The nature of the triple point singularity in the case of stationary reflection of weak shock waves

The structure of flow in the vicinity of a triple point in the problem of stationary irregular reflection of weak shock waves is numerically investigated within the framework of the Euler model, including the von Neumann paradox range. To improve the accuracy of the solution near singular points a new technology including a grid contracted toward the triple point and the discontinuity fitting is applied. It is shown that in the four-wave flow pattern the curvatures of the tangential discontinuity and the Mach front at the triple point are finite. The singularity is concentrated only in a sector between the reflected wave front and the expansion fan. When the three-wave flow pattern is realized, the curvatures of the tangential discontinuity and both wave fronts at the triple point are infinite. On the range of weak and moderate shock waves the logarithmic singularity in subsonic sectors near the triple point conserves up to transition to the regular reflection.     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

On the Von Neumann Paradox of Weak Mach Reflection*

Recent experimental and numerical studies of weak Mach reflections are examined. It is shown that the fundamental reason for the von Neumann paradox is that his theory of Mach reflection is based on the assumption that the flow downstream of the reflected wave and the Mach shock near the wave triple point is uniform. The assumption is shown to be valid for strong Mach reflection which agrees with experiment, but invalid for weak Mach reflection which does not agree with experiment. It is also shown that viscous effects are dominant when the incident shock is within about 100 mean free path lengths of the corner, but not otherwise. The analytical theory of the entire subsonic region supports these conclusions.     

Surface and interfacial waves in ionic crystals

The continuum linear theory of ionic crystals is applied to develop a two-dimensional eigenvalue problem in the Stroh formalism. An integral approach is exploited to study the occurrence of surface waves along a free boundary of the crystal. Dispersion relations are obtained by separating real and imaginary parts of the governing system and various boundary conditions are examined. The problem of interfacial waves along the separation boundary between two different crystals is also outlined. Numerical computations are performed for a centrosymmetric crystal (KCl) in order to evaluate bulk wave speeds, limiting speed of surface waves and solutions to the dispersion equations for different boundary conditions.     

非线性弹性介质中冲击波斜反射的研究

本文讨论了各向同性非线性超弹性介质在平面小应变下的冲击波斜反射问题。给出了本构关系、简单波解和冲击波解,并作为例子求解了入射冲击波在自由面的斜反射问题。     

Reflection of a spherical blast wave from a planar surface

Numerous authors have carried out rather extensive studies in the last twenty to thirty years of the problem of the interaction of shock and blast waves with obstacles in their paths. Owing to the complexity of the problem, they assumed certain limiting cases for the shock wave interactions in which the parameters behind the shock wave were usually taken to be constants. The first wave diffraction studies involving variable parameters behind the front were presented in [1, 2], wherein a development of the theory of “short waves” (blast waves at a substantial distance from the center of an explosion) and their reflection from a planar surface was given. The theory of short waves assumes that the jump in pressure at the wave front and the region over which the parameters vary are small. The problem concerning reflection of a blast wave from a surface was also considered in [3, 4], wherein a solution in the region behind the reflected wave was obtained at initial times. The initial stage in the reflection of a blast wave from a planar, cylindrical, or spherical surface (the one-dimensional case) was studied in [5]. In this paper we investigate the interaction of a spherical blast wave, resulting from a point explosion, with a planar surface; we consider both regular and non-regular reflection stages. In solving this problem we use S. L. Godunov's finite-difference method. We obtain numerical solutions for various values of the shock strength at the instant of its encounter with the surface. We present the pressure fields in the flow regions, the pressure distribution over the surface at various instants of time, and the trajectories of the triple point. The parameter values at the front of the reflected wave are compared with results obtained from the theory of regular reflection of shock waves.     

A fluid-saturated porous medium under the action of a moving concentrated load

The problem of motion of a concentrated load along the surface of a fluid-saturated porous medium is studied for a subsonic range of speeds. An analytical solution is found. It is shown that there exists a critical speed equal to the speed of the Rayleigh-type surface waves in a porous elastic medium. If this critical speed is exceeded, then the behavior of the solution and the free surface shape are changed. The free surface shape is analyzed at different speeds.     

Prediction of wave pattern and wave resistance of surface piercing bodies by a boundary element method

An iterative boundary element method, which was originally developed for both two‐ and three‐dimensional cavitating hydrofoils moving steadily under a free surface, is modified and extended to predict the wave pattern and wave resistance of surface piercing bodies, such as ship hulls and vertical struts. The iterative nonlinear method, which is based on the Green theorem, allows the separation of the surface piercing body problem and the free‐surface problem. The free‐surface problem is also separated into two parts; namely, left and right (with respect to x axis) free‐surface problems. Those all (three) problems are solved separately, with the effects of one on the other being accounted for in an iterative manner. The wetted surface of the body (ship hull or strut, including cavity surface if exists) and the left and right parts with respect to x axis of free surface are modelled with constant strength dipole and constant strength source panels. In order to prevent upstream waves, the source strengths from some distance in front of the body to the end of the truncated upstream boundary are enforced to be zero. No radiation condition is enforced for downstream and transverse boundaries. A transverse wave cut technique is used for the calculation of wave resistance. The method is first applied to a point source and a three‐dimensional submerged cavitating hydrofoil to validate the method and a Wigley hull and a vertical strut to compare the results with those of experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

On theory of generalized thermoelastic solids with voids and diffusion

Governing equations of thermoelastic diffusion material with voids are modified with the help of Lord and Shulman theory of generalized thermoelasticity. These governing equations are then solved in two-dimension to show the existence of four coupled longitudinal waves and a shear wave. The complex absolute values of the speeds of the coupled longitudinal waves are computed numerically against the frequency for Magnesium material. The reflection of these plane waves from a stress free thermally insulated boundary is also studied, where the dependence of the reflection coefficients on angle of incidence is shown graphically for the incidence of coupled longitudinal wave only. The speeds and reflection coefficients of plane waves are also computed numerically in the absence of voids and diffusion parameters, which are shown graphically to observe the effects of voids and diffusion.     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

Reflection of plane waves under initial stresses at a free surface

The paper deals with the phenomenon of reflection of plane elastic waves in a free surface when the medium is initially stressed. It has been shown analytically that the reflected P and SV waves depend on initial stresses present in the medium. The numerical values of reflection coefficients for different initial stresses and the angle of incidence have been calculated by the Computer I.C.L. 1901-A and the results are given in the form of graphs. Many interesting results are found in the paper which are not seen in an initially stress-free medium.     

Elastic waves in an electro-microelastic solid

The present study is concerned with the wave propagation in an electro-microelastic solid. The reflection phenomenon of plane elastic waves from a stress free plane boundary of an electro-microelastic solid half-space is studied. The condition and the range of frequency for the existence of elastic waves in an infinite electro-microelastic body are investigated. The constitutive relations and the field equations for an electro-microelastic solid are stemmed from the Eringen’s theory of microstretch elasticity with electromagnetic interactions. Amplitude ratios and energy ratios of various reflected waves are presented when an elastic wave is made incident obliquely at the stress free plane boundary of an electro-microelastic solid half-space. It has been verified that there is no dissipation of energy at the boundary surface during reflection. Numerical computations are performed for a specific model to calculate the phase speeds, amplitude ratios and energy ratios, and the results obtained are depicted graphically. The effect of elastic parameter corresponding to micro-stretch is noticed on reflection coefficients, in particular. Results of Parfitt and Eringen [Parfitt, V.R., Eringen, A.C., 1969. Reflection of plane waves from a flat boundary of a micropolar elastic half-space. J. Acoust. Soc. Am. 45, 1258–1272] have also been reduced as a special case from the present formulation.     

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.     

Wave Reflection in Triclinic Crystalline Medium

In this paper, the reflection of a plane wave at a traction free boundary of a half -space composed of triclinic crystalline material is considered. It is shown that an incident plane wave generates three plane waves, namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) waves governed by the propagation condition involving the acoustic tensor. A simple procedure is presented for the calculation of all the three phase velocities of these waves. It is demonstrated that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. A procedure is established for the calculation of the amplitude vector in terms of the phase velocity, the propagation vector, and the stiffness coefficients of the medium. Closed form solutions are obtained for the reflection coefficients of qP, qSV and qSH waves. Using the parameters of Vosges sandstone exhibiting triclinic symmetry, the graphical representations of the reflection coefficients due to an incident qP wave are given. It is observed that, in triclinic medium, the reflection coefficients are significantly different from those in an isotropic medium.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

Critical angles and grazing incidence: the breakdown of regular shock reflection in solids

The regular pattern for shock reflection in a nonlinear hyperelastic solid is a centered array of shocks and simple wave fans. As the angle of incidence approaches grazing incidence or a critical angle, the reflection pattern overtakes the incident wave until finally the regular pattern can no longer be sustained. By expanding the reflection solution in powers of amplitude about the linear reflection solution, it is possible to develop a procedure to solve the reflection problem for weak but finite shocks for any material symmetry. Explicit solutions can be exhibited for isotropic materials, including relationships between amplitude and limiting angle for various boundary conditions and incident waves. Some problems require consideration of a nonlinear boundary condition even in the first approximation. Typically, these cases lead to considerable amplification in the leading reflected wave.     

Hypersonic Viscous-Gas Flow past an Axisymmetric Blunt Body with a Groove in Its Frontal Surface

The hypersonic perfect-gas flow past the Mars Express Probe with a narrow annular groove in the conical surface is studied on the basis of a numerical analysis of the Navier-Stokes equations. Since the cone angle is greater than the limiting value, the narrow groove is located in a subsonic flow. The calculations are carried out with reference to a characteristic Martian atmosphere entry trajectory point (M=23.95) for various heat transfer conditions on the body surface. It is shown that the problem has two solutions (high- and low-temperature) for a groove with a thermally insulated surface.