On the theory of rolling waves

The problem of rolling waves in a sheet of fluid flowing in a vertical plane [1] is treated on the basis of the complete Navier-Stokes equations with conditions on the unknown free boundary. The existence of a one-parameter family of rolling waves, bifurcating from the Poiseuille flow, is proved.     

On the theory of long waves

A new method of studying plane steady wave motion of a gravity fluid is elucidated in this paper. This method succeeds in establishing the existence of a solitary wave, for example, and in giving the first complete foundation for the approximate Rayleigh theory [1], which concerns the theory of finite-amplitude long waves. Underlying the method are general boundary properties of univalent functions, used earlier by the author to construct a qualitative theory of jet fluid motions [2].     

Rayleigh surface waves in the theory of thermoelastic materials with voids

In this paper we analyze the behavior of plane harmonic waves and Rayleigh waves in a linear thermoelastic material with voids. We take into account the damped effects of the thermal field upon the propagation waves. Consequently, the propagation condition is established in the form of an algebraic equation of 9th degree whose coefficients are complex numbers while the eigensolutions of the thermoelastodynamic with voids system are explicitly obtained in terms of the characteristic solutions. We show that the transverse waves are undamped in time and they are not influenced by the thermal and porous effects while the longitudinal waves are all damped in time and they are coupled with the thermal and porous effects. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the eigensolutions in concern. The secular equation is established in an implicit form and afterwards an explicit form is written for an isotropic and homogeneous thermoelastic with voids half-space. Furthermore, we use the numerical methods and computations to solve the secular equation for a specific material.     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

ON THE STABILITY BOUNDARY OF HAMILTONIAN SYSTEMS

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On the interactions of surface waves with immersed structures

The fluid forces resulting from wave interaction with large submerged structures may be calculated using numerical procedures based on the solution of the associated boundary-value problem. In this paper, the analysis of wave interaction with a fixed submerged object of arbitrary cross-section and infinite length using a two-dimensional boundary value formation based on linear diffraction theory is summarized. Subsequently, the application of the boundary element method to obtain a solution is presented. The numerical considerations are emphasized with particular reference to computational efficiency. Numerical results are presented in the form of dimensionless wave force plots for various structural shapes. In the case of a bottom-seated half cylinder, for which there exists a closed-form solution, comparisons are made between results generated using both boundary element and equivalent finite element approaches. In the case of a submerged cylinder, comparisons are made between boundary element derived values and experimental results. The boundary element results compare well with both the closed-form solution and the experimental values.     

Nonlinear theory of forced surface waves in a circular basin

A nonlinear theory is developed to study surface waves excited by the prescribed horizontal oscillation of the side wall of a circular basin. It is assumed that the frequency of the forced oscillation is near either one of the resonance frequencies of the water in the basin or twice of it. A multiple-scale asymptotic expansion is constructed to derive an equation for the amplitude of an excited eigenmode and critical points of some parameters are found for primary and subharmonic resonance waves. Across these critical points the eigenmode amplitude increases abruptly but remains bounded except at certain values of the water radius to the depth ratio where internal resonance appears.     

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems. Foundation item: the National Natural Science Foundation of China (10072012); the National Natural Science Foundation of Russia Biography: QI Zhao-hui (1964-)     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

On the theory of plane inhomogeneous waves in anisotropic elastic media

In the theory of plane inhomogeneous elastic waves, the complex wave vector constituted by two real vectors in a given plane may be described with the aid of two complex scalar parameters. Either of those parameters may be taken as a free one in the characteristic condition assigned to the wave equation. This alternative underlies the two fundamental approaches in the theory, namely, one associated with the Stroh eigenvalue problem and the other with the generalized Christoffel eigenvalue problem. The two approaches are identical insofar as a partial nondegenerate wave solution (partial mode) is concerned, but they differ in the fundamental solution (wave packet) assembling, and their dissimilarity is also revealed in the presence of degeneracies, which may involve either of the two governing parameters or both of them. Therefore, use of both approaches is essential for studying the degeneracy phenomenon in the theory of inhomogeneous waves. The criteria for different types of degeneracy, related to a double eigenvalue of the Stroh matrix or the Christoffel matrix and at the same time to a repeated root of the characteristic condition, are formulated by appeal to the matrix algebra and to the theory of polynomial equations. On this basis, dimensions of the manifolds, associated with degeneracy of different types in the space of variables, are established for elastic media of unrestricted anisotropy. The relation to the boundary-value problems is discussed.     

On solitary waves. Part 2 A unified perturbation theory for higher-order waves

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ _m , at a specific order, n=n _m , both depending on the base adopted, e.g. n _{m , α} =11-12 based on parameter α (wave amplitude), n _{m , β} =15 on β (amplitude-speed square ratio), and n _{m , ∈} =17 on ∈ ( wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work. The project partly supported by the National Natural Science Foundation of China (19925414,10474045)     

On solitary waves. Part 2 A unified perturbation theory for higher-order waves

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ _m , at a specific order, n=n _m , both depending on the base adopted, e.g. n _{m , α} =11-12 based on parameter α (wave amplitude), n _{m , β} =15 on β (amplitude-speed square ratio), and n _{m , ∈} =17 on ∈ ( wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work.     

On hypo-elastic transverse surface waves in an internal stratum

Transverse surface waves in a stratum of uniform thickness, bounded on both sides by very deep layers of different materials are investigated in the context of hypo-elasticity of grade zero. It is found that the wave motion is possible and that the phase-velocity equation is similar to the one in the corresponding pure elastic problem. But unlike in the pure elastic problem, the waves in the present problem produce non-zero normal stresses.     

On the theory of plastic wave propagation in a bar— unloading waves

This is a continuation of a previous paper on the problem of plastic wave propagation in a semiinfinite bar subjected to an axially-applied impact stress. In the present paper the problem of unloading waves is considered. Malvern's strain-rate theory is applied and both the case of elastic unloading and the case of elasticviscoplastic unloading are considered. A theoretical relationship between the rate-independent and the ratedependent theories is established. A bilinear stress-strain curve is considered, valid for both loading, instantaneous unloading, and linear unloading. Both the elastic and elastic-viscoplastic unloading cases are considered and by using the method of characteristics and an IBM 7040-7094 digital computer it is shown that for the elasticviscoplastic case there exists a lower quasiplateau during unloading. This quasiplateau depends on the rate of unloading.