Fast volumetric integral-equation solver for acoustic wave propagation through inhomogeneous media

Elements are described of a volumetric integral-equation-based algorithm applicable to accurate large-scale simulations of scattering and propagation of sound waves through inhomogeneous media. The considered algorithm makes possible simulations involving realistic geometries characterized by highly subwavelength details, large density contrasts, and described in terms of several million unknowns. The algorithm achieves its competitive performance, characterized by O(N log N) solution complexity and O(N) memory requirements, where N is the number of unknowns, through a fast and nonlossy fast Fourier transform based matrix compression technique, the adaptive integral method, previously developed for solving large-scale electromagnetic problems. Because of its ability of handling large problems with complex geometries, the developed solver may constitute an efficient and high fidelity numerical simulation tool for calculating acoustic field distributions in anatomically realistic models, e.g., in investigating acoustic energy transfer to the inner ear via nonairborne pathways in the human head. Examples of calculations of acoustic field distribution in a human head, which require solving linear systems of equations involving several million unknowns, are presented.     

次声波在非均匀运动大气中非线性传播特性的研究

基于有限时域差分方法将大气中近似到二阶的非线性波动方程进行离散化,得到了数值模拟所采用的差分方程. 在此基础上,对线阵列辐射的脉冲声波在非均匀运动大气中的垂直和斜向传播进行了二维数值模拟,模拟了武汉地区(114:20°E, 30:37°N)在夏季和冬季UT=29000 s时开始传播的脉冲声波在不同时刻的声压分布. 模拟时通过采用Msise00和HWM93 两个大气模型,考虑了由于大气温度和密度变化以及大气风场存在所引起的大气不均匀性和运动性. 通过研究上述两季有风与无风条件下的声压差值p_r,可以发现:风场对次声波在传播中声压分布的影响较大;由于不同季节和不同传播距离上"有效声速"的不同,导致了两季p_r分布波形存在差异;风场对声波非线性传播的影响要远大于其对线性传播的影响. 关键词： 次声波传播 非均匀运动大气 有效声速     

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.     

Some analytical results on nonlinear acoustic wave propagation in diffusive media

Royal Institute of Technology, Stockholm. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 7, pp. 665–686, July, 1993.     

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.     

A contrast source method for nonlinear acoustic wave fields in media with spatially inhomogeneous attenuation

Experimental data reveals that attenuation is an important phenomenon in medical ultrasound. Attenuation is particularly important for medical applications based on nonlinear acoustics, since higher harmonics experience higher attenuation than the fundamental. Here, a method is presented to accurately solve the wave equation for nonlinear acoustic media with spatially inhomogeneous attenuation. Losses are modeled by a spatially dependent compliance relaxation function, which is included in the Westervelt equation. Introduction of absorption in the form of a causal relaxation function automatically results in the appearance of dispersion. The appearance of inhomogeneities implies the presence of a spatially inhomogeneous contrast source in the presented full-wave method leading to inclusion of forward and backward scattering. The contrast source problem is solved iteratively using a Neumann scheme, similar to the iterative nonlinear contrast source (INCS) method. The presented method is directionally independent and capable of dealing with weakly to moderately nonlinear, large scale, three-dimensional wave fields occurring in diagnostic ultrasound. Convergence of the method has been investigated and results for homogeneous, lossy, linear media show full agreement with the exact results. Moreover, the performance of the method is demonstrated through simulations involving steered and unsteered beams in nonlinear media with spatially homogeneous and inhomogeneous attenuation.     

Interference effects in wave propagation in periodically inhomogeneous media

Radiophysics Scientific-Research Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 9, pp. 1144–1151, September, 1989.     

Equation for wave processes in inhomogeneous moving media and functional solution of the acoustic tomography problem based on it

The paper considers the derivation of the wave equation and Helmholtz equation for solving the tomographic problem of reconstruction combined scalar-vector inhomogeneities describing perturbations of the sound velocity and absorption, the vector field of flows, and perturbations of the density of the medium. Restrictive conditions under which the obtained equations are meaningful are analyzed. Results of numerical simulation of the two-dimensional functional-analytical Novikov–Agaltsov algorithm for reconstructing the flow velocity using the the obtained Helmholtz equation are presented.     

Angular spectrum approach to electromagnetic wave propagation in inhomogeneous media

The angular spectrum representation of the electromagnetic wave field is employed to solve the wave propagation in a weakly inhomogeneous medium. Taking the two-dimensional spatial Fourier transform of the radiation field as well as of the dielectric constant, the angular amplitude is shown to satisfy an integro-differential equation. A similar equation is also applicable for the propagation of radiation in a non-linear medium. This integro-differential equation is solved for two specific cases of interest, namely that of a stratified medium and of a square-law medium.     

Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber

In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.     

Coherent acoustic wave propagation in media with pair-correlated spheres

Propagation of plane compressional waves in a non-viscous fluid with a dense distribution of identical spherical scatterers is investigated. The analysis is based on the multiple scattering approach proposed by Fikioris and Waterman, and is generalized to include arbitrary choice of the pair-correlation functions used to represent the distribution of the scatterers. A closed form solution for the effective wavenumber as a function of the concentration of pair-correlated finite-size spheres is derived up to the second order. In the limit of uncorrelated point-scatterers, this solution is identical to that obtained by Lloyd and Berry. Different pair-correlation functions are exemplified and compared, and the resulting differences discussed.     

Wave envelopes method for description of nonlinear acoustic wave propagation

A novel, free from paraxial approximation and computationally efficient numerical algorithm capable of predicting 4D acoustic fields in lossy and nonlinear media from arbitrary shaped sources (relevant to probes used in medical ultrasonic imaging and therapeutic systems) is described. The new WE (wave envelopes) approach to nonlinear propagation modeling is based on the solution of the second order nonlinear differential wave equation reported in [J. Wójcik, J. Acoust. Soc. Am. 104 (1998) 2654-2663; V.P. Kuznetsov, Akust. Zh. 16 (1970) 548-553]. An incremental stepping scheme allows for forward wave propagation. The operator-splitting method accounts independently for the effects of full diffraction, absorption and nonlinear interactions of harmonics. The WE method represents the propagating pulsed acoustic wave as a superposition of wavelet-like sinusoidal pulses with carrier frequencies being the harmonics of the boundary tone burst disturbance. The model is valid for lossy media, arbitrarily shaped plane and focused sources, accounts for the effects of diffraction and can be applied to continuous as well as to pulsed waves. Depending on the source geometry, level of nonlinearity and frequency bandwidth, in comparison with the conventional approach the Time-Averaged Wave Envelopes (TAWE) method shortens computational time of the full 4D nonlinear field calculation by at least an order of magnitude; thus, predictions of nonlinear beam propagation from complex sources (such as phased arrays) can be available within 30-60 min using only a standard PC. The approximate ratio between the computational time costs obtained by using the TAWE method and the conventional approach in calculations of the nonlinear interactions is proportional to 1/N2, and in memory consumption to 1/N where N is the average bandwidth of the individual wavelets. Numerical computations comparing the spatial field distributions obtained by using both the TAWE method and the conventional approach (based on a Fourier series representation of the propagating wave) are given for circular source geometry, which represents the most challenging case from the computational time point of view. For two cases, short (2 cycle) and long (8 cycle) 2 MHz bursts, the computational times were 10 min and 15 min versus 2 h and 8 h for the TAWE method versus the conventional method, respectively.     

On a fourth-order wave equation for EM propagation in chiral media

A fourth-order wave equation is derived to study the propagation of an electromagnetic (EM) wave in a medium composed of chiral objects. This approximate wave equation does not reflect the handedness of the medium directly, which is more readily apparent, however, in a pair of two coupled second-order wave equations obtaineden route.     

Waveguide propagation of intense electromagnetic radiation in slightly inhomogeneous nonlinear media

The propagation of self-localizing beams of electromagnetic waves in the form of nonlinear waveguides in a slightly inhomogeneous medium is studied analytically and numerically. The trajectories of the axial ray are studied as a function of its direction and the field strength at the initial point on the basis of a nonlinear scalar Helmholtz equation. Analytic expressions are derived. The longitudinal refractive index, the field intensity, and the waveguide radius are plotted as functions of the instantaneous position of the point on the axial ray. Deep penetration of the beam into the opaque region and the position of the screening surface are studied as functions of the parameters of the beam and the medium. A steady-state 3D problem is analyzed for a power-law nonlinearity with an arbitrary power. A 2D problem is analyzed for the case of a ponderomotive nonlinearity with saturation.     

Nonlinear and diffraction effects in propagation of N-waves in randomly inhomogeneous moving media

Finite amplitude acoustic wave propagation through atmospheric turbulence is modeled using a Khokhlov-Zabolotskaya-Kuznetsov (KZK)-type equation. The equation accounts for the combined effects of nonlinearity, diffraction, absorption, and vectorial inhomogeneities of the medium. A numerical algorithm is developed which uses a shock capturing scheme to reduce the number of temporal grid points. The inhomogeneous medium is modeled using random Fourier modes technique. Propagation of N-waves through the medium produces regions of focusing and defocusing that is consistent with geometrical ray theory. However, differences up to ten wavelengths are observed in the locations of fist foci. Nonlinear effects are shown to enhance local focusing, increase the maximum peak pressure (up to 60%), and decrease the shock rise time (about 30 times). Although the peak pressure increases and the rise time decreases in focal regions, statistical analysis across the entire wavefront at a distance 120 wavelengths from the source indicates that turbulence: decreases the mean time-of-flight by 15% of a pulse duration, decreases the mean peak pressure by 6%, and increases the mean rise time by almost 100%. The peak pressure and the arrival time are primarily governed by large scale inhomogeneities, while the rise time is also sensitive to small scales.     

Solutions for the propagation of light in nonlinear optical media with spatially inhomogeneous nonlinearities

In this paper, we present solutions for the nonlinear Schrödinger (NLS) equation with spatially inhomogeneous nonlinearities describing propagation of light in nonlinear media, under two sets of transverse modulation forms of inhomogeneous nonlinearity. The bright soliton solution and Gaussian solution have been obtained for one set of inhomogeneous nonlinearity modulation. For the other, bright soliton solution, black soliton solution and the train solution have been presented. Stability of the solutions has been determined by exact soliton solutions under certain conditions.