Nonlinear theory of the compressible gas flows over a nonuniform boundary in the gravitational field in the shallow-water approximation

A set of equations is derived for the motion of a compressible ideal gas over a nonuniform boundary in the gravitational field in the shallow-water approximation. Classical simple waves are shown not to be the solutions to this set of equations. Generalized simple waves are found to exist only in the case of a linear underlying-surface profile. All continuous and discontinuous solutions are obtained in an explicit form for the case of the boundary in the form of an inclined plane, and an analytical solution is found for the problem of the decay of an arbitrary discontinuity. This solution consists of four wave configurations. Necessary and sufficient conditions are determined for the existence of each configuration.     

On the Tidal Motion Around the Earth Complicated by the Circular Geometry of the Ocean's Shape Without Coriolis Forces

The Cauchy–Poisson free boundary problem on the stationary motion of a perfect incompressible fluid circulating around the Earth is considered in this paper. Rotation plays a significant role in the early stages of the formation of solitary waves. However, these effects are less important on the solitary waves once they are formed. Therefore, for simplicity, rotation is not included for these simulations. The main concern is to find the inverse conformal mapping of the unknown free boundary in the hodograph plane onto some fixed mapping in the physical domain. The approximate solution to the problem is derived as the application of such a method. The behaviour of tidal waves around the Earth is discussed. It is shown that one of the features of the positively curved bottom is that the problem admits two different higher-order systems of shallow water equations, while the classical problem for the flat bottom admits only one system.     

Electromagnetic waves in uniaxial crystals with metallized boundaries: Mode conversion,pure reflections,and bulk polaritons

A theory is constructed for the reflection of plane electromagnetic waves in uniaxial crystals with a positive definite permittivity tensor and an arbitrarily oriented metallized boundary. The problem is solved both for general-position orientations corresponding to three-partial reflection and for special conditions allowing two-partial reflections: mode conversions when the incident and reflected waves belong to different sheets of the refraction surface and “pure” reflections when both waves belong to the same sheet. The space of pure reflections is shown to be formed by two types of optical-axis orientations: arbitrary directions in the plane of the crystal surface and in the plane of incidence. The configurations of the conversion surface for optically positive and negative crystals are investigated. A subspace of pure reflections that transform into one-partial bulk polaritons with the energy flux parallel to the surface at grazing incidence has been found. The domain of existence of such bulk eigenmodes is bounded by two “lines” of solutions. These are any directions along the boundary containing the optical axis for ordinary polaritons and the direction along the projection of the optical axis onto the surface at an arbitrary orientation of the axis with respect to the boundary for extraordinary polaritons.     

Theoretical and numerical calculations for the time-averaged acoustic force and torque acting on a rigid cylinder of arbitrary size in a low viscosity fluid

In this paper, theoretical calculations as well as numerical simulations are performed for the time-averaged acoustic force and torque on a rigid cylinder of arbitrary size in a fluid with low viscosity, i.e., the acoustic boundary layer is thin compared to the cylinder radius. An exact analytical solution and its approximation are proposed in the form of an infinite series including Bessel functions. These solutions can be evaluated easily by a mathematical software package such as mathematica and matlab. Three types of incident waves, plane traveling wave, plane standing wave, and dual orthogonal standing waves, are investigated in detail. It is found that for a small particle, the viscous effects for an incident standing wave may be neglected but those for an incident traveling wave are notable. A nonzero viscous torque is experienced by the rigid cylinder when subjected to dual orthogonal standing waves with a phase shift even when the cylinder is located at equilibrium positions without imposed acoustic forces. Furthermore, numerical simulations are carried out based on the FVM algorithm to verify the proposed theoretical formulas. The theoretical results and the numerical ones agree with each other very well in all the cases considered.     

Long wave run-up on random beaches

The estimation of the maximum wave run-up height is a problem of practical importance. Most of the analytical and numerical studies are limited to a constant slope plain shore and to the classical nonlinear shallow water equations. However, in nature the shore is characterized by some roughness. In order to take into account the effects of the bottom rugosity, various ad hoc friction terms are usually used. In this Letter, we study the effect of the roughness of the bottom on the maximum run-up height. A stochastic model is proposed to describe the bottom irregularity, and its effect is quantified by using Monte Carlo simulations. For the discretization of the nonlinear shallow water equations, we employ modern finite volume schemes. Moreover, the results of the random bottom model are compared with the more conventional approaches.     

Simulation and synthesis of elastic wave absorbers at the boundary of a rigid body: Incidence of longitudinal waves

Results of a computer simulation of an impedance absorber for longitudinal plane elastic waves incident on a free boundary of a rigid body are presented. The absorbing elements are mechanical resonators (of the elasticity-mass type) with two degrees of freedom and, hence, with two resonance frequencies, which correspond to the normal and tangential oscillations of the resonator. Formulas for calculating the absorber efficiency as a function of frequency and angle of incidence of longitudinal waves are derived with allowance for the absorption of both longitudinal and shear waves at their reflection from the absorbing surface. These formulas are used to solve the problem of synthesizing optimal absorbers that are characterized by the maximal average value of the absorption coefficient in preset ranges of frequency and angle of incidence. The possibility of increasing this average value by increasing the loss coefficient of the resonators or by using two types of resonators with different resonance frequencies is studied. The results of the calculations are presented in graphic form.     

On some boundary value problems on a strip in the complex plane

The Sommerfeld integral inversion method for the Helmholtz equation in an angular region with different boundary values leads to boundary value problems in an infinite strip of the complex plane. We investigate a generic system for such boundary value problems and give the existence and uniqueness results with optimal growth estimates on the solution. We also give the solutions of the Dirichlet problem in a strip when the boundary functions grow exponentially.     

Subsonic leaky Rayleigh waves at liquid-solid interfaces

The paper is devoted to the study of leaky Rayleigh waves at liquid-solid interfaces close to the border of the existence domain of these modes. The real and complex roots of the secular equation are computed for interface waves at the boundary between water and a binary isotropic alloy of gold and silver with continuously variable composition. The change of composition of the alloy allows one to cross a critical velocity for the existence of leaky waves. It is shown that, contrary to popular opinion, the critical velocity does not coincide with the phase velocity of bulk waves in liquid. The true threshold velocity is found to be smaller, the correction being of about 1.45%. Attention is also drawn to the fact that using the real part of the complex phase velocity as a velocity of leaky waves gives only approximate value. The most interesting feature of the waves under consideration is the presence of energy leakage in the subsonic range of the phase velocities where, at first glance, any radiation by harmonic waves is not permitted. A simple physical explanation of this radiation with due regard for inhomogeneity of radiated and radiating waves is given. The controversial question of the existence of leaky Rayleigh waves at a water/ice interface is reexamined. It is shown that the solution considered previously as a leaky wave is in fact the solution of the bulk-wave-reflection problem for inhomogeneous waves.     

Scattering of plane waves by a wedge with different face impedances

The scattering process of plane waves by a wedge with different face impedances is examined in terms of the closed form series solution. A new boundary condition is derived using the solution of the reflection problem of plane waves by an impedance plane. The series solution is obtained for the wedge problem. The results are investigated numerically.     

Receptivity to free-stream disturbance waves for hypersonic flow over a blunt cone

A high-order shock-fitting finite difference scheme is studied and used to do direction numerical simulation (DNS) of hypersonic unsteady flow over a blunt cone with fast acoustic waves in the free stream, and the receptivity problem in the blunt cone hypersonic boundary layers is studied. The results show that the acoustic waves are the strongest disturbance in the blunt cone hypersonic boundary layers. The wave modes of disturbance in the blunt cone boundary layers are first, second, and third modes which are generated and propagated downstream along the wall. The results also show that as the frequency decreases, the amplitudes of wave modes of disturbance increase, but there is a critical value. When frequency is over the critial value, the amplitudes decrease. Because of the discontinuity of curvature along the blunt cone body, the maximum amplitudes as a function of frequencies are not monotone. Supported by the National Natural Science Foundation of China (Grant Nos. 10632050 and 10502052)     

A study of plasma filled parallel plate waveguide with one boundary corrugated

Warm plasma theory is used to investigate the propagation of electromagnetic waves in a plasma filled parallel plate waveguide with one boundary plate corrugated. Dispersion relations for TE- and TM-modes are derived and it is found that the propagation of TE-modes is unaffected by corrugation of one boundary plane while the propagation of TM-modes is affected by it. For TM-modes the wave numberk depends on the frequency ta as well as on the distance through which the wave is propagated.The authors take the opportunity to thank Professor M. L. Ghosh, Department of Mathematics, North Bengal University for suggesting the problem and one of the authors (S. K. Ghosh) wishes to express his thanks to the University of North Bengal for financial support.     

Reflection and transmission of waves in pyroelectric and piezoelectric materials

The reflection and transmission theories of waves in pyroelectric and piezoelectric medium are studied in this paper. In general in an infinite homogeneous pyroelectric medium there are four bulk wave modes: quasi-longitudinal, two quasi-transversal and temperature waves. In an infinite homogeneous piezoelectric medium there are three bulk wave modes: quasi-longitudinal and two quasi-transversal waves. In the reflection and transmission problem there are five complex boundary conditions in the pyroelectric medium and four complex boundary conditions for the piezoelectric medium. In this paper, we find that the surface waves will be revealed in the reflection and transmission wave problem. The surface waves have the same wave vector component with the incident waves on the interface plane. The two dimensional reflection problem of waves at the interface between the semi-infinite pyroelectric medium and vacuum is researched in greater detail and a numerical example is given.     

The effect of a surface impedance load on the properties of quasi-Rayleigh waves

The propagation of quasi-Rayleigh waves along an impedance-loaded plane boundary of an isotropic elastic half-space is studied theoretically. The dispersion equation of these waves is derived with allowance for the fact that an impedance load has both normal and tangential components. The conditions for the existence of such waves are analyzed depending on the magnitude and nature of each of these components. Specific examples of calculating the quasi-Rayleigh wave velocities are considered: for the models of surface and bulk cracked media, for a fluid layer in an elastic medium, and for a resonant load.     

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.     

Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications

The classical boundary element formulation for the Helmholtz equation is rehearsed, and its limitations with respect to the number of variables needed to model a wavelength are explained. A new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions. This is termed the plane wave basis boundary element method. The modifications needed to the classical procedures, in terms of integration of the element matrices, and location of collocation points are described. The well-known Singular Value Decomposition solution technique, which is adopted here for the solution of the system matrix equation in its complex form, is briefly outlined. The conditioning of the system matrix is analysed for a simple radiation problem. The corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions. The CHIEF method is adopted to enhance the quality of the solution, particularly in the vicinity of irregular frequencies. The plane wave basis boundary element method is then applied to two problems: scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem. In both cases results are compared with analytical solutions. The results clearly demonstrate that the new method is considerably more efficient than the classical approach. For a given number of degrees of freedom, the frequency for which accurate results can be obtained, using the new technique, can be up to three or four times higher than that of the classical method. This makes the method a powerful new addition to our tools for tackling high-frequency radiation and scattering problems.     

Transient scattering by a circular cylinder

The scattering of transient plane waves by a circular cylinder is studied by using the Kirchhoff time-retarded potential boundary integral equation method. Two distinct problems are solved: (i) surface velocity potentials (or pressures) are found for rigid cylinders scattering ramp, ramp-step and Gaussian incident potential (or pressure) waves and (ii) surface velocities are found for free boundary (pressure release) cylinders scattering ramp, ramp-step and Gaussian incident velocity waves. The numerical schemes for both boundary conditions are very similar; since the same influence coefficients are used they differ only by a sign in the final formulation. Numerical results are readily obtained for the first few transit times. This approach is complementary to the usual modal approach in that it is best suited to early time values where the modal solutions converge most slowly.     

Reflective optical bi-stability of antiferromagnetic films

We investigate one magnetically nonlinear response of antiferromagnetic (AF) films to incident electromagnetic waves, or the reflective optical bi-stability (ROB). Such geometry is used, where the AF anisotropy axis and external static magnetic field both are parallel to the film surfaces and normal to the incident plane. For TE incident waves with the electric component transverse to the incident plane, the ROB of the AF film with the absorption is calculated, but the case of TM incident waves is neglected since no magnetic nonlinearity is induced in this geometry. The bi-stability is completely different in the two resonant-frequency vicinities. Two kinds of bi-stability are found in the higher vicinity, and their features versus incident power are opposite. We also find that there are critical incident angle and critical film thickness for the existence of bi-stability. The bi-stability disappears when the film thickness or incident angle exceeds its critical value. Because the properties of bi-stable reflection sensitively depend on the external field and the incident angle, this bi-stability can be easily modulated by means of changing these quantities.     

Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.     

The Elliptic Sinh-Gordon Equation in the Quarter Plane

We study the elliptic sinh-Gordon equation formulated in the quarter plane by using the so-called Fokas method, which is a signi?cant extension of the inverse scattering transform for the boundary value problems. The method is based on the simultaneous spectral analysis for both parts of the Lax pair and the global algebraic relation that involves all boundary values. In this paper, we address the existence theorem for the elliptic sinh-Gordon equation posed in the quarter plane under the assumption that the boundary values satisfy the global relation. We also present the formal representation of the solution in terms of the unique solution of the matrix Riemann- Hilbert problem de?ned by the spectral functions.     

On the existence of longitudinal or flexural waves in rods at nonlinear higher harmonics

This article theoretically studies the conditions for existence of longitudinal or flexural waves in nonlinear, isotropic rods and presents numerical simulations corroborating the theoretical results. It has been known that the existence of guided waves at nonlinearity induced double harmonics is subject to constraints which arise from the potential of power flux transfer from the primary generating mode to the generated higher order modes. The knowledge about the behavior of waves in rods at harmonics higher than double is still limited. This gap was addressed here by the method of perturbation coupled with wavemode orthogonality and forced response. This reduces the nonlinear problem to a forced linear problem which was subsequently investigated to formulate an angular order-based constraint as the condition of existence/nonexistence of nonlinearity-driven higher harmonics of longitudinal and flexural waves in rods.