Propagation Of Small-amplitude Waves Research Articles

Dynamic response of cross-linked random fiber networks

Systems composed of fibers are ubiquitous in the living and artificial systems, including cartilage, tendon, ligaments, paper, protective clothing, and packaging materials. Given the importance of fiber systems, their static behavior has been extensively studied and it has been shown that network deformation is nonaffine for compliant, low-density networks and affine for stiff, high-density networks. However, little is known about the dynamic response of fibrous systems. In this work, we investigate numerically the propagation of small-amplitude elastic waves in these materials and characterize their dynamic response as a function of network parameters.

Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.

Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder

We consider torsional wave propagation in a pre-stressed annular cylinder. Hydrostatic pressure is applied to the inner and outer surfaces of an incompressible hyperelastic annular cylinder, of circular cross-section, whose constitutive behaviour is governed by a Mooney–Rivlin strain energy function. The pressure difference creates an inhomogeneous deformation field, which modifies the inner and outer radii of the annular cylinder. We wish to determine the effect that this pre-stress, and a given axial stretch, has on the propagation of small-amplitude torsional waves through the medium. We use the theory of small-on-large to deduce the linear wave equation that governs incremental torsional waves and then determine the dispersion relation for the pre-stressed annulus by using an approximation procedure (the Liouville–Green transformation). We show that this scheme compares well to numerical solutions except in regions very close to turning points (of the transformed ordinary differential equation). In particular, we find that the inhomogeneous deformation makes the coefficients of the governing ordinary differential equation spatially dependent and affects the location of the roots of the dispersion relation. We observe that, if the pressure on the outer surface of the annular cylinder is greater (smaller) than that on the inner, then the cut-on frequencies are spaced further apart (closer) than they would be in the stress-free case. This result could potentially be used to tune the propagation characteristics of the cylinder over a range of frequencies.

Painlevé Analysis, Lie Symmetries and Exact Solutions for Variable Coefficients Benjamin—Bona—Mahony—Burger (BBMB) Equation

In this paper, a variable-coefficient Benjamin—Bona—Mahony—Burger (BBMB) equation arising as a mathematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The integrability of such an equation is studied with Painlevé analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Furthermore different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.

Surface waves in nematic liquid crystals

The problem of small-amplitude harmonic wave propagation along the surface of an incompressible nematic liquid crystal is considered when the evolution of the orientation vector is specified by viscous stresses and the orientational elasticity can be ignored. An analytic solution and a dispersion relation are obtained in the case of the planar and homeotropic initial orientation of this vector.

A Mathematical Study on Three Layered Oscillatory Blood Flow Through Stenosed Arteries

A mathematical model is constructed to examine the characteristics of three layered blood flow through the oscillatory cylindrical tube (stenosed arteries). The proposed model basically consists three layers of blood (viscous fluids with different viscosities) named as core layer (red blood cells), intermediate layer (platelets/white blood cells) and peripheral layer (plasma). The analysis was restricted to propagation of small-amplitude harmonic waves, generated due to blood flow whose wave length is larger compared to the radius of the arterial segment. The impacts of viscosity of fluid in peripheral layer and intermediate layer on the interfaces, average flow rate, mechanical efficiency, trapping and reflux are discussed with the help of numerical and computational results. This model is the generalized form of the preceding models. On the basis of present discussion, it is found that the size of intermediate and peripheral layers reduces in expanded region and enhances in contracted region with the increasing viscosity of fluid in peripheral layer, whereas, opposite effect is observed for viscosity of fluid in intermediate layer. Final conclusion is that the average flow rate and mechanical efficiency increase with the increasing viscosity of fluid in both layers, however, the effects of the viscosity of fluid in both layers on trapping and reflux are opposite to each other.

Large acoustoelastic effect

Classical acoustoelasticity couples small-amplitude elastic wave propagation to an infinitesimal pre-deformation, in order to reveal and evaluate non-destructively third-order elasticity constants. Here, we see that acoustoelasticity can also be used to determine fourth-order constants, simply by coupling a small-amplitude wave with a small-but-finite pre-deformation. We present results for compressible weakly nonlinear elasticity, we make a link with the historical results of Bridgman on the physics of high pressures, and we show how to determine “D”, the so-called fourth-order elasticity constant of soft (incompressible, isotropic) solids by using infinitesimal waves.

Waves on the interface between a nematic liquid crystal and an ideal isotropic fluid

The problem of surface wave propagation over the interface between a nematic liquid crystal and an ideal isotropic fluid is considered. For the nematic liquid crystal the Frank-Oseen model with an isotropic viscous stress tensor is used. Anisotropic surface tension is described by the Rapini model. In this formulation, for the problem of harmonic small-amplitude surface wave propagation, in the case of infinite depths of both phases, an analytical solution is obtained. The dispersion relation is derived and its properties are investigated.

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.

Dynamics of a rotating layer of an ideal electrically conducting incompressible inhomogeneous fluid in an equatorial region

The equations describing the three-dimensional equatorial dynamics of an ideal electrically conducting inhomogeneous rotating fluid are studied. The magnetic and velocity fields are represented as superpositions of unperturbed steady-state fields and those induced by wave motion. As a result, after introducing two auxiliary functions, the equations are reduced to a special scalar one. Based on the study of this equation, the solvability of initial-boundary value problems arising in the theory of waves propagating in a spherical layer of an electrically conducting density-inhomogeneous rotating fluid in an equatorial zone is analyzed. Particular solutions of the scalar equation are constructed that describe small-amplitude wave propagation.

KP solitons in shallow water

The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev–Petviashvili (KP) equation. The KP equation describes weakly dispersive and small amplitude wave propagation in a quasi-two-dimensional framework. Recently, a large variety of exact soliton solutions of the KP equation has been found and classified. These solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and are called line-solitons. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety which can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial value problems indicate that certain classes of initial waves asymptotically approach to these exact solutions of the KP equation. We discuss an application of our theory to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, and it predicts an extraordinary fourfold amplification of the wave at the wall. There are several numerical studies confirming the prediction, but all indicate disagreements with the KP theory. Contrary to those previous numerical studies, we find that the KP theory actually provides an excellent model to describe the Mach reflection phenomena when the higher order corrections are included in the quasi-two-dimensional approximation. We also present laboratory experiments of the Mach reflection recently carried out by Yeh and his colleagues, and show how precisely the KP theory predicts this wave behavior.

Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect

Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third- and fourth-order elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for ρv(2), where ρ is the mass density and v the wave speed, in terms of the elongation e of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: ρv(2) = a+be+ce(2), say, a depends linearly on μ; b on μ and A; and c on μ, A, and D, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves.

Electroelastic waves in a finitely deformed electroactive material

In this paper, the coupling between a finite deformation and an electric field is examined with particular reference to the propagation of small amplitude waves in a non-linear electroelastic material based on the quasi-electrostatic approximation. The general equations governing the linearized response of electroelastic solids superimposed on a state of finite deformation in the presence of an electric field are derived along with incremental forms of the electroelastic constitutive laws and boundary conditions. Both unconstrained and incompressible materials are considered. Without restriction on the electroelastic constitutive law, the theory is first applied to the analysis of plane waves propagating in a homogeneously deformed material with an underlying uniform electric field and illustrated in the case of an isotropic material. The general equations governing 2D incremental motions are then derived and applied to the study of surface waves in a homogeneously deformed half-space of incompressible isotropic material with the electric field normal to the surface of the half-space. The dependence of the wave speed on the deformation, the electric field and the electromechanical coupling parameters is illustrated for a prototype electroelastic constitutive law.

A KdV-type Boussinesq system: From the energy level to analytic spaces

Considered here is the well-posedness of a KdV-type Boussinesq system modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid when the motion is sensibly two dimensional. Solutions are obtained in a range of Sobolev-type spaces, from the energy level to the analytic Gevrey spaces. In addition, a criterion for detecting the possibility of blow-up in finite time in terms of loss of analyticity is derived.

The influence of boundary conditions on dispersion in an elastic plate

The propagation of waves along an elastic layer of uniform thickness has been an area of active research for many years. Many contributions have been made to the study of small amplitude wave propagation in a linear isotropic elastic layer, almost all in respect of traction-free boundary conditions. In this paper the associated dispersion relation is briefly reviewed. We also consider two other non-classical types of the boundary conditions, the so-called fixed and free-fixed problems. The associated dispersion relations are first investigated numerically, from which it is shown that no analogues of classical bending or extension exists. The fixed-free case is of particular interest as, unlike the other two cases, it does not decompose into symmetric and anti-symmetric parts. A representation of the associated dispersion relation is established in terms of the fixed and free dispersion relations. Long wave approximations of the phase are obtained; it is envisaged that these approximations will allow future establishment of appropriate long wave models.

Ondas superficiais de gravidade

Neste artigo discutimos a propagação de ondas de pequena amplitude num líquido de profundidade constante com base em uma equação de ondas. A solução geral mostra que a velocidade de propagação é dependente do comprimento de onda de modo que ocorre o fenômeno da dispersão. Soluções para os casos de pequena e grande profundidades são também obtidas.

A Coaxial Tube Model of the Cerebrospinal Fluid Pulse Propagation in the Spinal Column

The dynamics of the movement of the cerebrospinal fluid (CSF) may play an important role in the genesis of pathological neurological conditions such as syringomyelia, which is characterized by the presence of a cyst (syrinx) in the spinal cord. In order to provide sound theoretical grounds for the hypotheses that attribute the formation and growth of the syrinx to impediments to the normal movement of the CSF, it is necessary to understand various modes through which CSF pulse in the spinal column propagates. Analytical models of small-amplitude wave propagation in fluid-filled coaxial tubes, where the outer tube represents dura, the inner tube represents the spinal cord, and the fluid is the CSF, have been used to that end. However, so far, the tendency was to model one of the two tubes as rigid and to neglect the effect of finite thickness of the tube walls. The aim of this study is to extend the analysis in order to address these two potentially important issues. To that end, classical linear small-amplitude analysis of wave propagation was applied to a system consisting of coaxial tubes of finite thickness filled with inviscid incompressible fluid. General solutions to the governing equations for the case of harmonic waves in the long wave limit were replaced with the boundary conditions to yield the characteristic (dispersion) equation for the system. The four roots of the characteristic equation correspond to four modes of wave propagation, of which the first three are associated with significant motion of the CSF. For the normal range of parameters the speeds of the four modes are c(1)=13 ms, c(2)=14.7 ms, c(3)=30.3 ms, and c(4)=124.5 ms, which are well within the range of values previously reported in experimental and theoretical studies. The modes with the highest and the lowest speeds of propagation can be attributed to the dura and the spinal cord, respectively, whereas the remaining two modes involve some degree of coupling between the two. When the thickness of the spinal cord, is reduced below its normal value, the first mode becomes dominant in terms of the movement of the CSF, and its speed drops significantly. This suggests that the syrinx may be characterized by an abnormally low speed of the CSF pulse.

A boundary element method for the hydrodynamic analysis of floating bodies in variable bathymetry regions

In this work, a hybrid technique is presented for the hydrodynamic analysis of floating bodies in variable bathymetry regions. Our method is based on the coupled-mode theory for the propagation of water waves in general bottom topography, developed by Athanassoulis and Belibassakis [Athanassoulis GA, Belibassakis KA. A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J Fluid Mech 1999;389:275–301.] and extended to 3D by Belibassakis et al. [Belibassakis KA, Athanassoulis GA, Gerostathis TP. A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry. Appl Ocean Res 2001;23:319–336.], which is free of any mild-slope assumption, in conjunction with a boundary integral representation of the near field in the vicinity of the floating body. Both 2D and 3D problems have been considered. In all cases the near field is represented by boundary integral representation involving simple (Rankine) sources. In the 2D case, the far-field is modelled by complete (normal-mode) series expansions derived by separation of variables in the constant-depth half-strips. In the 3D case, the far-field is modelled by an integral representation involving the appropriate Green's function for harmonic water waves over a bottom with different depths at infinity, developed by Belibassakis and Athanassoulis [Belibassakis KA, Athanassoulis GA. Three-dimensional Green's function for harmonic water waves over a bottom with different depths at infinity. J Fluid Mech 2004;510:267–302.]. The numerical solution is obtained by means of a low-order panel method materialising the hybrid technique. Numerical results are presented concerning floating bodies of simple geometry, lying over sloping and undulating seabeds. With the aid of systematic comparisons with benchmark solutions the convergence and accuracy of the present method in 3D has been studied, and the effects of bottom slope and curvature on the hydrodynamic characteristics (hydrodynamic coefficients and responses) of the floating bodies are illustrated and discussed.

Small amplitude waves and stability for a pre-stressed viscoelastic solid

We study the propagation of small amplitude waves superimposed on a large static deformation in a nonlinear viscoelastic material of differential type. We use bulk waves and surface waves to address the questions of dissipation and of material and geometric stability. In particular, the analysis provides bounds on the constitutive parameters and on the predeformation that ensure linearized stability in the neighbourhood of a large pre-stretch. This type of result is relevant to the imaging of biological soft tissues using acoustical techniques, where pre-deformation is known to increase contrast and reduce de-correlation noise.

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.