered. The plates satisfy the so-called “Rawlins” boundary conditions. This problem is formulated as a pair of simultaneous integral equations of the Wiener-Hopf type and solved by a method proposed by A. E. Heins in 1950.     

Vibrations of the surface of an elastic double-layered half-space with a periodic system of cracks

The two-dimensional problem of the normal incidence of a plane transverse wave from the far field on to the free surface of an elastic double-layered half-space, comprising a homogeneous layer attached to a semi-infinite base of a different elastic material, is considered. At the boundary between the two media there is a system of plane cracks, arranged periodically along the separation line, which models the fracture zone at the interface between dense solid rock and soft sedimentary rock. The effect of the fractures on the transmission of a transverse seismic wave generated by a deep-focus earthquake, and of the type of vibrations of the free surface of the ground that result, is studied. It is difficult to predict whether the seismic wave is strengthened or weakened by the fracture zone. The effect of the system of cracks on vibrations of the free surface largely depends on the physical and geometrical parameters and, primarily, on the vibration frequencies.     

2D Green’s functions for semi-infinite orthotropic thermoelastic plane

Based on the 2D general solutions of orthotropic thermoelastic material, the Green’s function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic plane is constructed by three newly introduced harmonic functions. All components of coupled field in semi-infinite thermoelastic plane are expressed in terms of elementary functions. Numerical results are given graphically by contours.     

Diffraction of a non-stationary linearly inhomogeneous acoustic wave which slides off a semi-infinite soft screen

An exact analytical solution of a new non-stationary scalar diffraction problem is obtained and analysed. A plane acoustic wave with a profile in the form of a delta function propagates along a semi-infinite soft screen. The wave amplitude varies linearly along the wave front. After reaching the end of the screen it “slides” off the screen, generating a diffraction field. A special modification of the Smirnov–Sobolev method is used to find this field. The solution is obtained in the form of an elementary function. It is shown that the sliding wave excites a travelling perturbation that is unlimited along the length of the screen. A similar phenomenon obviously also occurs when elastic waves slide from a cut (crack), which must be taken into account, in particular, in fracture theory.     

Aerodynamic Sound due to a Point Source near a Half-plane

In this paper we extend Jones's analysis of the diffractionof sound in three dimensions by a semi-infinite plane with aplane vortex sheet attached in the two cases when the wave equationis in the form for still air and when convection is present.It is found that in so far as the moving medium is concernedthe imposition or otherwise of the Kutta-Joukowski conditiondoes not have much influence on the scattered field away fromthe diffracting plane; when the source is near the edge thefield has the same directionality and order of magnitude. Onthe other hand, near the wake the Kutta-Joukowski conditionproduces a much stronger field than elsewhere even when thesource is not near the edge. We also conclude that the samephenomenon occurs for arbitrary sources and not just for theline source discussed by Jones.     

Diffraction in a cylindrically orthotropic elastic solid containing a stress free crack

A cylindrically orthotropic elastic solid is excited by a point impulsive body force. The solid contains a semi-infinite stress free crack. The resulting anti-plane wave motion problem has been solved in the form of a finite series representing the incident and reflected pulses plus an integral representing the diffraction pulse. The series part of the solution has been previously treated. In the present investigation the diffraction integral is integrated when λ (which measures the anisotropy of the solid) is an odd integer number. The diffraction integral is also integrated when λ is half an odd integer, for the special case in which the source lies in the plane of the crack and parallel to the crack edge. The displacement jump across the circular diffraction wave front is given for unrestricted (positive) values of λ.     

Diffraction in a cylindrically orthotropic elastic solid containing a stress free crack

A cylindrically orthotropic elastic solid is excited by a point impulsive body force. The solid contains a semi-infinite stress free crack. The resulting anti-plane wave motion problem has been solved in the form of a finite series representing the incident and reflected pulses plus an integral representing the diffraction pulse. The series part of the solution has been previously treated. In the present investigation the diffraction integral is integrated when λ (which measures the anisotropy of the solid) is an odd integer number. The diffraction integral is also integrated when λ is half an odd integer, for the special case in which the source lies in the plane of the crack and parallel to the crack edge. The displacement jump across the circular diffraction wave front is given for unrestricted (positive) values of λ.     

Nonlinear Propagation of Wave Packets for Near-Critical Wave Numbers in a Liquid that Is Piecewise Nonuniform with Depth

This is a study of the evolution of finite amplitude wave packets at the contact surface between a semi-infinite liquid medium and a liquid located above it near a solid lid. The limiting case of near-critical wave numbers for propagation of wave packets in the liquid system is discussed. Perturbation expansions are obtained for the deviation of the contact surface from its unperturbed position and for wave numbers near the critical value. This is compared with the solution for the analogous limiting cases of one or two semi-infinite liquids. A version in which the wave numbers are far from critical is examined.     

Diffraction of the Plane Wave on a Periodic Grating Composed of Thin Chiral Half-Planes

The diffraction of the normally incident plane wave by a grating consisting of thin semi-infinite chiral slabs is considered. The chiral slabs are simulated by appropriate transition boundary conditions. The problem is simplified by decoupling the and components with the help of a similarity transformation. Then the problem is reduced to scalar Riemann–Hilbert problems and is solved in explicit form. An expansion of the diffracted field in plane waves is obtained, and numerical results are discussed. Bibliography: 9 titles.     

Scattering of antiplane shear wave by a propagating crack at the interface of two dissimilar elastic media

An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.     

波系在液体中的弹性管梁上的反射和辐射

本文研究入射波系在液体中的半无限弹性管梁的开口端的反射和辐射问题,此波系由管梁上的挠曲波和管内、管外液体中相应的表面波(声波)所组成.利用Fourier变换,将这个半无限问题严格地归结为求解Wiener-Hopf型方程.然后将液体和管梁的密度比作为小参数,用摄动法求近似解.文章着重研究了反射系数的计算,还给出了远场的辐射型式曲线.     

An eigenvalue search method using the Orr–Sommerfeld equation for shear flow

A physically-based computational technique was investigated which is intended to estimate an initial guess for complex values of the wavenumber of a disturbance leading to the solution of the fourth-order Orr–Sommerfeld (O–S) equation. The complex wavenumbers, or eigenvalues, were associated with the stability characteristics of a semi-infinite shear flow represented by a hyperbolic-tangent function. This study was devoted to the examination of unstable flow assuming a spatially growing disturbance and is predicated on the fact that flow instability is correlated with elevated levels of perturbation kinetic energy per unit mass. A MATLAB computer program was developed such that the computational domain was selected to be in quadrant IV, where the real part of the wavenumber is positive and the imaginary part is negative to establish the conditions for unstable flow. For a given Reynolds number and disturbance wave speed, the perturbation kinetic energy per unit mass was computed at various node points in the selected subdomain of the complex plane. The initial guess for the complex wavenumber to start the solution process was assumed to be associated with the highest calculated perturbation kinetic energy per unit mass. Once the initial guess had been approximated, it was used to obtain the solution to the O–S equation by performing a Runge–Kutta integration scheme that computationally marched from the far field region in the shear layer down to the lower solid boundary. Results compared favorably with the stability characteristics obtained from an earlier study for semi-infinite Blasius flow over a flat boundary.     

Acoustic Radiation from a Circular Cylinder in a Subsonic Stream

The problem of acoustic radiation from a semi-infinite circularcylinder is discussed when a plane wave is propagated out ofthe mouth of the cylinder at which a vortex layer is attached.The effects of convection are included at low Mach numbers anda comparision is made of the radiation in the far field forthe case of propagation in (a) still air, (b) moving air withouta vortex layer and (c) moving air with a vortex layer. It isshown that in general the magnitude of the sound is much greaterdownstream than it is upstream. This directionality becomesmore pronounced in case (b) than in case (a) whilst in the presenceof a vortex layer the field is strengthened in the immediateregion downstream and weakened elsewhere.     

A relaxed cutting plane method for semi-infinite semi-definite programming

In this paper, we develop two discretization algorithms with a cutting plane scheme for solving combined semi-infinite and semi-definite programming problems, i.e., a general algorithm when the parameter set is a compact set and a typical algorithm when the parameter set is a box set in the m-dimensional space. We prove that the accumulation point of the sequence points generated by the two algorithms is an optimal solution of the combined semi-infinite and semi-definite programming problem under suitable assumption conditions. Two examples are given to illustrate the effectiveness of the typical algorithm.     

Propagation of perturbations in fluids excited by moving sources

A large series of A.A. Dorodnicyn’s works deals with rigorous mathematical formulations and development of efficient research techniques for mathematical models used in inhomogeneous fluid dynamics. Numerous problems he studied in these directions are closely related to stratified fluid dynamics, which were addressed in a series of works having been published in this journal by this paper’s authors and their coauthors since 1980. This paper describes the results of a series of works analyzing the propagation of small perturbations in various stratified and/or uniformly rotating inviscid fluids. It is assumed that each of the fluids either occupies an unbounded lower half-space with a free surface or is a semi-infinite two-component fluid layer. The perturbations are excited by a moving source specified as a periodic plane wave traveling along the interface of the fluids. Problems for five mathematical fluid models are formulated, their explicit analytical solutions are constructed, and their existence and uniqueness are discussed. The asymptotics of the solution as t → +∞ are studied, and the long-time wave patterns developing in five fluid models are compared.     

Two-dimensional Green’s functions for semi-infinite isotropic thermoelastic plane

Based on the two-dimensional steady-state governing equations of isotropic thermoelastic material and the compact general solution expressed in three harmonic functions, the corresponding three harmonic functions contain nine undetermined constants are constructed for a line heat source applied in the interior of a semi-infinite thermoelastic plane. All components of thermoelastic field in the semi-infinite plane can be derived by substituting the harmonic functions into the general solution. And the undetermined constants can be obtained by the compatibility conditions, equilibrium conditions and the different boundary conditions for extended Mindlin problem and extended Lorentz problem. Thus, the Green’s functions in above two cases are obtained, and the numerical results are given graphically by contours.     

Aerodynamic Sound Due to a Source Near a Half-Plane

The effect of applying a Kutta-Joukowski condition at the edgeof a semi-infinite plane which is generating noise in a turbulentfluid at low Mach numbers is examined. It is found that in somecircumstances the noise is increased and the intensity of thedistant sound field may depend upon the third power of a typicalfluid velocity. When the sound field is convected the orders of magnitude ofthe acoustic far field are the same whether or not the Kutta-Joukowskicondition is applied, provided that the point of observationis not near the wake. Near the wake there is an acoustic "surface"wave which is much stronger than the distant field elsewhere.     

On the connection of the superposition and homogeneous-solutions methods in wave propagation problems in a semi-infinite rigidly-clamped elastic layer

The process of harmonic wave propagation is investigated in a semi-infinite rigidly-clamped elastic layer. An analytic solution of the problem is obtained by the superposition method. The wave field expansion in the form of a normal mode series for a corresponding infinite waveguide is established. According to residue theory, the explicit form of the expansion coefficients is established with physical requirements of the radiation conditions taken into account.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 3–10, 1988.