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.     

Frequency-domain wave equation and its time-domain solutions in attenuating media

Our purpose in this paper is to describe the wave propagation in media whose attenuation obeys a frequency power law. To achieve this, a frequency-domain wave equation was developed using previously derived causal dispersion relations. An inverse space and time Fourier transform of the solution to this algebraic equation results in a time-domain solution. It is shown that this solution satisfies the convolutional time-domain wave equation proposed by Szabo [J. Acoust. Soc. Am. 96, 491-500 (1994)]. The form of the convolutional loss operator contained in this wave equation is obtained. Solutions representing the propagation of both plane sinusoidal and transient waves propagating in media with specific power law attenuation coefficients are investigated as special cases of our solution. Using our solution, comparisons are made for transient one-dimensional propagation in a medium whose attenuation is proportional to frequency with recently obtained numerical solutions of Szabo's equation. These show good agreement.     

Reflexion of a spherical wave by the plane interface between a perfect fluid and a porous medium

By using the Fourier transform of the system of equations and continuity conditions, it is easy to obtain the Fourier transform of the solution. This last function is decomposed into several terms which are identified as Fourier images of known functions. An exact representation of the scattered sound pressure field is obtained as a combination of the radiation of the image source and layer potentials. Approximations are given when the point spherical source is located on the interface.     

Regularity of Solutions to Vorticity Navier–Stokes System on ℝ<Superscript>2</Superscript>

The Cauchy problem for the Navier–Stokes system for vorticity on plane is considered. If the Fourier transform of the initial data decays as a power at infinity, then at any positive time the Fourier transform of the solution decays exponentially, i.e. the solution is analytic.     

Calculating the forced response of two-dimensional homogeneous media using the wave and finite element method

The forced response of two-dimensional, infinite, homogenous media subjected to time harmonic loading is treated. The approach starts with the wave and the finite element (WFE) method where a small segment of a homogeneous medium is modelled using commercial or in-house finite element (FE) packages. The approach is equally applicable to periodic structures with a periodic cell being modelled. This relatively small model is then used, along with periodicity conditions, to formulate an eigenvalue problem whose solution yields the wave characteristics of the whole medium. The eigenvalue problem involves the excitation frequency and the wavenumbers (or propagation constants) in the two directions. The wave characteristics of the medium are then used to obtain the response of the medium to a convected harmonic pressure (CHP). Since the Fourier transform of a general two-dimensional excitation is a linear combination of CHPs, the response to a general excitation is a linear combination of the responses to CHPs. Thus, the response of a two-dimensional medium to a general excitation can be obtained by evaluating an inverse Fourier transform. This is a double integral, one of which is evaluated analytically using contour integration and the residue theorem. The other integral can be evaluated numerically. Hence, the approach presented herein enables the response of an infinite two-dimensional or periodic medium to an arbitrary load to be computed via (a) modelling a small segment of the medium using standard FE methods and post-processing its model to obtain the wave characteristics, (b) formulating the Fourier transform of the response to a general loading, and (c) computing the inverse of the Fourier transform semi-analytically via contour integration and the residue theorem, followed by a numerical integration to find the response at any point in the medium. Numerical examples are presented to illustrate the approach.     

Acoustic radiation of cylindrical elastic shells subjected to a point source: investigation in terms of helical acoustic rays.

The paper deals with the acoustic radiation of a cylindrical elastic shell with no internal loading surrounded by a fluid medium when its external surface is subjected to a point source. The problem is addressed via the use of the spatial Fourier transform. An expression is obtained for the radiated pressure that is evaluated for the far field using both the stationary phase method and the fast Fourier transform (FFT). The acoustic field calculated from the FFT is much more complicated than that obtained by using only the stationary phase method. In agreement with the geometrical theory of diffraction (GTD), alternative interpretations of the radiated field in terms of helical acoustic rays allows one to understand the reason for this result. The outstanding phenomenon underlined by the use of the FFT is the emergence of an infinite number of spatial dispersion curves associated with each leaky wave propagating in shells when excited by a point source.     

Analytical representation of the surface wave generated by an antenna at the interface between two homogeneous media

Theoretical approach to the solution of georadiolocation (inverse) problem of reconstructing the antenna current and the permittivity of the underlying surface, based on the measurements of the field propagating along the Earth℉s surface, is proposed. An expression for the tangential component of the electric field of the surface wave from a transmitting antenna lying in the plane of discontinuity of two homogeneous media is expressed in terms of the Laplace function Ŵ(q, ζ) of two complex arguments for any discontinuity parameters. The function Ŵ(q, ζ) is presented as series, which make it possible to calculate rapidly the values of Ŵ(q, ζ) and investigate its analytic properties. The inverse Laplace transform is performed by integrating the function Ŵ(q, ζ) over closed contours. The corresponding integrals are proper in this case, due to which the calculation time greatly decreases in comparison with the Fourier transform. When the permittivity dispersion can be neglected, the tangential component of the surface wave is algebraically expressed in terms of the function Ŵ(q, ζ). This circumstance allows one to decrease the time for calculating the surface wave at a specified point to several milliseconds and determine both the current through the transmitting antenna and the permittivity of the underlying surface for a few seconds.     

声测井中的声压-速度有限差分方法

为了解决薄互储层的声测井问题,提出了声压-速度有限差分方法:用声压和速度矢量做为场变量,分别描述井内流体和井外弹性固体或双相介质。这样选择场变量的优点是:处理脉冲点源(或线源)与套网格技术相比简单得多;在内边界上的差分格式稳定,精度得到了改进;人为边界上的吸收效果较好。用柱坐标分别给出了井壁上流体与弹性固体、流体与双相介质的声压-速度边界条件,并用守恒积分方法处理了井壁上的边界条件。通过用声压-速度有限差分方法模拟弹性固体和双相介质地层的声场,证明了声压-速度有限差分方法的有效性。     

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.     

Transient propagation in media with classical or power-law loss

This paper addresses the problem of small-signal transient wave propagation in media whose absorption coefficient obeys power-law frequency dependence, i.e., alpha infinity omega n. Our approach makes use of previously derived relations between the absorption and dispersion based on the Kramers-Kronig relations. This, combined with a recently obtained solution to a causal convolution wave equation enable expressions to be obtained for one-dimensional transient propagation when n is in the range 0 < n < 3. For n = 2, corresponding to no dispersion, straightforward analytical expressions are obtained for a delta-function and a sinusoidal step function sources and these are shown to correspond to relations previously derived. For other values of n, the effects of dispersion are accounted for by using Fourier transforms. Examples are used to illustrate the results for normal and anomalous dispersive media and to examine the question as to the conditions under which the effects of dispersion should be accounted for, especially for wideband ultrasound pulses of the type used in B-mode tissue imaging. It is shown that the product of the attenuation and total propagation path can be used as a criterion for judging whether dispersion needs to be accounted for.     

弛豫媒质中有限束超声波的非线性传播畸变理论

文中提出弛豫媒质中有限束非线性声波方程,并采用微扰法求得由非线性传播畸变产生的高次谐波的一般解.研究表明,对高斯型声波,其谐波畸变解可以解析给出,而且其径向分布始终维持高斯函数.虽然其频散量大小会影响各次谐波的振幅,但其相速的变化却仍与对在频率的小振幅波相同.文中还用Blackstock算子将所得的结果应用于任何吸收-频散媒质,包括只能从经验得到其吸收与频率关系的一些生物媒质.     

Nonlinear pulsed ultrasound beams radiated by rectangular focused diagnostic transducers

A numerical model for simulating nonlinear pulsed beams radiated by rectangular focused transducers, which are typical of diagnostic ultrasound systems, is presented. The model is based on a KZK-type nonlinear evolution equation generalized to an arbitrary frequency-dependent absorption. The method of fractional steps with an operator-splitting procedure is employed in the combined frequency-time domain algorithm. The diffraction is described using the implicit backward finite-difference scheme and the alternate direction implicit method. An analytic solution in the time domain is employed for the nonlinearity operator. The absorption and dispersion of the sound speed are also described using an analytic solution but in the frequency domain. Numerical solutions are obtained for the nonlinear acoustic field in a homogeneous tissue-like medium obeying a linear frequency law of absorption and in a thermoviscous fluid with a quadratic frequency law of absorption. The model is applied to study the spatial distributions of the fundamental and second harmonics for a typical diagnostic ultrasound source. The nonlinear distortion of pulses and their spectra due to the propagation in tissues are presented. A better understanding of nonlinear propagation in tissue may lead to improvements in nonlinear imaging and in specific tissue harmonic imaging. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 4, pp. 560–570. This article was translated by the authors.     

A PMD-supported 100-Gb/s optical frequency-domain IM-DD transmission system

A novel method for distortion-free optical pulse transmission is theoretically proposed and simulated, in which two time lenses formed by dispersion fibers and quadratic phase modulations are utilized. One is used as an optical inverse Fourier transformation （OIFT） device to transform the initial time-domain data to frequency-domain one at the transmitter and the other as an optical Fourier transformation （OFT） device to recover the data at the receiver. By using the unchanged spectral envelope in linear optical fiber communication, the initial data can be recovered. Through simulations, a 10× 100 Gb/s intensity-modulated direct-detection （IM-DD） dense wavelength division multiplexing （DWDM） system over 20000 km transmission without the compensation for polarization mode dispersion （PMD） and dispersion slope is achieved, which can be used to upgrade the current 100Gb/s IM-DD system to a 100-Gb/s one directly.     

基于PCA的变压器故障红外监测应用研究

变压器作为电力系统中非常重要的输变电设备,其运行状况直接影响到整个系统运行的安全水平。根据红外光谱仪所测得信息,通过傅里叶变换可得到各故障特征气体红外吸收信息;依据Lambert-Beer定律并结合所测信息,采用主成分分析法就可以对各故障特征气体进行定性与定量分析,判别当前变压器的运行状况。对于故障运行的电力变压器,及时发现并解除存在的故障或故障隐患对整个系统的安全运行具有重要意义。     

Numerical implementation of static Field Dislocation Mechanics theory for periodic media

This paper investigates the implementation of Field Dislocation Mechanics (FDM) theory for media with a periodic microstructure (i.e. the Nye dislocation tensor and the elastic moduli tensor are considered as spatially periodic continuous fields). In this context, the uniqueness of the stress and elastic distortion fields is established. This allows to propose an efficient numerical scheme based on Fourier transform to compute the internal stress field, for a given spatial distribution of dislocations and applied macroscopic stress. This numerical implementation is assessed by comparison with analytical solutions for homogeneous as well as heterogeneous elastic media. A particular insight is given to the critical case of stress-free dislocation microstructures which represent equilibrium solutions of the FDM theory.     

Counterpropagation of waves with shock fronts in a nonlinear tissue-like medium

A numerical model for describing the counterpropagation of one-dimensional waves in a nonlinear medium with an arbitrary power law absorption and corresponding dispersion is developed. The model is based on general one-dimensional Navier-Stokes equations with absorption in the form of a time-domain convolution operator in the equation of state. The developed algorithm makes it possible to describe wave interactions in the presence of shock fronts in media like biological tissue. Numerical modeling is conducted by the finite difference method on a staggered grid; absorption and sound speed dispersion are taken into account using the causal memory function. The developed model is used for numerical calculations, which demonstrate the absorption and dispersion effects on nonlinear propagation of differently shaped pulses, as well as their reflection from impedance acoustic boundaries.     

三能级钾原子气体三维傅里叶变换频谱的解析解

利用投影切片定理、傅里叶位移定理和误差函数给出三能级钾原子气体三维傅里叶变换频谱在T=0界面的解析解.固定均匀线宽,非均匀展宽和对角线相关系数可以定量地识别,通过在适当方向上拟合三维傅里叶变换频谱谱峰的切片来确定.结果表明,非均匀展宽增大,频谱图沿着对角线方向延伸,对角线相关系数增大,频谱图逐渐变圆,振幅也逐渐变小.     

Implementation of nonreflecting boundary conditions for the nonlinear Euler equations

Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier–Bessel modes. This leads to an ‘exact’ nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier–Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.     

Radiation of surface Stoneley waves by distributed force sources acting at the solid earth-atmosphere interface

Using the Fourier transform, we find an integral solution describing the excitation of seismoacoustic waves in the solid Earth and the atmosphere by time-dependent forces arbitrarily distributed over the interface between the media. The solid Earth and the atmosphere are modeled by an isotropic solid half-space and a homogeneous gaseous half-space, respectively. Depending on the types of the excited surface and bulk waves, classification of the corresponding force distributions is performed. In the case of harmonic sources, an expression for the period-averaged radiated power of the Stoneley wave is obtained. For arbitrary time dependence of the forces, we find an expression describing the the Stoneley-wave energy radiated during the entire time of the source operation. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 7, pp. 624–637, July 2007.     

单色光在损耗介质中激励的受激布里渊散射

研究了单色光在损耗介质中激励的受激布里渊散射，运用分布起伏噪声源模型描述了受激布里渊散射的产生过程，通过傅里叶变换，得到了输出斯托克斯光波的功率谱密度。