Nonlinear seismic waves: A model for site effects

The aim of this paper is to propose a possible mathematical model of site effects that occur when seismic waves propagate through a sediment filled basin. The model is based on the mechanical properties of the medium (that we consider as a granular material) through which the seismic waves propagate. By looking for asymptotic solutions having the features of a progressive wave, we derive an evolution equation which is a modified Korteweg–deVries–Burgers equation containing also a nonlinear dissipative term. This equation is integrated numerically and the modelled site amplification is evaluated by using the smoothed spectral ratio between the propagated profile of the wave and the initial one.     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.     

Non-linear waves in a viscous fluid contained in an elastic tube with variable cross-section

In the present work, treating the large arteries as a thin-walled, long and circularly cylindrical, prestressed elastic tube with variable cross-section and using the reductive perturbation method, we have studied the amplitude modulation of non-linear waves in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, the evolution equation is obtained as the dissipative non-linear Schrödinger equation with variable coefficients. It is shown that this type of equations admit a solitary wave solution with a variable wave speed. It is observed that, the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Propagation of weakly nonlinear magnetoacoustic waves

Using the reductive perturbation method we show that the modified Burgers equation governs the propagation of a weakly nonlinear slow magnetoacoustic wave near the equilibrium state with zero transverse magnetic field.     

Wave Dynamics of Saturated Porous Media and Evolutionary Equations

Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Biot approach. Mathematical models under research are the Biot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Biot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg–de Vries–Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Biot model and its generalization are shown.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Weakly nonlinear propagation of small-wavelength,impulsive acoustic waves in a general atmosphere

Multiple-scale asymptotic analysis is applied to small-wavelength, weakly nonlinear propagation of an impulsive acoustic wave in a general (3D, in-motion and time dependent) atmosphere. In keeping with previous work on sonic booms and nonlinear acoustics in general, the result is a combination of ray tracing and a generalised Burgers equation describing evolution of the waveform carried by a ray. This is nonetheless, to our knowledge, the first derivation of such a model based on asymptotic analysis of the governing equations for a general atmosphere. Results are given, discussed and compared with measurements for the particular example of the test explosion known as Misty Picture.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling

A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem.     

The Benjamin-Ono-Burgers equation: An application in solar physics

Equations describing nonlinear, long wavelength motions in a ducted weakly dissipative mhd system are derived and found to belong to a class of which the Benjamin-Ono-Burgers equation is a particular example. The slow decay of the solitary wave solutions is investigated. An illustration of the theory is provided by photospheric flux tubes on the Sun.     

Set of steady-state traveling waves on the surface of a liquid film flowing down an inclined plane

The nonlinear evolution equation often encountered in modeling the behavior of perturbations in various nonconservative media, for example, in problems of the hydrodynamics of film flow, is examined. Steady-state traveling periodic solutions of this equation are found numerically. The stability of the solutions is investigated and a bifurcation analysis is carried out. It is shown how as the wave number decreases ever new families of steady-state traveling solutions are generated. In the limit as the wave number tends to zero a denumerable set of these solutions is formed. It is noted that solutions which also oscillate in time may be generated from the steadystate solutions as a result of a bifurcation of the Landau-Hopf type.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1989.     

Evolution of cylindrical waves of finite amplitude diverging in heated gas

The behavior of weak cylindrical and spherical waves of finite amplitude in a dissipative gas close to the wave front is described by a generalized Burgers equation [1]. The construction of various types of solution of this equation for large Reynolds numbers is known [1–3]. For the evolution of diverging perturbations in heated gas, a study of this equation in the region Re < 1, where Re is the effective Reynolds number at the initial time, is of interest. The direct application of the method of successive approximations to this problem is restricted by the condition Re 1, and becomes more and more difficult as the Reynolds number grows and the form of the initial wave becomes more complex. This paper describes in explicit form the construction of an approximate solution of the Cauchy problem for the generalized Burgers equation in the case of cylindrical symmetry in the region Re < 1. The initial wave selected is the arbitrary perturbation represented by a function which is absolutely integrable on the real axis. An integral estimate of the error as a function of Re is given. The question of how the structure of the solution corresponds to the Cole-Hopf transformation is discussed. All the treatment can easily be extended to the spherically symmetric case.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–153, July–August, 1985.     

位势流动和非均匀介质中声波方程的36种形式

声波方程是对大多数声学问题进行数学描述的出发点. 那些得到 广泛应用的经典波动方程及对流波动方程都存在苛刻的适用条件, 即仅适用于描述处于静态或匀速运动状态的定常 均匀介质中的线性无耗散声波. 然而, 很多实际场合并不满足这些严格的适用条件. 本文对经典声波方程和对流声波 方程进行推广, 导出了编号为W1$\sim$W36的36种不同形式的声波方程, 涵盖了处于静止、势流或旋涡流状态下的非均匀 和/或非定常介质中的声波传播问题. 所考虑的声波传播情形包括: (1) 线性波, 即具有小梯度(小振幅)性质; (2)非线性波, 即具有陡峭梯度性质, 包括``波纹'(小振幅大梯度)或者大振幅波. 本文仅考虑非耗散声波, 即排除了由剪切、体积黏度及热传导所引起的耗散. 对具有匀熵或等熵(熵沿流线守恒)性质的均匀介质和非均匀介质中的声传播进行了研究但非等熵(即耗散)情况除外; 另外, 对非定常介质中的 声波问题也进行了分析. 所涉及的介质可以处于静止、匀速运动状态, 或者是非匀速的和/或非定常的平均流动, 包括: (1)低Mach数的势平均流(即不可压缩的平均态), 或高速势平均流(即非均匀可压缩的平均流); ② 变截面管 道中的准一维传播, 包括无平均流的号管和具有低或高Mach数平均流的喷管; 或③平面的、空间的、或轴对称的单 向剪切平均流. 本文没有探讨其他类型的旋涡平均流(将与耗散及其他情形一起留待下一步研究), 例如, 可能与剪切效应相结合的轴对称旋转平均流. 通过对流体力学的一般方程进行消元处理或根据声学变分原理, 导出了36种波动方程, 对一些波动方程还采用这两种方法进行相互校验. 尽管声波方程的36种形式没有涵盖非线性、非均匀与非定常及非匀速运动介质 这3个效应的所有可能的组合情形, 但它们的确包括了孤立状态下的各种效应, 并包括了多种多重效应组合的 情形. 虽然经典波动方程和对流波动方程仅适用于处于静止(或匀速运动)的均匀定常介质中的线性无耗散声波, 但它们在 相关文献中已被广泛采用; 本文给出的36种声波方程提供了它们多种有用的推广形式. 在许多实际应用中, 经典波动方 程和对流波动方程仅是粗略的近似, 声波方程的更一般形式可提供更令人满意的理论模型. 本文每节末尾给出了这些应用 的众多范例. 在这篇评论文章中引用了240篇参考文献.     

Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.     

A finite-strain model for anisotropic viscoplastic porous media: I – Theory

In this work, we propose an approximate homogenization-based constitutive model for estimating the effective response and associated microstructure evolution in viscoplastic (including ideally-plastic) porous media subjected to finite-strain loading conditions. The proposed model is based on the “second-order” nonlinear homogenization method, and is constructed in such a way as to reproduce exactly the behavior of a “composite-sphere assemblage” in the limit of hydrostatic loading and isotropic microstructure. However, the model is designed to hold for completely general three-dimensional loading conditions, leading to deformation-induced anisotropy, whose development in time is handled through evolution laws for the internal variables characterizing the instantaneous “ellipsoidal” state of the microstructure. In Part II of this study, results will be given for the instantaneous response and microstructure evolution in porous media for several representative loading conditions and microstructural configurations.     

Packet of gravity surface waves at high Reynolds numbers

Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the third power of the steepness. The conditions, under which the packet envelope evolution is described by the nonlinear Schrödinger equation with a dissipative term linear in the amplitude, are determined. The rule, in accordance with which the term of this type can be correctly added in the evolutionary equation of an arbitrary order is formulated.     

On the effectiveness of the approximate inertial manifold—a computational study

We present a computational study evaluating the effectiveness of the nonlinear Galerkin method for dissipative evolution equations. We begin by reviewing the theoretical estimates of the rate of convergence for both the standard spectral Galerkin and the nonlinear Galerkin methods. We show that the rate of convergence in both cases depends mainly on how well the basis functions of the spectral method approximate the elements in the space of solutions. This in turn depends on the degree of smoothness of the basis functions, the smoothness of the solutions, and on the level of compatibility at the boundary between the basis functions of the spectral method and the solutions. When the solutions are very smooth inside the domain and very compatible with the basis functions at the boundary, there may be little advantage in using the nonlinear Galerkin method. On the other hand, for less smooth solutions or when there is less compatibility at the boundary with the basis functions, there is a significant improvement in the rate of convergence when using the nonlinear Galerkin method. We demonstrate the validity of our assertions with numerical simulations of the forced dissipative Burgers equation and of the forced Kuramoto-Sivashinsky equation. These simulations also demonstrate that the analytical upper bounds derived for the rates of convergence of both the standard Galerkin and the nonlinear Galerkin are nearly sharp.This work was supported in part by the National Science Foundation, AFOSR Grant No. F49620-92-J-0287, and the Joint University of California-Los Alamos National Laboratory Institute for Cooperative Research (INCOR) Program for Climate Modeling. Partial support has also come from the Department of Energy Computer Hardware, Advanced Mathematics, Model Physics (CHAMMP) research program as part of the U.S. Global Change Research Program.     

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.     

Experimental Study of Nonlinear Resonances and Anti-Resonances in a Forced,Ordered Granular Chain

We experimentally study a one-dimensional uncompressed granular chain composed of a finite number of identical spherical elastic beads with Hertzian interactions. The chain is harmonically excited by an amplitude- and frequency-dependent boundary drive at its left end and has a fixed boundary at its right end. Such ordered granular media represent an interesting new class of nonlinear acoustic metamaterials, since they exhibit essentially nonlinear acoustics and have been designated as “sonic vacua” due to the fact that their corresponding speed of sound (as defined in classical acoustics) is zero. This paves the way for essentially nonlinear and energy-dependent acoustics with no counterparts in linear theory. We experimentally detect time-periodic, strongly nonlinear resonances whereby the particles (beads) of the granular chain respond at integer multiples of the excitation period, and which correspond to local peaks of the maximum transmitted force at the chain’s right, fixed end. In between these resonances we detect a local minimum of the maximum transmitted forces corresponding to an anti-resonance in the stationary-state dynamics. The experimental results of this work confirm previous theoretical predictions, and verify the existence of strongly nonlinear resonance responses in a system with a complete absence of any linear spectrum; as such, the experimentally detected nonlinear resonance spectrum is passively tunable with energy and sensitive to dissipative effects such as internal structural damping in the beads, and friction or plasticity effects. We compare the experimental results with direct numerical simulations of the granular network and deduce satisfactory agreement.     

Ideal Gas Outflow from a Cylindrical or Spherical Source into a Vacuum

The solutions of initial and boundary value problems of the outflow of an ideal (inviscid and non-heat-conducting) gas from cylindrical and spherical sources into a vacuum are obtained. Time is measured from the moment, when the source is turned on; at this moment the source is surrounded by a vacuum. The entropy, flow rate, and the Mach number of the gas outflowing from the source are given, together with the source radius; the Mach number can be greater of or equal to unity. If the source radius is greater than zero, then the flow domain in the “radial coordinate–time” plane consists of the stationary source flow and adjoining non-self-similar centered expansion wave consisting of C^?-characteristics. The stationary flow is described by the known formulas, while the expansion wave is calculated by the method of characteristics. The calculations by this method confirm the earlier obtained laws for large values of the radial coordinate. The interface between the vacuum and the expansion wave is the straight trajectory of particles and, at the same time, a unique rectilinear C^?-characteristic. For the source of zero radius (“pointwise” source) the velocity, density, and speed of sound of the outflowing gas are infinite. The gas velocity remains infinite everywhere, while the density and speed of sound become zero for any non-zero values of the radial coordinate. For the pointwise source the problem of outflow into a vacuum is self-similar. In the plane of the “self-similar” velocity and speed of sound its solution is given by three singular points of a differential equation in these variables. At one of these points the self-similar velocity is infinite, the self-similar speed of sound is zero, and the self-similar independent variable varies from zero to infinity, with the exception of the extreme values.