Justification of the Courant-Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

As is well known, two solutions of the problem of a supersonic stationary inviscid nonheatconducting gas flow onto a planar infinite wedge are theoretically possible: the solution with a strong shock (the flow speed behind the shock is subsonic) and the solution with a weak shock (the flow speed behind the shock is supersonic). Unlike the well-studied case of a strong shock that is generically unstable [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Study of the stability in the problem on flowing around a wedge. The case of strong wave, J. Math. Anal. Appl. 319 (2006) 248-277; A.M. Blokhin, D.L. Tkachev, Yu.Yu. Pashinin, Stability condition for strong shock waves in the problem of flow around an infinite plane wedge, Nonlinear Anal. Hybrid Syst. 2 (2008) 1-17], R. Courant and K.O. Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948] assumed that the solution with a weak shock is asymptotically stable by Lyapunov. Presentation of classical solution to the corresponding problem which is found in the present paper is the first step on the way to justification of Courant-Friedrichs hypothesis on linear level.     

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).     

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.     

The Nonlinear Initial-Boundary Value Problem and the Existence of Multi-Dimension al Shook Wave for Quasilinear Hyperbolic-Parabolic Coupled Systems

For the quasilinear hyperbolie-parabolio coupled system, the nonlinear initial- boundary value problem and the shook wave free boundary problem are considered. By linear iteration, the existence and uniqueness of the local H^m (m\geq [N+1/2]+4) solution are obtained under the assumption that for the fixed boundary problem, the boundary conditions are uniformly Lopatinski well-posed with respect to the hyperbolic and parabolic part, and for the free boundary problem, there exists a linear stable shock front structure. In particular, the local existence of the isothermal shock wave solution for radiative hydrodynamic eqations is proved.     

Solution of a matrix Wiener–Hopf equation connected with the plane wave diffraction by an impedance loaded parallel plate waveguide

A matrix Wiener–Hopf equation connected with a new canonical diffraction problem is solved explicitly. We consider the diffraction of a plane electromagnetic wave by an impedance loaded parallel plate waveguide formed by a two‐part impedance plane and a parallel perfectly conducting half‐plane. The representation of the solution to the boundary‐value problem in terms of Fourier integrals leads to a matrix Wiener–Hopf equation. The exact solution is obtained in terms of two infinite sets of unknown coefficients satisfying two infinite systems of linear algebraic equations. These systems are solved numerically and the influence of the parameters such as the waveguide spacing and the surface impedances of the two‐part plane on the diffraction phenomenon is shown graphically. Copyright © 2005 John Wiley & Sons, Ltd.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

On the properties of solutions of the harmonic Dirichlet problem in a two-dimensional domain with cuts

We consider the Dirichlet problem for the Laplace equation in a plane domain with smooth cuts of arbitrary form for the case in which the solution is not continuous at the endpoints of the cuts. We present a well-posed statement of the problem, prove the existence and uniqueness theorems for the classical solution, obtain an integral representation of the solution, and use it to analyze the properties of the solution. We show that, as a rule, the Dirichlet problem in this setting has no weak solutions, even though there exists a classical solution.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

Reflection of plane waves from a rigid wall and a free surface in a transverse isotropic medium

For the general solution of two-dimensional equations of dynamics of a transverse isotropic medium with the Carrier–Gassmann condition, we give a representation in terms of two resolvent functions satisfying two separate wave equations. The problem of reflection of plane waves from a rigid wall and a free surface is solved. The coefficients of reflection and transformation of the plane waves are found. These formulas yield a solution for isotropic media too. Some special cases are consideredwhere the shapes (amplitudes) of the reflectedwaves are not uniquely determined, but linearly related with the shape of the incident wave.     

An integral equation method for the electromagnetic scattering from a scatterer on an absorbing plane

Consider a time-harmonic electromagnetic plane wave incident on a scatterer on a grounded absorbing plane modelized as an infinite impedance plane. In this paper, a new integral representation formula is rigorously derived. Existence and uniqueness of weak solutions for the model problem are also established. The proof of existence is based on an extension of the Hodge decomposition technique to open boundaries. The results reported in this paper form a basis for numerical solutions of the electromagnetic scattering problem from a scatterer on an absorbing plane.     

Diffraction of a plane wave by an infinite elastic plate stiffened by a doubly periodic set of rigid ribs

The diffraction of a plane wave by an infinite elastic plate stiffened by a doubly periodic set of rigid ribs of moderate wave dimensions is studied. The problem is reduced to an infinite quasiregular system of linear algebraic equations, and their solution describes the amplitudes of the waves propagating from the plate into the fluid.     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 5–18.     

Higher Order Conditions for Weak Shocks: Modified Prandtl Relation

The classical equations for the jumps in the state variables across a planar shock wave in an inviscid flow ‐ which can be solved by starting with the Prandtl relation for the jump in the normal velocity ‐ have been applied locally in gas dynamics to a tangent plane of a curved shock. This tangent plane approximation has been validated for real fluids using matched asymptotics with a small parameter ϵ ≔ δ/l, where δ is the effective shock thickness and l is a typical radius of curvature of the shock; ϵ is of order 1/Re where Re is the Reynolds number based on l. The leading order inner solution in the scale δ yields the classical shock structure resolving the discontinuities across the shock in the larger scale l, while the O(ϵ) corrections to the classical shock conditions account for the shock structure, shock curvature and the flow field gradients behind and ahead of the shock. We prove that there is an analog (correction) of the classical Prandtl relation that shows that the first order correction to the tangent plane approximation is O(ϵ/Δ), where Δ denotes the scaled shock strength. Hence, the correction to the Prandtl relation is of paramount importance in the analysis of weak shocks.     

Torsional Oscillations of a Rigid Convex Inclusion Embedded in an Elastic Solid

The problem discussed in this paper concerns a rigid axi-symmetricbody of convex form embedded in an infinite isotropic elasticsolid. When the inclusion is set in motion by an impulsive torqueapplied about the axis of symmetry it executes a damped torsionaloscillation and generates a shear pulse in the surrounding elasticmaterial. A representation of the elastic wave field in theform of a progressing wave expansion is shown to lead to anintegral equation of Volterra type for the angular motion ofthe inclusion. The exact solution obtained by Chadwick &Trowbridge (1967) for the case of a spherical inclusion is thenre-derived and used in developing an approximate method of solutionof the more general problem described above. Detailed resultsare worked out for spheroidal inclusions of oblate and prolateforms and numerical results are presented and discussed.     

An exact far-field inversion for the Born approximation and plane wave decompositions for Hankel functions

We consider the Born approximation (representative for first-order approximations) of the scattering problem for the scalar Helroholtz equation with a fixed real-valued free-space wavenumber and a complex-valued compactly supported potential. The boundary condition is the Sommerfeld radiation condition. We derive an exact series-integral representation of the potential from the Fourier coefficients of its far-field pattern, suitable for discussion of the connected stability problem. Furthermore we stress the connection between this representation and some plane wave decompositions for Hankel functions. Without loss of generality we restrict ourselves to the case of two space dimensions.     

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.     

Instability of a Vortex Sheet Leaving a Semi-Infinite Plate

The growth of undulations along an infinite vortex sheet is a classical problem of stability theory. Here we modify that problem by including the effects of a boundary: the vortex sheet is assumed to leave a rigid semi-infinite plate and to undergo spatially growing undulations downstream. The usual solution for a doubly infinite sheet is corrected by the Wiener-Hopf technique to account for the presence of the plate. The correction depends sensitively on whether a Kutta condition is enforced at the trailing edge. Two Kutta conditions, called rectified and full, are suggested to apply depending on conditions in the unperturbed flow. In either case, the correction due to the plate becomes negligible half a wavelength downstream from the trailing edge.