Parabolic Wave Equation Research Articles

Acoustic streaming at high Reynolds numbers due to focused piston radiation in real fluids

In an earlier presentation [Hamilton et al., J. Acoust. Soc. Am. 97, 3376(A) (1995)], a theoretical model for acoustic streaming at high Reynolds numbers (based on streaming velocities) in thermoviscous fluids was used for analytical and numerical predictions based on the assumption of weakly nonlinear focused Gaussian beams. Here, numerical calculations of streaming are presented for radiation from circular pistons, both focused and unfocused, and at amplitudes for which shock formation occurs. The forcing function for the streaming equations is evaluated with time-domain numerical solutions of the nonlinear parabolic wave equation. Effects of thermoviscous dissipation on the acoustic waveforms, and in particular on shock structures, are taken into account explicitly. The calculations provide a consistent description of acoustic streaming generated in real fluids by sound beams at moderate amplitudes, either with or without shocks. Because the shocks are not considered to be discontinuous features of the solutions, uniformly valid predictions of streaming are obtained in the transition region encompassing the birth and death of shocks amid the competing effects of dissipation and diffraction. The model accommodates both continuous and pulsed sources. Comparisons are made with experiments reported in the literature. [Work supported by the Office of Naval Research.]

Numerical Analysis of Long Range Acoustic Propagation Based on Wide Angle Parabolic Wave Equation

The purpose of this paper is to describe an accurate computational method for underwater acoustic propagation problems based on the wide angle wave equation with the transparent boundary condition. In order to evaluate the present method, we calculate typical examples of a deep underwater propagation model with Munk's sound velocity profile and shallow underwater with the sound profile increasing exponentially from the sea surface to the bottom including the loss in the seabed material. The results are almost the same as the semi-analytical solutions obtained using rays or normal modes. The validity and usefulness of the present method are confirmed.

A line‐of‐sight propagation model for calculating atmospheric, sea, terrain multipath, and clutter at microwave frequencies

The two‐dimensional parabolic wave equation (PE) is solved asymptotically in order to derive a propagation/scattering model for propagation in a stratified atmosphere, together with scattering and diffraction from the sea/terrain surface, within line of sight. The asymptotic solution to the PE begins by expanding the unknown field in a Luneberg‐Kline expansion. The first‐order solution to the Luneberg‐Kline expansion yields the eikonal equation for the ray trajectory, and the second‐order solution yields the transport equation for the energy density of the rays along the trajectory. The solution to the eikonal and transport equations is used to derive an asymptotic result for the plane wave spectrum solution for the PE, referred to as the wave‐theoretic model. The asymptotic solution for the field shows the relationship between a properly directed tube of rays in the atmosphere satisfying the eikonal and conservation of energy equations and the same tube of rays striking an edge and being diffracted, as in the geometrical theory of diffraction (GTD). The solution for the eikonal yields the ray trajectory in the stratified atmosphere and therefore the grazing angle on the sea/terrain profile necessary for calculating multipath and clutter. Also, a new derivation for the backscatter from a deterministic sea/terrain profile is given which agrees with Collin's [1992] new full wave theory and at grazing reduces to the perturbation result with the correct polarization dependence. Incorporating the various electromagnetic propagation/scattering mechanisms in one model yields a prediction tool that can be applied to a large class of propagation/scattering problems such as the analysis of radar performance in a littoral scenario, or the performance of a line‐of‐sight communication link operating over irregular terrain.

Forward propagation in a half-space with an irregular boundary

The parabolic wave equation (PWE) has been used extensively for modeling the propagation of narrow beams in weakly inhomogeneous random media. Corrections have been developed to accommodate wider scattering angles and boundaries have been introduced. Nonetheless, the formalism remains approximate and irregular surfaces with general boundary conditions present difficulties that have yet to be overcome. This paper presents an alternative approach to the entire class of propagation problems that strictly involve forward propagation. Forward-backward iteration has been shown to be a powerful procedure for computing the source functions that support propagation over irregular boundaries at low grazing angles. We show that the source functions for any unidirectional sweep can be computed by using a marching solution. This is not only more efficient than the single-sweep computation, but it facilitates accommodation of inhomogeneities in the propagation media. An exact equation for forward propagation in unbounded inhomogeneous media is used to derive a correction term that is applied at each forward-marching step. Results that combine ducting atmospheres and rough-surface scattering effects are presented for both the Dirichlet and Neumann boundary conditions.

Accurate modeling of transparent boundaries in quasi‐optics

A general approach to reduction of the infinite computational domain to a strip or a cylinder is developed for two‐ and three‐dimensional narrow‐angle diffraction problems. Our method is based on constructing exact solutions of the parabolic wave equation in open subdomains free of nonuniform diffractive objects. As a result, an exact nonlocal boundary condition is derived providing full transparency of the artificially introduced computational border. This technique leads to an essential reduction of the computing time without any loss of accuracy, which is illustrated by practical examples from radio propagation and X ray diffractive optics. Extension of the method onto quasi‐stratified environments and modified parabolic equations is discussed.

Diffraction phenomena inside thick Fresnel zone plates

The parabolic wave equation method is used to describe the complex wave field inside the body of a thick zone plate used to focus X ray radiation. Two analytical approaches are applied: (1) Diffraction of a plane wave incident onto a separate interface between opaque and open zones is considered. We construct an approximate analytical solution to the classical problem of diffraction by a dielectric wedge in terms of the Fresnel integral and a new special function. (2) Coupled wave theory is used to describe collective effects of diffraction by many interfaces. The zone plate is considered as thick diffraction grating with slowly varying period. An analytical solution is found for three interacting modes. A possibility to optimize the zone plate performance is shown. Taken together, these results describe all the main features observed in the output field of realistic X ray zone plates.

Experimental results for transmission of a finite-amplitude, focused sound beam at a curved interface between two media

Transmission of an intense focused sound beam through two different liquids separated by a curved interface was investigated experimentally. The acoustic field was generated by a 2.25-MHz tone burst from a focused source of radius 19 mm and focal length 159 mm. Measurements of propagation curves and beam patterns were made using a PVDF needle hydrophone. The source was placed in water, the first liquid, and either lamp oil (cwater/coil=1.08) or a 50% glycerin–water mixture (cwater/cmixture =0.85) was used as the second liquid. Curvature at the interface between the water and the second liquid was controlled by using a thin (≊40 μm) polyether polyurethane membrane. Both concave or convex curvatures, with radii varying from 90 mm to infinity (plane interface), were achieved by varying the hydrostatic pressure between the two liquid chambers. Measurements were compared with theory, presented earlier [Makin, J. Acoust. Soc. Am. 95, 2864(A) (1994)], based on the nonlinear parabolic (KZK) wave equation. Reasonably good comparison between theory and experiment was observed for the first four harmonics generated in the incident and transmitted fields for interfaces with various curvatures, and located in either the pre-focal or post-focal regions. [Work supported by NSF.]

The small-amplitude, continuous wave reflected field produced by a spherical source placed at the focus of an ellipsoidal reflector

Ellipsoidal reflectors are widely used in electrohydraulic lithotripters. The small-amplitude, continuous wave (cw) reflected field of a small spherical source that is placed at the first focus (f1) of an ellipsoidal reflector is investigated with both theory and experiment. This configuration is equivalent to a low-amplitude (linear regime), cw-driven lithotripter. To model the acoustic field reflected from an ellipsoidal reflector, geometrical acoustics is used to define a directivity function at the mouth of the reflector. An equivalent focused source with amplitude shading defined by the directivity function is then assumed as the boundary condition for the linear, lossless parabolic wave equation which is used to diffractively propagate the field outside the reflector. The spherical source is driven with a long tone burst at 470 kHz. The ellipsoidal reflector is made out of nickel and has a rhodium coating. The major and minor radii are 6.2 cm and 3.5 cm, respectively. Good agreement is found between experiment and theory. A useful property of ellipsoidal reflectors is that one can quickly and accurately position the source at f1 and the hydrophone at f2 by a simple iteration technique. [Work supported by NIH.]

Comparison of algorithms for solving parabolic wave equations

The three types of algorithms that have been developed for solving parabolic wave equations are compared. Until recently, it was necessary to choose between the finite-difference and split-step Fourier algorithms and make a trade-off between efficiency and capability. Test problems are presented to illustrate the efficiency of the split-step Padé algorithm, which provides the capability of the finite-difference algorithm. For deep water problems, the split-step Padé algorithm provides efficiency comparable to the split-step Fourier algorithm. For the shallow water problems that are currently of interest, the split-step Padé algorithm can be more than an order of magnitude faster than the split-step Fourier algorithm.

A probability distribution for the complex field of waves propagating in random media

A model for the joint probability density function is proposed for the quadrature components of waves propagating in random media (WPRM). The model is based on a further generalization of the model for the intensity distribution for WPRM proposed by Ewart [J. Acoust. Soc. Am. 86, 1490–1498 (1989)]. Both this distribution model and the intensity distribution model apply to the full range of scattering strengths and ranges. That is, from short ranges where the intensity probability density function (pdf) is lognormal, through the region of the medium focus, where the intensity moments 〈Iq〉 can be well above q!, to the saturation regime, where the moments approach exponential. Simulation of the complex fields used to test the model was accomplished by parabolic wave equation marching of the field through a medium with a fourth-order power law transverse spectrum (usually termed the modified exponential). The fourth moment of the fields are accurately predicted using parabolic fourth moment theory—giving some confidence in the higher moments. The resulting joint pdf’s are parametrized in terms of the scattering strength γ and the scaled range X that parametrize the simulations. This work provides a necessary first step in the important quest to include highly non-Gaussian quadrature component statistics in signal processing formulations.

An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation

A new implementation of the previously published mixed Fourier transform (MFT) method for including impedance boundaries in split-step parabolic equation solutions is described and demonstrated. The new algorithm is formulated entirely in the discrete domain which results in extended applicability and increased computation speed. A brief review of the original MFT solution is followed by a detailed description of the discrete formulation. The performance of the new algorithm is then demonstrated with a few examples which rely heavily on the accuracy of the impedance boundary. These examples include 10 MHz surface wave propagation over smooth and rough sea surfaces and 10 GHz calculations utilizing an effective rough surface impedance.

Measurements of harmonic generation in a focused finite-amplitude sound beam

Measurements of harmonic generation in the sound field radiated by a focused circular piston source are compared with numerical calculations based on the nonlinear parabolic wave equation. The large dynamic range and high spatial resolution of the measurements enable precise comparisons to be made with the theoretical predictions.

Applications of optimized rational approximations to parabolic equation modeling

Rational approximations designed using least-squares constraints are easy to obtain and have useful applications. The coefficients are defined in terms of accuracy constraints on the propagating part of the spectrum and stability constraints on the rest of the spectrum. Among the applications are a filtering operator as well as improved approximations for the split-step Padé solution [J. Acoust. Soc. Am. 93, 1736–1742 (1993)] and radiation boundary conditions for outgoing wave equations [Clayton and Engquist, Geophysics 45, 895–904 (1980)]. An efficient operator filter is obtained by designing a rational approximation that is the identity function (or a weighted function) over the propagating (or desired) spectrum and decays to zero elsewhere. Operator approximations have previously been applied to replace the operator in the interior of the domain (the parabolic wave equation), generate radiation boundary conditions, solve scattering problems, and generate initial conditions. With the improved approximations, it is possible to use range steps of more than a hundred wavelengths with the split-step Padé solution. By enforcing both accuracy and stability constraints, it should be possible to overcome stability problems [Howell and Trefethen, Geophysics 53, 593–603 (1988)] and obtain useful higher-order radiation boundary conditions for outgoing wave equations.

A numerical solution of the parabolic elastic wave equation

Based on the parabolic equation approximation, a set of equations has been developed for three-dimensional time-harmonic wave propagation in an elastic medium. The elastic equations for the scalar and vector potentials are written in a matrix form which is a direct counterpart to previous work on the scalar wave equation for a fluid medium. An ordinary differential equation (ODE) method in conjunction with a finite-difference scheme leads to a stable marching procedure. One feature of this approach is that every finite-difference discretization results in a tridiagonal system of equations; these equations can be solved efficiently by recursive formulas. This paper reports the computational results which are used to check (1) the stability of the marching scheme, and (2) the accuracy of the elastic model. Accuracy and validity are verified by comparing the numerical results of the finite-difference method with a far-field analytic solution in an unbounded medium.

High amplitude, bifrequency experiments in porous rocks

Fluid-filled, porous, sedimentary rocks have the general characteristic that the sound speeds are strongly dependent on the state of stress and therefore, the third-order elastic coefficients (i.e., A, B, C) are generally much larger than for normal solids. However, the linear attenuation can also be very large such that in nonlinear wave propagation, the Gol’dberg number is small. Experiments studying harmonic generation in porous rocks have been made using an ultrasonic immersion system (i.e., water/solid/water) where a high amplitude (e.g., 500 kPa), bifrequency (0.95 and 1.05 MHz), tone burst is emitted and then sum and difference frequencies are detected after propagation through the water/solid/water system. A frequency domain numerical implementation of the KZK (Khokhlov–Zabolotskaya–Kuznetsov) nonlinear parabolic wave equation is used to describe the nonlinear propagation in the three layer system. Experiments on solids with well-known acoustic properties (linear and nonlinear) were used to validate the model. Measurements were then made on several sandstones and limestones and the model used to derive the nonlinear propagation parameter, β=f(A,B,C). Finally, these results are compared with independent measurements of A, B, C for these rocks.

Application of the parabolic wave equation to X-ray diffraction optics

A new computational approach is suggested to the analysis of imaging properties of realistic X-ray Fresnel zone plates with high aspect ratio. Its mathematical basis is the parabolic wave equation (PWE) describing diffraction inside the zone plate body as well as wave propagation and focusing throughout the optical system. A finite-difference method is used for solving PWE in a truncated free-space domain with a perfectly transparent artificial boundary. The results of global wave field calculation are visualized by means of color graphics. Numerical estimates of resolution, diffraction efficiency and field of view enables one to compare imaging performance of realistic X-ray zone plates with an idealized plane Fresnel diffraction lens.

Modified finite-difference beam propagation method based on the generalized Douglas scheme for variable coefficients

The accuracy of the implicit finite-difference beam propagation method (FD-BPM), in which the phase term is not split, is improved using the generalized Douglas scheme. The propagation error of the fundamental mode in two- and three-dimensional waveguides is evaluated by the mode-mismatch loss calculation. It Is demonstrated that the truncation error is reduced to O(/spl Delta/x)/sup 4/ in the transverse direction, even when the parabolic wave equation contains variable coefficients, The computational time is almost identical to the conventional FD-BPM based on the Crank-Nicholson scheme.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Horizontal multipaths caused by mesoscale structure and their effects on global tomography with near‐axial paths

The optimistic and questionable assumption that the lowest acoustic normal modes are adiabatic is used. Then, for each low mode a parabolic wave equation in the horizontal plane is derived that contains lateral variability of sound speed near the sound channel axis caused by mesoscale structure that is modeled by homogeneous and isotropic fluctuations having a single scale length, L∼100 km. In the geometrical acoustics limit, horizontal rays are found to be chaotic with the growth rate (Lyapunov exponent) given by ν∼ε2/3/L, where ε∼5×10−3 is the rms sound speed fluctuation. Horizontal multipaths begin at the focal range, rf∼ν−1∼3 Mm, and thereafter the number of horizontal multipaths per mode increases exponentially at the rate ν until diffraction effects limit this ray chaos. Full saturation is found to occur at the ‘‘log range,’’ rs∼rf ln(L2/λrf)∼14 Mm, for the acoustic wavelength λ∼30 m. Since mode coupling is neglected, this saturation range is a hard upper bound for doing global ocean acoustic tomography with near‐axial paths. [Work supported by ONR and ARPA.]

FINITE DIFFERENCE TREATMENT OF IRREGULAR INTERFACES: AN ERROR ANALYSIS

The application of finite difference schemes to handle a horizontal interface between two media gives numerically adequate results, as evidenced in the solution of parabolic wave equations with density variations. Using the same finite difference schemes for handling a stair-step approximation to an irregular interface gives satisfactory results provided that the slope angle is small, but a small error is introduced. In the event that the slope angle is not small, this error has to be handled carefully since it may influence the results. Using an analysis of the error, this paper derives a closed form expression for a correction term. An irregular interface can then be handled adequately by the same numerical treatment used for the horizontal interface by applying this correction term. A test case is presented to demonstrate the inaccuracy of using one standard numerical horizontal interface treatment and to show how our new development improves the accuracy.

Time-domain modeling of pulsed finite-amplitude sound beams

A time-domain algorithm that solves the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation is described. The algorithm models the propagation of pulsed finite amplitude sound beams radiated from axisymmetric sources in homogeneous, thermoviscous fluids. Numerical results are presented for waveform distortion and shock formation in directive beams radiated by pulsed circular pistons. Waveforms are calculated through the shock region and out to far-field locations where they are dominated by the nonlinearly generated low-frequency components. Effects of pulse duration, frequency modulation, and noise are examined. Methods for including relaxation and focusing are described.