Nonlinear surface waves in soft, weakly compressible elastic media

Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.     

Guided and standing Bloch waves in periodic elastic strips

Elastic waveguiding structures are investigated for guides constructed of periodically alternating media thereby forming a striped waveguide. The guide has finite width with either homogeneous traction-free or clamped boundary conditions on the guide walls, or a combination of these. The band spectrum and associated Floquet–Bloch eigensolutions for these in-plane elastic waveguides are identified. Several features of this guiding structure emerge, and are of interest; in some cases a total stop band at zero frequency is identified providing space for low frequency localized modes: such modes also appear when we create defects in the structured waveguide. The dispersion curves often have maxima and minima of the spectral edges far from the edges of the Brillouin zone and these are related to slow sound or standing waves within the structure. Numerical and asymptotic techniques are developed and discussed, the latter are based on weak contrast, and weak geometric changes, or utilize jump conditions in limits where one medium is thin relative to the other. There are technological applications that could utilize this theory and we demonstrate that imaging is possible using anomalous dispersion.     

Guided waves in a stratified half-space

The dispersion and excitation mechanisms and the energy distribution of guided waves in a stratified half-space are studied. All possible guided waves excited by a symmetric point source in two or three-layer medium models and their relation to the medium parameters are analyzed in detail. The excitation and propagation characteristics, as well as the energy distribution along the depth direction, of all modes of the surface waves and trapped waves are numerically investigated and analyzed thoroughly not only in the case when the shear wave velocity increases from up to down layers but also when a low-velocity layer is contained in halfspace, especially when the shear wave velocity decreases from up to down layers. It is found that there exist many guided wave modes in the case where the shear wave velocity of each layer increases from up to down layers. However, there is less than one guided wave mode in the case where the shear wave velocity of each layer decreases from up to down layers. The trapped waves exist and propagate along the low-velocity structure in the stratified half-space. It is also found that the characteristic of a mode is related to the source frequency. It is possible that a surface wave at one value of frequency is like a trapped wave at another value of frequency. Finally, the relation of the characteristics of all guided waves (surface waves and trapped waves) to the parameters of media is studied.     

Guided waves in elastic plates with Gaussian section variation: experimental and numerical results

Experimental and numerical results are presented on the behavior of guided waves in elastic plates in plane strain that include a Gaussian variation of their section, located between two areas of constant thickness. The area of varying section is wide compared to the used wavelengths, which allows wave propagation inside this area. The experimental results show that an incident Lamb wave is indeed converted into an adiabatic wave inside the varying section domain. A trapped wave in the Gaussian domain is also observed, depending on the incident mode and on the Gaussian maximum height. Outside the varying section domain, conversion into different Lamb waves is observed. This conversion phenomenon is experimentally quantified by the measurement of the Lamb wave normal displacement and of its carried energy. A numerical study, based on the Finite Elements Method is performed, and successfully compared to the experimental results.     

Guided waves in elastic rectangular bars:A solution using partial plate mode decomposition

Similar to Auld's solution for Lamb waves,the wave modes in elastic rectangular bar are solved by partial wave decomposition method.The partial waves are composed of plate modes with the same wavenumber component in waveguide longitudinal direction,thus free boundary conditions on one pair of opposite surfaces are automatically satisfied.Based on completeness assumption and orthogonality of the plate modes,four independent eigenequations are eventually derived for dispersion curve and mode shape investigation.Numerical evaluation shows the calculated results are in consistent with the FEM results.It is then verified that the plate modes which obliquely bounced back and forth between the two opposite surfaces compose the guided modes traveling in the rectangular waveguides with certain wave numbers in transversely resonant cases.     

Hydromagnetic surface waves in a moving compressible plasma column

Summary The dispersive characteristic of hydromagnetic surface waves along a plasma-plasma interface when one of the fluids has a relative motion has been studied as a function of the compressibility factors ₁/V ₁, wheres ₁ andV ₁ are the acoustic and Alfvén wave speed in one of the media. Both slow and fast magnetosonic surface waves for each symmetric and asymmetric modes can exist. The nature and existence of these modes depend on the values ofs ₁/V ₁ and ϑ, the angle of wave propagation. The phase velocity of the slow wave increases whereas for the fast wave it decreases with increase in the angle ϑ. The authors of this paper have agreed to not receive the proofs for correction.     

Guided waves in anisotropic media

This paper investigates the propagation of electromagnetic waves in a dielectric anisotropic medium containing conducting planes. A plane and rectangular waveguide is considered for certain particular cases of the orientation of the principal axis of the anisotropic medium relative to the waveguide coordinates. The dispersion equations for the propagation constants are deduced.In conclusion we thank D. A. Dobrotin for his advice.     

Guided waves in a transversely isotropic cylinder immersed in a fluid

Propagation of flexural guided waves in a fluid-loaded transversely isotropic cylinder is studied. Numerical results are presented for a cobalt cylinder immersed in water. The phase velocities are not significantly affected except for several modes in which the energy leakage occurs into the fluid over certain frequency ranges. Attenuation spectra for the leaking modes are plotted.     

Internal waves in a compressible two-layer model atmosphere: Hamiltonian description

We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential in each of the layers (v _{1, 2} = ▿ϕ_{1, 2}) and neglecting the acoustic degrees of freedom (div($ \bar \rho $ \bar \rho (z)▿ϕ_{1, 2}) ≈ 0, where $ \bar \rho $ \bar \rho (z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ₁ − ψ₂. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$ \bar \rho $ \bar \rho (η)η _t ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential vorticity. We investigate a system with a model dependence $ \bar \rho $ \bar \rho (z) ∝ e ^−2αz with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion corrections and derive an approximate nonlinear equation of the form u _t + auu _x − b[−$ \hat \partial _x^2 $ \hat \partial _x^2 + α²]^1/2 u _x = 0 (Smith’s equation) for the slow evolution of a traveling wave.     

Harmonics of Tollmien — Schlichting waves in a compressible boundary layer

The harmonics of Tollmien—Schlichting waves in a compressible boundary layer of a plate are computed with the aid of nonlinear parabolized stability equations. At the (downstream) growth of the second harmonic amplitude up to the values of the order of the basic harmonic amplitude, the amplification rate of the latter is shown to increase abruptly. A similar rapid deviation from the results of the linear theory characterizes the onset of the boundary-layer transition to turbulent state. Computations are carried out for the Mach numbers M = 0.01 and 2. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 05-01-00079a).     

Dispersion of elastic waves in a triangular bar

It is shown that in a bar with an equilateral triangular cross-section longitudinal, torsional, and bending modes of wave propagation are possible. The first few branches of the dispersion curves for each of these modes have been calculated using the collocation method. The first branch of the longitudinal mode shows excellent agreement with experiment. The existence of an “end-resonance” is inferred from the experimental results.     

Helical waves in a fluid-filled cylindrical elastic shell

Helical normal waves of a waveguide that has the form of a fluid-filled thin elastic shell are considered. A dispersion equation for this kind of wave is derived, and its solutions are presented for certain values of parameters characterizing the problem.     

Guided waves in a monopile of an offshore wind turbine

We study the guided waves in a structure which consists of two overlapping steel plates, with the overlapping section grouted. This geometry is often encountered in support structures of large industrial offshore constructions, such as wind turbine monopiles. It has been recognized for some time that the guided wave technology offers distinctive advantages for the ultrasonic inspections and health monitoring of structures of this extent. It is demonstrated that there exist advantageous operational regimes of ultrasonic transducers guaranteeing a good inspection range, even when the structures are totally submerged in water, which is a consideration when the wind turbines are deployed off shore.     

Nonlinear waves in elastic media

We ask about the possible existence of solitary waves in infinite, homogeneous, isotropic, elastic media. Namely, can a nonlinear localized wave packet propagate without altering its shape in such materials? We consider one- dimensional propagation both of body and surface waves. In the first case we show, under rather general assumptions, that if a wave packet propagates without altering its shape it must, of necessity, be a solution of a linear wave equation and in this sense, (body) solitary waves do not exist. Surface solitary waves may however exist: a model equation is derived in which nonlinear and dispersive effects balance each other to allow for waves-both periodic and solitary-of constant shape. It is conceivable they are of some relevance in seismology.     

Modelling wave dynamics of compressible elastic materials

An Eulerian conservative hyperbolic model of isotropic elastic materials subjected to finite deformation is addressed. It was developed by Godunov [S.K. Godunov, Elements of continuum mechanics, Nauka, Moscow, 1978 (in Russian) and G.H. Miller, P. Colella, A high-order Eulerian Godunov method for elastic–plastic flow in solids, J. Comput. Phys. 167 (2001) 131–176]. Some modifications are made concerning a more suitable form of governing equations. They form a set of evolution equations for a local cobasis which is naturally related to the Almansi deformation tensor. Another novelty is that the equation of state is given in terms of invariants of the Almansi tensor in a form which separates hydrodynamic and shear effects. This model is compared with another hyperbolic non-conservative model which is widely used in engineering sciences. For this model we develop a Riemann solver and determine some reference solutions which are compared with the conservative model. The numerical results for different tests show good agreement of both models for waves of very small and very large amplitude. However, for waves of intermediate amplitude important discrepancies between results are clearly visible.     

Propagation of elastic waves in a nonideal layered medium

The peculiarities of propagation of acoustic excitations through an imperfect 1D superlattice are studied in the virtual crystal approximation. The dependence of the lowest acoustic band gap of a nonideal (disordered in composition) two-sublattice 1D phonon crystal on the concentration of impurity layers is simulated numerically.     

The anisotropy of elastic waves in a tellurium crystal

For acoustic waves propagating in an acoustooptic tellurium crystal, the dependence of their polarization on the propagation direction with respect to the crystal axes is discussed. The characteristic features of waves propagating in the crystal are considered; these features manifest themselves in an excess of the phase velocity of shear acoustic modes over the velocity of longitudinal modes. The change in the wave type from quasi-longitudinal to quasi-transverse as a result of the variation in the propagation direction of ultrasound is investigated. It is shown that such a behavior of bulk acoustic waves is caused by the specific relation between the elastic moduli, which differs from the corresponding relations observed in other acoustooptic materials.