Wide-angle one-way wave equations

A one-way wave equation, also known as a paraxial or parabolic wave equation, is a differential equation that permits wave propagation in certain directions only. Such equations are used regularly in underwater acoustics, in geophysics, and as energy-absorbing numerical boundary conditions. The design of a one-way wave equation is connected with the approximation of (1-s2)1/2 on [-1,1] by a rational function, which has usually been carried out by Padé approximation. This article presents coefficients for L2, L infinity, and other alternative classes of approximants that have better wide-angle behavior. For theoretical results establishing the well posedness of these wide-angle equations, see the work of Trefethen and Halpern ["Well-posedness of one-way wave equations and absorbing boundary conditions," Math. Comput. 47, 421-435 (1986)].     

Nonlinear propagation of spark-generated N-waves in air: modeling and measurements using acoustical and optical methods

The propagation of nonlinear spherically diverging N-waves in homogeneous air is studied experimentally and theoretically. A spark source is used to generate high amplitude (1.4 kPa) short duration (40 μs) N-waves; acoustic measurements are performed using microphones (3 mm diameter, 150 kHz bandwidth). Numerical modeling with the generalized Burgers equation is used to reveal the relative effects of acoustic nonlinearity, thermoviscous absorption, and oxygen and nitrogen relaxation on the wave propagation. The results of modeling are in a good agreement with the measurements in respect to the wave amplitude and duration. However, the measured rise time of the front shock is ten times longer than the calculated one, which is attributed to the limited bandwidth of the microphone. To better resolve the shock thickness, a focused shadowgraphy technique is used. The recorded optical shadowgrams are compared with shadow patterns predicted by geometrical optics and scalar diffraction model of light propagation. It is shown that the geometrical optics approximation results in overestimation of the shock rise time, while the diffraction model allows to correctly resolve the shock width. A combination of microphone measurements and focused optical shadowgraphy is therefore a reliable way of studying evolution of spark-generated shock waves in air.     

Diffraction-free propagation of a focused delta-pulsed beam

The propagation of short broadband pulses in a homogeneous medium is analyzed on the basis of the wave equation and in the parabolic approximation of that equation. It is shown that the diffraction spreading of a focused optical pulsed beam does not occur in the limiting case of a δ pulse.     

New approaches to nonlinear diffractive field propagation

In many domains of acoustic field propagation, such as medical ultrasound imaging, lithotripsy shock treatment, and underwater sonar, a realistic calculation of beam patterns requires treatment of the effects of diffraction from finite sources. Also, the mechanisms of loss and nonlinear effects within the medium are typically nonnegligible. The combination of diffraction, attenuation, and nonlinear effects has been treated by a number of formulations and numerical techniques. A novel model that incrementally propagates the field of baffled planar sources with substeps that account for the physics of diffraction, attenuation, and nonlinearity is presented. The model accounts for the effect of refraction and reflection (but not multiple reflections) in the case of propagation through multiple, parallel layers of fluid medium. An implementation of the model for axis symmetric sources has been developed. In one substep of the implementation, a new discrete Hankel transform is used with spatial transform techniques to propagate the field over a short distance with diffraction and attenuation. In the other substep, the temporal frequency domain solution to Burgers' equation is implemented to account for the nonlinear accretion and depletion of harmonics. This approach yields a computationally efficient procedure for calculating beam patterns from a baffled planar, axially symmetric source under conditions ranging from quasilinear through shock. The model is not restricted by the usual parabolic wave approximation and the field's directionality is explicitly accounted for at each point. Useage of a harmonic-limiting scheme allows the model to propagate some previously intractable high-intensity nonlinear fields. Results of the model are shown to be in excellent agreement with measurements performed on the nonlinear field of an unfocused 2.25-MHz piston source, even in the near field where the established parabolic wave approximation model fails. Next, the model is used to compare the water path and in situ fields of a medical ultrasound device. Finally, the model is used to calculate the spatial heating rate associated with a nonlinear field and to simulate the phenomenon of saturation-induced beam broadening.     

Wide-Angle Parabolic Approximation for Modeling High-Intensity Fields from Strongly Focused Ultrasound Transducers

A novel numerical algorithm based on the wide-angle parabolic approximation is developed for modeling linear and nonlinear fields generated by axially symmetric ultrasound transducers. An example of a strongly focused single-element transducer is used to compare the results of ultrasound field simulations based on the Westervelt equation, Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with differently modified boundary condition, and nonlinear wide-angle parabolic equation. It is demonstrated that having a computational speed comparable to modeling the KZK equation, the use of wide-angle parabolic approximation makes it possible to obtain solutions for highly focused ultrasound beams that are closer in accuracy to solutions based on the Westervelt equation.     

Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 6, pp. 725–735. This article was translated by the authors.     

Modeling of an electrohydraulic lithotripter with the KZK equation.

The acoustic pressure field of an electrohydraulic extracorporeal shock wave lithotripter is modeled with a nonlinear parabolic wave equation (the KZK equation). The model accounts for diffraction, nonlinearity, and thermoviscous absorption. A numerical algorithm for solving the KZK equation in the time domain is used to model sound propagation from the mouth of the ellipsoidal reflector of the lithotripter. Propagation within the reflector is modeled with geometrical acoustics. It is shown that nonlinear distortion within the ellipsoidal reflector can play an important role for certain parameters. Calculated waveforms are compared with waveforms measured in a clinical lithotripter and good agreement is found. It is shown that the spatial location of the maximum negative pressure occurs pre-focally which suggests that the strongest cavitation activity will also be in front of the focus. Propagation of shock waves from a lithotripter with a pressure release reflector is considered and because of nonlinear propagation the focal waveform is not the inverse of the rigid reflector. Results from propagation through tissue are presented; waveforms are similar to those predicted in water except that the higher absorption in the tissue decreases the peak amplitude and lengthens the rise time of the shock.     

Statistical characteristics of an intense wave behind a two-dimensional phase screen

An analysis of the parameters of nonlinear waves transmitted through a layer of a randomly inhomogeneous medium is carried out. The layer is modeled by a two-dimensional phase screen. Passing through the screen plane, the wave acquires a random phase shift. The wave front becomes distorted, and randomly located regions of ray convergence and divergence are formed, in which the nonlinear evolution of the wave alters profoundly. The problem is solved in the approximation of geometrical acoustics. The ray pattern of a plane wave transmitted through the regular screen is constructed. The solution that describes the spatial structure of the field and the evolution of an arbitrary temporal wave profile behind the screen is obtained. Statistical characteristics of the discontinuity amplitude are calculated for different distances from the screen. A random modulation is shown to result in a faster (in comparison with the case of a homogeneous medium) nonlinear attenuation of the wave and in the smoothing of the shock profile. The distribution function of the wave field parameters becomes broader because of random focusing effects.     

Small-amplitude nonlinear dust acoustic wave a magnetized dustyplasma with charge fluctuation

Some properties of nonlinear dust acoustic waves in magnetized dusty plasma with variable charges by reductive perturbation technique have been studied. The effect of adiabatic dust charge variations under the assumption that the ratio of dust charging time to the dust hydrodynamical time is very small, and the nonadiabatic dust charges variations under the assumption that the same ratio is small but finite, are also incorporated. It is seen that the magnetic field and the dust charge variations significantly modify the wave amplitude. It is also seen that in case of adiabatic charge variations, the Korteweg-de Vries (KdV) equation governs the nonlinear dust acoustic wave, whereas in case of nonadiabatic dust charge variations, the wave is governed by the KdV Burger equation. Nonadiabaticity generated anomalous dissipative effect causes generation of the dust acoustic shock wave. Numerical integration of KdV Burger equation shows that the dust acoustic wave admits oscillatory (dispersion dominant) or monotone (dissipation dominant) shock solutions depending on the magnitude of the coefficient of the Burger term     

Phased Array Beam Fields of Nonlinear Rayleigh Surface Waves

This study concerns calculation of phased array beam fields of the nonlinear Rayleigh surface waves based on the integral solutions for a nonparaxial wave equation.Since the parabolic approximation model for describing the nonlinear Rayleigh waves has certain limitations in modeling the sound beam fields of phased arrays,a more general model equation and integral forms of quasilinear solutions are introduced.Some features of steered and focused beam Gelds radiated from a linear phased array of the second harmonic Rayleigh wave are presented.     

Numerical model for nonlinear standing waves and weak shocks in thermoviscous fluids.

Nonlinear standing waves in a one-dimensional tube are studied numerically by using a finite-difference algorithm. The numerical code models the acoustic field in resonators for homogeneous, thermoviscous fluids. Calculations are performed exclusively in the time domain, and all harmonic components are obtained by one resolution. The fully nonlinear differential equation is written in Lagrangian coordinates. It is solved without truncation. Effects of absorption are included. Displacement and pressure wave forms are calculated at different locations and results are shown for different excitation levels and tube lengths. Amplitude distributions along the resonator axis for every harmonic component are also evaluated. Simulations are performed for amplitudes ranging from linear to strongly nonlinear and weak shock. A very good concordance with classic experimental and analytical results is obtained.     

Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons

The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg–de Vries–Burgers (mKdV–Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV–Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior.     

Dynamics of high-frequency modulated waves in a nonlinear dissipative continuous bi-inductance network

This paper presents intensive investigation of dynamics of high frequency nonlinear modulated excitations in a damped bimodal lattice. The effects of the dissipation are considered through a linear dissipation coefficient whose evolution in terms of the carrier wave frequency is checked. There appears that the dissipation coefficient increases with the carrier wave frequency. In the linear limit and for high frequency waves, study of the asymptotic behavior of plane waves reveals the existence of two additional regions in the dispersion curve where the modulational phenomenon is observed compared to the lossless line. Based on the multiple scales method exploited in the continuum approximation using an appropriate decoupling ansatz for the voltage of the two different cells, it appears that the motion of modulated waves is described by a dissipative complex Ginzburg–Landau equation instead of a Korteweg–de Vries equation. We also show that this amplitude wave equation admits envelope and hole solitons in the high frequency mode. From basic sources, we design a programmable electronic generator of complex signals with desired characteristics, which delivers signals exploited as input waves for all our numerical simulations. These simulations are performed in the LTspice software that uses realistic components and give the results that corroborate perfectly our analytical predictions.     

Exact solutions of equation of transverse vibrations for a bar with a specific cross section variation law

Flexural wave propagation along a bar whose thickness smoothly decreases down to zero within its end piece is considered. The propagation velocity tends to zero as the tapered end of the bar is approached, and the time of wave propagation to the tapered end is infinite. As a consequence, waves propagating along the bar are not reflected from the end. Previous quantitative study of the effect in the WKB approximation shows that, in the case of parabolic tapering, the WKB approximation yields a uniform asymptotics, which is valid (or invalid) for any of the bar’s cross sections. In the case of a bar with parabolic tapering, the equation of flexural vibrations of the bar has exact analytic solutions in the form of power functions. Based on these solutions, a modified WKB approximation is proposed to solve equations for bars with nonparabolic thickness variation laws. The input impedance of a bar with a parabolic tapering is calculated and analyzed.     

Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber.     

Scattering of a plane inhomogeneous electromagnetic wave by a spherical particle

We extend an analytical solution for the problem of diffraction of a plane wave by a spherical particle to the case of an inhomogeneous wave. Numerical examples showing a significant change in the scattered-wave structure compared with the case of diffraction of a homogeneous wave are presented. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 1, pp. 72–81, January 2006.     

Stochastic functional analysis of propagation and localisation of cylindrical waves in a two-dimensional random medium

Propagation and localisation of cylindrical waves in a two-dimensional (2D) isotropic and homogeneous random medium is studied using the stochastic functional approach. By expanding the random permittivity fluctuation in the form of a Wiener integral equation, and representing the wave fields by a linear combination of outgoing and incoming waves, the scalar Helmholtz equation is solved in the cylindrical coordinates system. An analytical expression of the cylindrical wave is derived and demonstrates the localisation phenomenon, as well as the wavenumber fluctuation in the random medium. Comparing with the waves in non-random medium, the wave transfer equation between plane wave and cylindrical wave in random medium shows an additional exponential factor to indicate the modulation effect owing to the medium randomness in both the amplitude and phase. Numerical simulations are presented to illustrate the functional dependence of the localisation phenomena.     

Propagation of Electron Acoustic Soliton,Periodic and Shock Waves in Dissipative Plasma with a q-Nonextensive Electron Velocity Distribution

The nonlinear properties of small amplitude electron-acoustic(EA) solitary and shock waves in a homogeneous system of unmagnetized collisionless plasma with nonextensive distribution for hot electrons have been investigated.A reductive perturbation method used to obtain the Kadomstev-Petviashvili-Burgers equation.Bifurcation analysis has been discussed for non-dissipative system in the absence of Burgers term and reveals different classes of the traveling wave solutions.The obtained solutions are related to periodic and soliton waves and their behavior are shown graphically.In the presence of the Burgers term,the EXP-function method is used to solve the Kadomstev-Petviashvih-Burgers equation and the obtained solution is related to shock wave.The obtained results may be helpful in better conception of waves propagation in various space plasma environments as well as in inertial confinement fusion laboratory plasmas.     

Propagation of Electron Acoustic Soliton,Periodic and Shock Waves in Dissipative Plasma with a q-Nonextensive Electron Velocity Distribution

The nonlinear properties of small amplitude electron-acoustic (EA) solitary and shock waves in a homogeneous system of unmagnetized collisionless plasma with nonextensive distribution for hot electrons have been investigated. A reductive perturbation method used to obtain the Kadomstev-Petviashvili-Burgers equation. Bifurcation analysis has been discussed for non-dissipative system in the absence of Burgers term and reveals different classes of the traveling wave solutions. The obtained solutions are related to periodic and soliton waves and their behavior are shown graphically. In the presence of the Burgers term, the EXP-function method is used to solve the Kadomstev-Petviashvili-Burgers equation and the obtained solution is related to shock wave. The obtained results may be helpful in better conception of waves propagation in various space plasma environments as well as in inertial confinement fusion laboratory plasmas.     

二维随机介质中柱面波传播及其局域性的随机泛函分析

分析了二维各向同性均匀随机介质中柱面波的传播特性及局域化现象.用随机泛函理论,在频域内将随机介电起伏展开成柱坐标系下的Wiener积分式,将波场表示为内外行柱面波的线性和,求解二维Helmholtz波动方程,得到随机介电起伏对柱面波幅度与相位调制的解析表达.由柱面波能量的空间分布验证了波的局域化现象,并求解局域化长度.二维随机介质中平面波按柱面波展开的波转换方程与非随机介质中的情形有相似的表达,但具有随机介电起伏对幅度和相位的调制,并给出数值模拟结果.