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.     

Coupled surface and grain boundary motion: a travelling wave solution

Existence and uniqueness are proven for a travelling wave solution for a problem in which motion by mean curvature is coupled with surface diffusion. This problem pertains to a bicrystal in a “quarter-loop” geometry in which one grain grows at the expense of the other, and the internal grain boundary between the two crystals contacts the exterior surface at a “groove root” or “tri-junction” where various balance laws hold. Far in front and behind the groove root the overall height of the bicrystal is assumed to be unperturbed. Whereas in a previous paper (Acta Mater. 51 (2003) 1981) a partially linearized formulation was considered for which explicit solutions could be found, here we treat the fully nonlinear problem. Employing an angle formulation and a scaled arc-length parameterization, we reduce the problem to the solution of a third order ODE with a jump condition at the origin. Existence is proven if m, the ratio of the exterior surface energy to the surface energy of the grain boundary, is less than about $\approx 0.92$. Uniqueness of these solutions is demonstrated within the class of single-valued solutions. A numerical comparison is made with the solution of the partially linearized formulation found earlier for the sake of illustration.     

Shock Wave Tracking for Hyperbolic Systems Exhibiting Local Linear Degeneracy

Extending the results of our previous study [2], we now investigate the propagation of interior shocks corresponding to the signaling problem of small-amplitude, high-frequency type. We derive a formula for the shock front and show that the previously constructed asymptotic solution is valid on both sides of this front. This solution is further distinguished to a higher order in which the effects of material inhomogeneity are accounted for. Moreover, if λ = λ( u , x) represents the eigenvalue under consideration, we show that the single-wave-mode boundary disturbance of [2] can lead only to a λ-shock. We also derive an entropy condition for the shock wave. As an application of our theory, the fluid-filled hyperelastic tube problem of [7] is further examined and an example calculation made in which we show that a compressive shock wave is generated at the shock-initiation point. This demonstration is effected as a particular example of the solution to a general bifurcation problem.     

Numerical inversion of Laplace transform by the method of asymptotic extension of the interval in dynamic viscoelasticity problems

Conclusions 1. A numerical investigation into the process of wave propagation in infinite and finite viscoelastic rods has been carried out by means of the method of asymptotic extension of the interval.2. It is established that use, in the calculations, of kernels of relaxation with weak singularity does not give rise to a stress jump at the wave front.3. The effect of parameters of the Rzhanitsyn kernel on the "erosion" of the wave front has been investigated. It is discovered that the instant of occurrence of a stress that is appreciably different from zero, at points far away from the beginning of the rod, is determined by the long-term modulus of elasticity of the material of the rod.4. The solution of the problem concerned with the propagation of a load impulse of finite duration shows a decrease of the maximum value of the stress in the rod, when the duration of the applied impulse decreases, and an intense change in the shape of the impulse while it propagates along the rod, as a consequence of wave dispersion.P. Stuchka Latvian State University, Riga. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 864–870, September–October, 1976.     

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Stability condition for strong shock waves in the problem of flow around an infinite plane wedge

We consider the thermodynamical equilibrium state flow of an inviscid non-heat-conducting gas flowing around a plane infinite wedge, and study the stationary solution to this problem–the so-called strong shock wave; the flow behind the shock front is subsonic.We find the solution to the linear analog of the original mixed problem, prove that the solution trace on the shock wave is the superposition of the direct and reflected waves, and (the main point) justify the Lyapunov asymptotical stability of the strong shock wave provided that the uniform Lopatinsky condition is fulfilled. The initial data have a compact support, and the solvability conditions occur.     

Study of the stability in the problem on flowing around a wedge. The case of strong wave

We consider the flow of an inviscid nonheatconducting gas in the thermodynamical equilibrium state around a plane infinite wedge and study the stationary solution to this problem, the so-called strong shock wave; the flow behind the shock front is subsonic.We find a solution to a mixed problem for a linear analog of the initial problem, prove that the solution trace on the shock wave is the superposition of direct and reflected waves, and, the main point, justify the Lyapunov asymptotical stability of the strong shock wave provided that the angle at the wedge vertex is small, the uniform Lopatinsky condition is fulfilled, the initial data have a compact support, and the solvability conditions take place if needed (their number depends on the class in which the generalized solution is found).     

A numerical algorithm for computing tsunami wave amplitudes

A numerical multistep algorithm for computing tsunami wave front amplitudes is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved with Godunov’s approach and a bicharacteristic method. A qualitative comparison of the two methods is done. A change of variables is made with the eikonal equation solution at the second step. At the last step, using an expansion of the fundamental solution to the shallow water equations in the new variables, we obtain a Cauchy problem of lesser dimension for the leading edge wave amplitude. The results of numerical experiments are presented.     

Bistable traveling waves around an obstacle

We consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity. The goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space ?^N. As a first step, we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle. This causes disturbance on the shape of the front, but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction, leaving the obstacle behind. Whether the recovery is uniform in space is shown to depend on the shape of the obstacle. © 2008 Wiley Periodicals, Inc.     

Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

In this work, we study the problem of mean-variance hedging with a random horizon T∧τ, where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.     

Dynamic reaction of a pseudo-ceramic half-space with a longitudinal shear crack to concentrated forces

The dynamic problem of electroelasticity for a piezoelectric half-space with tunnel cavities-cuts is examined. The problem is reduced to the solution of an integrodifferential singular equation in terms of the amplitude of the jump in displacements at the cut. Equations are obtained for the dynamic mechanial-stress intensity factor. The influence exerted by the curvature of the crack, its orientation, and the normalized wave number on the intensity factor is studied numerically.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 50–55, 1989.     

Focusing of shock waves in a highly viscous fluid

The method of combined asymptotic expansions is used to solve the problem of the focusing of a shock wave (in a weakly compressible medium of high viscosity. Asymptotic forms of the solution are constructed in a number of spatial zones. The focusing zone is described by its asymptotic form obtained by combining it with the solution corresponding to viscous geometrical acoustics. The reflection of a shock wave formed as a result of velocity jump near one of the foci of the ellipsoid of revolution is discussed as an example. Analytical relationships descrbing the focusing zone around the second focus are obtained. It is shown that at the focus itself the wave profile has an antisymmetric form, and the compression wave is followed by a rarefaction wave of the same form.     

Diffraction of a plane elastic wave by a wedge with a special form of boundary conditions

An exact solution of the antiplane problem of the diffraction of a plane elastic SH-wave with a step profile by a wedge is obtained. The stresses on the wedge sides are assumed to be proportional to a linear combination of the displacements, velocities and higher derivatives with respect to time of the displacements along the wedge axis. A solution of the problem is obtained using integral transformations with subsequent transformation using Cagniard's method. Solutions of the corresponding problems with boundary conditions of the Winkler and inertial types are considered. When a wave with a linear profile is incident on the wedge the stresses suffer a discontinuity of the second kind on the diffraction wave front; the same type of feature is observed in the problem with the inertial condition.     

Front propagation for discrete periodic monostable equations

This paper deals with front propagation for discrete periodic monostable equations. We show that there is a minimal wave speed such that a pulsating traveling front solution exists if and only if the wave speed is above this minimal speed. Moreover, in comparing with the continuous case, we prove the convergence of discretized minimal wave speeds to the continuous minimal wave speed.     

Diffraction on a semi-infinite screen of a plane nonstationary wave the amplitude of which increases linearly along the front set

An exact analytic solution of the two-variables nonstationary problem of diffraction on an ideal half-infinite screen is obtained by the Smirnov-Sobolev method. The source of the field is an incident plane acoustic wave with a δ-function profile. The wave amplitude is a linear function increasing along the front set. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 138–152.     

Applications of Distributional Derivatives to Wave Propagation

First the concepts of the surface distributions are explained.Thereafter the first and higher-order distributional derivativesare derived for a function of several variables. These conceptsare then used to study the propagation of wave fronts in continuummechanics. Explicit formulas for the jump relations for thefirst and second-order partial derivatives are obtained acrossthe wave front. A systematic method is then developed such thatthe algebraic work for the study of waves becomes very simple.A few illustrations in wave propagation are presented. At theend it is pointed out how this method can be effectively appliedin the derivation of the jump relations for single and double-layerpotentials in the theory of harmonic functions.     

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.     

On the wave front solution of a competition–diffusion system in population dynamics

In this paper, we consider a competition–diffusion system of two equations [Zhou and Pao, Asymptotic behavior of a competition–diffusion system in population dynamics, Nonlinear Anal. 6 (11) (1982) 1163–1184]. The diffusion coefficients of the system are not equal. We prove existence of a wave front solution which connects two nonzero restpoints of the system. In the proof, we rely essentially on the results of Kolmogorov et al. [A study of diffusion with increase in the quantity of matter, and its application to a biological problem, Bull. Moscow State Univ. 17 (1937) 1–72]. We also estimate the wave speed.