Shear to longitudinal mode conversion via second harmonic generation in a two-dimensional microscale granular crystal

Shear to longitudinal mode conversion via second harmonic generation is studied theoretically and computationally for plane waves in a two-dimensional, adhesive, hexagonally close-packed microscale granular medium. The model includes translational and rotational degrees of freedom, as well as normal and shear contact interactions. We consider fundamental frequency plane waves in all three linear modes, which have infinite spatial extent and travel in one of the high-symmetry crystal directions. The generated second harmonic waves are longitudinal for all cases. For the lower transverse–rotational mode, an analytical expression for the second harmonic amplitude, which is derived using a successive approximations approach, reveals the presence of particular resonant and antiresonant wave numbers, the latter of which is prohibited if rotations are not included in the model. By simulating a lattice with adhesive contact force laws, we study the effectiveness of the theoretical analysis for non-resonant, resonant, and antiresonant cases. This work is suitable for the analysis of microscale and statically compressed macroscale granular media, and should inspire future studies on nonlinear two- and three-dimensional granular systems in which interparticle shear coupling and particle rotations play a significant role.     

On diffraction of normally incident SH-waves by a line crack in micropolar elastic medium

The problem of diffraction of waves due to plane harmonic SH-waves incident normally on a line crack situated in an infinite micropolar elastic medium has been considered. The solution of the problem is obtained for both low and high frequencies for small coupling parameter. The stress-intensity factors in micropolar elastic medium have been derived. The stress-intensity factor for such problem in an elastic medium can be deduced from results obtained in this paper. It is also found that the effect of micropolarity in the propagation of waves is more significant in high frequencies than low frequencies.     

Wave propagation in transversely isotropic porous piezoelectric materials

Wave propagation in porous piezoelectric material (PPM), having crystal symmetry 6 mm, is studied analytically. Christoffel equation is derived for the propagation of plane harmonic waves in such a medium. The roots of this equation give four complex wave velocities which can propagate in such materials. The phase velocities of propagation and the attenuation quality factors of all these waves are described in terms of complex wave velocities. Phase velocities and attenuation of the waves in PPM depend on the phase direction. Numerical results are computed for the PPM BaTiO₃. The variation of phase velocity and attenuation quality factor with phase direction, porosity and the wave frequency is studied. The effects of anisotropy and piezoelectric coupling are also studied. The phase velocities of two quasi dilatational waves and one quasi shear waves get affected due to piezoelectric coupling while that of type 2 quasi shear wave remain unaffected. The phase velocities of all the four waves show non-dispersive behavior after certain critical high frequency. The phase velocity of all waves decreases with porosity while attenuation of respective waves increases with porosity of the medium. The characteristic curves, including slowness curves, velocity curves, and the attenuation curves, are also studied in this paper.     

水陆两栖飞机波浪水面上降落耐波性数值分析

在规定的气象水文条件下,水陆两栖飞机起飞和降落的能力是决定其性能的重要因素,即耐波性能。采用ALE方法对流体域进行描述,运用基于微幅波理论的动边界数值造波方法模拟了不同波高和不同波长的动态海平面波浪,通过添加质量阻尼的消波方法抑制了固壁边界反射波对造波结果的影响,并采用罚函数耦合方法描述飞机与水体的耦合作用,研究了水陆两栖飞机在不同海情条件下波浪面上降落的纵摇运动、升沉运动以及底部压力等运动学和动力学特性,分析了水陆两栖飞机入水波浪的波长及波高对水陆两栖飞机耐波性能的影响,为飞机结构设计、水上降落操作规则制订及水陆两栖飞机耐波性物理水池试验提供参考。     

A note on diffraction of normally incident P-wave by a line crack in micro-polar elastic medium

The problem of diffraction of waves due to plane harmonic P-wave incident normally on a line crack situated in an infinite micro-polar elastic medium has been studied in this paper. The problem has been solved for both low and high frequencies for small coupling parameter. The stress intensity factors (SIF) have been obtained in micro-polar elastic medium from which the corresponding stress intensity factor for classical elastic medium can be deduced.     

A state space formalism and perturbation method for generalized thermoelasticity of anisotropic bodies

A state space formalism for generalized anisotropic thermoelasticity accounting for thermomechanical coupling and thermal relaxation is developed, which includes the classical thermoelasticity as a special case. By properly grouping the field variables using matrix notations, the basic equations of thermoelasticity are formulated into a state equation and an output equation in terms of the state vector. To obtain the solution for a specific problem it suffices to solve the state equation under the prescribed conditions. For weak thermomechanical coupling an asymptotic solution can be obtained by using the method of perturbation with multiple scales. Propagation of plane harmonic thermoelastic waves in an anisotropic medium is studied within the context.     

Scattering of elastic waves by three-dimensional surface topographies

Diffraction of plane harmonic waves by three-dimensional surface irregularities is investigated through the use of an indirect boundary integral equation method. The irregularity of an arbitrary shape is embedded in an elastic half-space and subjected to incident P, SV, SH, and Rayleigh waves. The material of the half-space is assumed to be linear, weakly anelastic, homogeneous and isotropic.

The accuracy of the method is demonstrated through comparison of the results with existing axisymmetric solutions. Several numerical examples for non-axisymmetric canyons are presented. The resulting amplification patterns exhibit strong sensitivity on type and angle of the incident waves and on the location of the observation point. Systematic comparisons of three-dimensional and corresponding two-dimensional models demonstrate similarity of the amplification pattern. The amplification is larger in some three-dimensional models than in two-dimensional ones. Strong coupling between SH and P-SV modes is observed for off-azimuthal incident waves. This phenomenon is specially pronounced for incident SH waves and it is intrinsic to three-dimensional scattering.     

Rayleigh-type waves in a water-saturated porous half-space with an elastic plate on its surface

The paper deals with the plane problem of steady-state time harmonic vibrations of an infinite elastic plate resting on a water-saturated porous solid. The displacements of the plate are described by means of the linear theory of small elastic oscillations. The motion of the two-phase medium is studied within the framework of Biot's linear theory of consolidation. The main interest is focused on the investigation of properties of the Rayleigh-type waves propagating alongside of the contact surface between the plate and the porous half-space. In particular, the dependence of the phase velocity and attenuation of the waves on the plate stiffness, mass coupling coefficient, and degree of saturation of the medium is studied. Besides, for the limiting case of an infinitely thin plate, the comparison of the wave characteristics is carried out with those of the pure Rayleigh waves.     

Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides

The paper describes a passive time-independent setting for non-reciprocal wave transmission in mechanical and acoustic systems with strong nonlinearities. In the proposed system vibro-impact elements with pre-defined clearances are used to couple two non-dispersive waveguides. The asymmetry necessary for the non-reciprocal behavior is realized through unequal grounding springs of the vibro-impact elements. We show that, for appropriate selection of the parameters, the proposed system acts as a mechanical diode, allowing the transmission of acoustic waves in one direction and completely preventing reverse transmission. Two different designs of the coupling elements are suggested, with the possibility of single-sided or double-sided impacts. A unique feature of the proposed non-reciprocal acoustic system is that minimal distortion of the harmonic content of the transmitted wave occurs, in contrast to current designs where nonlinear non-reciprocity is achieved at the expense of a rather strong distortion of the transmitted signals. For both designs, we derive exact solutions for propagation and reflection of the harmonic waves, and demonstrate the possibility for strong non-reciprocity. Stability properties of the observed solutions in the space of parameters are also explored.     

Vortex structure simulation for supersonic mixing layers using nonlinear PSE method

The method of nonlinear parabolized stability equations (PSE) is applied in the simulation of vortex structures in compressible mixing layer. The spatially-evolving unstable waves, which dominate the vortex structure, are investigated through spatial marching method. The instantaneous flow field is obtained by adding the harmonic waves to basic flow. The results show that T-S waves do not keep growing exponentially as the linear evolution, the energy transfer to high order harmonic modes, and that finally all harmonic modes get saturated due to nonlinear interaction. The mean flow distortion induced by the nonlinear interaction between the harmonic modes and their conjugate harmonic ones, makes great change of the average flow and increases the thickness of mixing layer. PSE methods can well capture the two- and three-dimensional large scale nonlinear vortex structures in mixing layers such as vortex roll-up, vortex pairing, and Λ vortex.     

Nonlinear heave-roll coupling and ship rolling

A nonlinear model for simulating the heave-roll motions of ships in following waves is presented. The parametric excitation is modeled by a Hill's type equation, instead of the conventional Mathieu's equation. The model includes not only the linear but also the quadratic coupling term. Instability conditions for parametrically excited rolling motions are derived using the harmonic balance method. The results are verified by numerical analyses. The effects of including the quadratic coupling term on the instability conditions and nonlinear responses are studied. The complex dynamic behaviour of the coupled system in the various instability regions is also investigated. Bifurcations of the flip, fold and pitchfork types are observed in the Poincaré mapping of the numerically simulated responses. Chaotic motions leading to boundary crises and inevitable capsize are also reported.     

Are waves with negative spatial damping unstable?

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.     

Laboratory experiments on internal wave interactions with a pycnocline

Laboratory experiments have been performed to investigate the interaction of internal waves with a pycnocline. An oscillating cylinder generated internal wave beams, which were observed using the synthetic schlieren technique. Internal waves incident on the pycnocline layer excited higher-frequency modes. In the absence of shear, a discrete spectrum of harmonic modes was generated due to nonlinear effects. These harmonic modes might play a role in the formation of internal solitary waves which have been observed in ocean pycnoclines. With shear, a continuous spectrum of excited modes was found.     

An intermediate wavelength, weakly nonlinear theory for the evolution of capillary gravity waves

A new nonlinear evolution equation is derived for surface solitary waves propagating on a liquid-air interface where the wave motion is induced by a harmonic forcing. Instead of the traditional approach involving a base state of the long wave limit, a base state of harmonic waves is assumed for the perturbation analysis. This approach is considered to be more appropriate for channels of length just a few multiples of the depth. The dispersion relation found approaches the classical long wave limit. The weakly nonlinear dispersive waves satisfy a KdV-like nonlinear evolution equation with steeper nonlinearity.     

Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids

The present paper concerns with the linear theory of micropolar thermoelasticity for materials with voids. Some basic properties of wave numbers of the longitudinal and transverse plane harmonic waves are treated. The existence theorems of non-trivial solutions and eigenfrequencies of the interior homogeneous boundary value problems of steady vibrations are proved. The connection between plane harmonic waves and eigenfrequencies of the aforementioned problems is established.     

The dynamics of a compressible viscous liquid (review). II

This article is the second part of a review of the dynamics of rigid and elastic bodies in a compressible viscous liquid in a linearized formulation. The following processes are investigated: the forced harmonic vibrations of rigid bodies in moving and resting compressible viscous liquids, the nonstationary motion of rigid bodies in a compressible viscous liquid at rest, the movement of rigid bodies in a resting compressible viscous liquid under the action of radiation forces that are due to the interaction with propagating acoustic harmonic waves, the propagation of harmonic waves in thin-walled cylindrical elastic shells containing a compressible viscous liquid, and the propagation of harmonic waves in hydroelastic systems consisting of a resting compressible viscous liquid and elastic compressible or incompressible bodies with initial stresses. Publications concerning the above problems are analyzed. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 3–30, March, 2000.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.