Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian

The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many three-dimensional problems of interest. Here, a wave equation that utilizes two lossy derivative operators based on the fractional Laplacian is derived. These operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into Fourier based pseudospectral and k-space methods without the increase in memory required by their time-domain fractional counterparts. A framework for encoding the developed wave equation using three coupled first-order constitutive equations is discussed, and the model is demonstrated through several one-, two-, and three-dimensional simulations.     

A k-space Green's function solution for acoustic initial value problems in homogeneous media with power law absorption

An efficient Green's function solution for acoustic initial value problems in homogeneous media with power law absorption is derived. The solution is based on the homogeneous wave equation for lossless media with two additional terms. These terms are dependent on the fractional Laplacian and separately account for power law absorption and dispersion. Given initial conditions for the pressure and its temporal derivative, the solution allows the pressure field for any time t>0 to be calculated in a single step using the Fourier transform and an exact k-space time propagator. For regularly spaced Cartesian grids, the former can be computed efficiently using the fast Fourier transform. Because no time stepping is required, the solution facilitates the efficient computation of the pressure field in one, two, or three dimensions without stability constraints. Several computational aspects of the solution are discussed, including the effect of using a truncated Fourier series to represent discrete initial conditions, the use of smoothing, and the properties of the encapsulated absorption and dispersion.     

k-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics

Biomedical applications of photoacoustics, in particular photoacoustic tomography, require efficient models of photoacoustic propagation that can incorporate realistic properties of soft tissue, such as acoustic inhomogeneities both for purposes of simulation and for use in model-based image reconstruction methods. k-space methods are well suited to modeling high-frequency acoustics applications as they require fewer mesh points per wavelength than conventional finite element and finite difference models, and larger time steps can be taken without a loss of stability or accuracy. They are also straightforward to encode numerically, making them appealing as a general tool. The rationale behind k-space methods and the k-space approach to the numerical modeling of photoacoustic waves in fluids are covered in this paper. Three existing k-space models are applied to photoacoustics and demonstrated with examples: an exact model for homogeneous media, a second-order model that can take into account heterogeneous media, and a first-order model that can incorporate absorbing boundary conditions.     

A unified treatment of nonclassical nonlinear effects in the propagation of ultrasound in heterogeneous media

Recent experiments on rocks and other materials, such as soil, cement, concrete and damaged elastic materials, have led to the discovery of nonlinear (NL) hysteretic effects in their elastic behaviour. These observations suggest the existence of a NL mesoscopic elasticity universality class, to which all the aforementioned materials belong. The purpose of the present contribution is to search for the basic mathematical roots for nonclassical nonlinearity, in order to explain its universality, classify it and correlate it with the underlying meso- or microscopic interaction mechanisms. In our discussions we explicitly consider two quite different kinds of specimens: a two-bonded-elements structure and a thin multigrained bar. It is remarkable that, although the former includes only one interface and the latter very many interstices, the same "interaction box" formalism can be applied to both. Another important result of the proposed formalism is that the spectral contents of an arbitrary system for any input amplitude may be predicted, under certain assumptions, from the result of a single experiment at a higher amplitude.     

Modeling of nonlinear ultrasound propagation in tissue from array transducers

A computationally efficient model capable of simulating finite-amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described. It is based on a second-order operator splitting approach [Tavakkoli et al., J. Acoust. Soc. Am. 104, 2061-2072 (1998)], with a fractional step-marching scheme, whereby the effects of diffraction, attenuation, and nonlinearity can be computed independently over incremental steps. This approach is an extension to that of Christopher and Parker [J. Acoust. Soc. Am. 90, 507-521; 90, 488-499 (1991)], wherein linear and nonlinear effects are propagated separately over incremental steps, and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources. Results of the model are compared with published data. Predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented.     

Modeling the propagation of nonlinear three-dimensional acoustic beams in inhomogeneous media

A three-dimensional model of the forward propagation of nonlinear sound beams in inhomogeneous media, a generalized Khokhlov-Zabolotskaya-Kuznetsov equation, is described. The Texas time-domain code (which accounts for paraxial diffraction, nonlinearity, thermoviscous absorption, and absorption and dispersion associated with multiple relaxation processes) was extended to solve for the propagation of nonlinear beams for the case where all medium properties vary in space. The code was validated with measurements of the nonlinear acoustic field generated by a phased array transducer operating at 2.5 MHz in water. A nonuniform layer of gel was employed to create an inhomogeneous medium. There was good agreement between the code and measurements in capturing the shift in the pressure distribution of both the fundamental and second harmonic due to the gel layer. The results indicate that the numerical tool described here is appropriate for propagation of nonlinear sound beams through weakly inhomogeneous media.     

Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method

An analytical solution is presented for nonlinear surface wave propagation. A variational iteration method (VIM) was employed and time-dependent profiles of the surface elevation level and velocity obtained analytically for different initial conditions. It is shown that the VIM used here is a flexible and accurate approach and that it can rapidly converge to the same results obtained by the Adomian decomposition method.     

Wave propagation in nonlinear disordered media

We analyze mechanisms and regimes of wave packet spreading in nonlinear disordered media. We discuss resonance probabilities, and predict a dynamical crossover from strong to weak chaos. The crossover is controlled by the ratio of nonlinear frequency shifts and the average eigenvalue spacing of eigenstates of the linear equations within one localization volume. We consider generalized models in higher lattice dimensions and obtain critical values for the nonlinearity power, the dimension, and norm density, which influence possible dynamical outcomes in a qualitative way.     

Wave propagation in heterogeneous bistable and excitable media

Two examples for the propagation of traveling waves in spatially non-uniform media are studied: (a) bistable media with periodically varying excitation threshold and (b) bistable and excitable media with randomly distributed diffusion coefficient and excitation properties. In case (a), we have applied two different singular perturbation techniques, namely averaging (first and second order) and a projection method, to calculate the averaged front velocity as a function of the spatial period L of the heterogeneity for the Schlögl model. Our analysis reveals a velocity overshoot for small values of L and propagation failure for large values of L. The analytical predictions are in good agreement with results of direct numerical simulations. For case (b), effective medium properties are derived by a self-consistent homogenization approach. In particular, the resulting velocities found by direct numerical simulations of the random medium are reproduced well as long as the diffusion lengths in the medium are larger than the heterogeneity scale. Simulations reveal also that complex irregular dynamics can be triggered by heterogeneities.     

Acoustic wave propagation in two-phase heterogeneous porous media

The propagation of an acoustic wave through two-phase porous media with spatial variation in porosity is studied. The evolutionary wave equation is derived, and the propagation of an acoustic wave is numerically analyzed in application to marine sediments with various physical parameters.     

The propagation of ultrasound in porous media

The equations of Ying and Truell, and Waterman and Truell, describing the propagation of ultrasound in two-phase materials are solved numerically for porous solids, and are found to give unphysical results for high porosity. A new self-consistent theory, which can be solved analytically, is presented and is shown to have reasonable behaviour at high porosity.     

Soliton propagation in heterogeneous and random media of a specific class

The motion of solitons in a medium whose parameters vary randomly but so that a stochastic nonlinear equation remains fully integrable is considered. It is found that, in this case, the position of the soliton maximum executes Brownian motion, while its phase becomes random.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 91–97, April, 1987.     

Light propagation in media with a highly nonlinear response: An analytical study

The problem of light propagation in highly nonlinear media is studied with the help of a recently introduced systematic approach to the analytical solution of equations of nonlinear optics [L.L. Tatarinova, M.E. Garcia, Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics, Phys. Rev. A 78 (2008) 021806(R)(1—4)]. Numerous particular cases of media exhibiting high-order nonlinear refractive indices are considered. We obtain analytical expressions for determining the self-focusing position and a new exact expression for calculating the filament intensity. The constructed solutions allowed us to revise a so-called self-focusing scaling law, i.e., the functional dependence of the self-focusing position on the initial light peak intensity. It was demonstrated that this dependence is governed by the form of the nonlinear refractive index and not by the laser beam shape at the boundary.     

Simulations of optoacoustic wave propagation in light-absorbing media using a finite-difference time-domain method

Optoacoustic (OA) imaging is an emerging technology that combines the high optical contrast of tissues with the high spatial resolution of ultrasound. Taking full advantage of OA imaging requires a better understanding of OA wave propagation in light-absorbing media. Current simulation methods are mainly based on simplified conditions such as thermal confinement, negligible viscosity, and homogeneous acoustic properties throughout the image object. In this study a new numerical approach is proposed based on a finite-difference time-domain (FDTD) method to solve the general OA equations, comprising the continuity, Navier-Stokes, and heat-conduction equations. The FDTD code was validated using a benchmark problem that has an approximate analytical solution. OA experiments were also conducted and data were in good agreement with those predicted by the FDTD method. Characteristics of simulated OA waveforms and OA images were discussed. The simulator was also employed to study wavefront distortion in OA breast imaging.     

Modeling of wave propagation in inhomogeneous media

We present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time-reversal invariance and reciprocity in generally inhomogeneous media. The approach relies on a representation theorem of the wave equation to express the Green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium. Following a predictable initial computational effort, Green's functions between arbitrary points in the medium can be computed as needed using a simple cross-correlation algorithm.     

Spatio-temporal pulse propagation in nonlinear dispersive optical media

We discuss state-of-art approaches to modeling of propagation of ultrashort optical pulses in one and three spatial dimensions. We operate with the analytic signal formulation for the electric field rather than using the slowly varying envelope approximation, because the latter becomes questionable for few-cycle pulses. Suitable propagation models are naturally derived in terms of unidirectional approximation.     

Chirped optical pulse propagation in saturating nonlinear media

Using a variational method, we have investigated the propagation characteristics of a chirped optical pulse in anomalously dispersive media possessing saturating nonlinearity. For the special case of uniform loss less media, the dynamics of the temporal width of the pulse is shown to be equivalent to an oscillator of unit mass which is executing its motion under some effective potential well. The potential is examined and four different types of behavior of the pulse width are noticed. The role of saturation parameter and the initial chirp in determining the propagation characteristics have been examined. It is found that, both high value of chirp and saturation are detrimental to stable pulse propagation. Particularly, the effect of chirp becomes severe with the increase in the value of saturation. We have shown that incorporation of saturation in the nonlinearity leads to the existence of bistable soliton. For the case of a lossy medium, net broadening of width takes place over many cycles of oscillation. The net broadening decreases with the increase in the value of saturation.     

Dark optical solitons in power law media with time-dependent coefficients

This Letter talks about the dynamics of dark optical solitons that are governed by the nonlinear Schrödinger's equation with power law nonlinearity. The solitons are considered in presence of linear attenuation, third order dispersion and self-steepening terms, all with time-dependent coefficients. The solitary wave ansatz is used to carry out the integration and an exact soliton solution is obtained. It is only necessary that these time-dependent coefficients are Riemann integrable.     

Experimental study of surface wave propagation in strongly heterogeneous media

In the present paper, the propagation of Rayleigh waves in a strongly heterogeneous medium is discussed. Scattering of stress waves is a difficult scientific problem. Specifically, the interaction of surface waves with distributed inhomogeneity seems highly complicated due to the existence of two displacement components. Rayleigh waves undergo significant attenuation and velocity change depending on the frequency and the inhomogeneity content. The aim of this study is to highlight the dispersive behavior of concrete, especially when damaged, and increase the experimental data in an area where the work is limited.