Parabolic Wave Equation Research Articles

Fluctuations of spherical waves in a turbulent atmosphere: effect of the axisymmetric approximation in computational methods.

The validity of the axisymmetric parabolic-equation (PE) method for line-of-sight sound propagation in a turbulent atmosphere is investigated. The axisymmetric PE method is a finite-difference method for solving a 2D parabolic wave equation, which follows from the 3D wave equation by the assumption of axial symmetry around the vertical axis through the source. It is found that this axisymmetric approximation has a considerable spurious effect on the fluctuations of the sound field. This is concluded from analytical expressions for the log-amplitude and phase variances, derived both for isotropic turbulence and for axisymmetric turbulence. The expressions for axisymmetric turbulence are compared with the results of numerical computations with the PE method.

Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

Observations of seismic waves from earthquakes at depths between 100 and 200 km beneath Japan show that the initial portion of the S-wave arrival has greater duration than can be accounted for by earthquake source-time duration. The observed long duration of S waves has been explained as being caused by multiple forward scattering around the ray path between source and receiver. Array obser- vations of Lg waveforms have also shown that multiple forward scattering along the path between the source and receiver are important influences on the character of Lg waveforms and that the scattering cannot be explained only by vertical variations in velocity. Multiple forward scattering in 3D has been modeled using the Markov approximation for the parabolic-wave equation, which allows the calculation of seis- mogram envelopes in statistically characterized random media. To test the range of validity of the Markov approximation-derived solutions, we made 2D numerical calculations of wavefields in random media using approximations to the parabolic wave equation, which only models forward scattering, and by finite-difference so- lution of the scalar-wave equation, which gives complete wavefields. Media of back- ground velocity 4 km/sec are characterized using a Gaussian autocorrelation function with a5k mcorrelation distance and 5% rms fractional fluctuation. To compare with envelopes obtained from the Markov approximation, we calculated wavefields for source-receiver distances ranging from 50 to 200 km for several statistically identical realizations of random media. We obtain ensemble average envelopes by averaging envelopes from the realizations. We find a good agreement between ensemble- average envelopes obtained from the numerical solution of the parabolic-wave equa- tion and envelopes obtained from the Markov approximation. The later portion of the ensemble-average envelopes calculated using finite difference have larger am- plitudes than those from the Markov approximation, which is probably due to the late-arriving energy that has been scattered at wide angles from the global propa- gation direction between the source and the receiver. We observe that the variations among the numerically calculated envelopes of individual realizations of random media are well fit by a Rayleigh distribution, which describes the distribution of envelope amplitude when signals of one frequency but random phase are summed. Our results show that the Markov approximation provides reliable information about envelope shapes for forward-scattered wavefields but that the influence of wide-angle scattering and backscattering have some influences on envelope shapes and should be considered when analyzing data using random media models.

Large fluctuation of wave amplitude produced by small fluctuation of velocity structure

Seismic velocity structure of the real Earth is described by a combination of deterministic large scale structure and small scale fluctuation. When waves propagate in media having the small scale fluctuation, the fluctuation makes the rays bend, introduces focussing and defocusing, and produces fluctuations of wave amplitude. To estimate the size of wave amplitude fluctuation that is caused by velocity fluctuation, the variances of wave amplitudes are evaluated as a function of propagation distance from numerical simulations of scalar wave propagation in 3D random media having Gaussian or exponential autocorrelation functions. The simulations are performed using a phase screen method, which is a numerical procedure for solving the parabolic wave equation. The following conclusions are obtained from the simulations that use two types of incident wave: stationary monochromatic sine wave and transient Ricker wavelet. For velocity fluctuation as small as several percent, the variance of wave amplitude is larger than the square of its average in some cases of high frequency waves. These results show the importance of small error and fine spatial resolution of velocity structure model for proper modeling of wave amplitude. When the fluctuation of the velocity structure is large, the wave amplitude variance increases with the propagation distance until it reaches some peak value after which it decreases with the increase of propagation distance. The variance for the Ricker wavelet incidence is smaller than that for a monochromatic sine wave. Because of the stochastic dispersion due to the small fluctuation of velocity structure, the maximum amplitude decreases with the propagation distance even if there is neither an intrinsic absorption nor back-scattering for the Ricker wavelet incidence.

Path Loss Predictions in Urban Areas with Irregular Terrain Topography

A method based on the parabolic wave equation is used to predict path loss in a two dimensional urban setting where buildings sprawl over irregular terrain. Propagation is assumed to take place in ...

Low-grazing angle scattering from rough surfaces in a duct formed by a linear-square refractive index profile

The problem of rough surface scattering and propagation over rough terrain in a ducting environment has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. An alternative method is the boundary integral equation (BIE) method. The implementation of the BIE in inhomogeneous media (ducting environments) is not straightforward, however, since the Green's function for such a medium is not usually known. In this paper, a closed-form approximation of the Green's function for a two-dimensional (2-D) ducting environment formed by a linear-square refractive index profile is derived using asymptotic techniques. This Green's function greatly facilitates the use of the BIE approach to study low-grazing angle (LGA) rough surface scattering and propagation over rough surfaces in the aforementioned ducting environment. This paper demonstrates how the BIE method can model the combined effects of surface roughness and medium inhomogeneity in a very rigorous fashion. Furthermore, it illustrates its capability of accurately predicting scattering in all directions including backscattering. The boundary integral equation of interest is solved via the method of ordered multiple interactions (MOMI), which eliminates the requirements of matrix storage and inversion and, hence, allows the application of the BIE method to very long rough surfaces.

Propagation modeling over terrain using the parabolic wave equation

We address the numerical solution of the parabolic wave equation over terrain using the Fourier/split-step approach. The method, referred to as a shift map, generalizes that of Beilis and Tappert (1979) who introduced a coordinate transformation technique to flatten the boundary. This technique is extended to a wide-angle form, allowing larger propagation angles with respect to the horizon. A new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave (PWE) using the previously developed mixed Fourier transform. It is also shown by example that in many cases of interest, the boundary may be approximated by discrete piecewise linear segments without affecting the field solution. A more accurate shift map solution of the PWE for a piecewise linear boundary is, therefore, developed for modeling propagation over digitally sampled terrain data. The shift-map solution is applied to various surface types, including ramps, wedges, curved obstacles, and actual terrain. Where possible, comparisons are made between the numerical solution and an exact analytical form. The examples demonstrate that the shift map performs well for surface slopes as large as 10-15/spl deg/ and discontinuous slope changes on the order of 15-20/spl deg/. To accommodate a larger range of slopes, it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well-known terrain masking (knife-edge diffraction) approximation.

Study of horizontal multipaths and ray chaos due to ocean mesoscale structure

Long-range ocean acoustic propagation in the presence of idealized mesoscale structure is studied by first deriving a two-dimensional horizontal-plane parabolic wave equation that follows from the adiabatic mode approximation. In the geometric limit, a nonautonomous Hamiltonian dynamical system having one degree of freedom is derived. A stochastic formalism is developed to analyze this nonintegrable dynamical system. The main result is that on average two rays that are initially separated by an infinitesimal amount diverge exponentially at a rate given by the Lyapunov exponent that has been calculated theoretically and compared to numerical experiments with agreement to two decimal places. The practical implication of this result is that tomographic inversions based on assumed pointwise accurate ray predictions might not be possible beyond the "predictability horizon" of many thousands of kilometers, due to horizontal-plane multipaths induced by naturally occurring mesoscale activity.

Transparent boundaries for the parabolic wave equation

The boundary conditions ensuring the full absorption of arbitrary paraxial wave packets, described by the parabolic wave equation, are derived in exact form. This provides a mathematically strict method for reducing an infinite domain to a strip or a cylinder when performing numerical calculations.

Differences between the narrow-angle and wide-angle propagators in the split-step Fourier solution of the parabolic wave equation

For tropospheric electromagnetic propagation, Maxwell's equations can be reduced to a parabolic wave equation, which is solved by marching over range steps. In each step, the solution is split into a product of three operators. The first and third account for atmospheric and surface variation, while the center operator propagates the field as though in vacuum. This center operator is the object of interest here. Older versions of the method used the narrow-angle propagator, while more recent versions use the wide-angle propagator. It was thought that the wide-angle propagator was entirely superior to the narrow-angle propagator, but some artifacts observed in experiments have led to the present investigation. The two propagators are examined numerically and analytically and found to exhibit subtle differences at large angles from the horizontal. This has required modifications to the way in which sources are created for starting the split-step solution. The narrow- and wide-angle propagators are also compared on two problems with analytic solutions to quantify the improvement of the wide-angle over the narrow-angle.

Angular Broadening: Effects of Nonzero, Spatially Varying Plasma Frequency between the Source and Observer

Angular broadening of radiation due to scattering by density irregularities is usually described using geometric optics (GO) or the parabolic wave equation (PWE) with the assumptions that the radiation frequency f greatly exceeds the local plasma frequency f(sub p0) or that f(sub p0)/f is constant along the path. These assumptions are inappropriate for many solar system radio phenomena. Here the PWE and GO formalisms are extended to treat angular broadening in plasmas with nonzero, spatially varying ratios, f(sub p0)(z)/f depends on both f(sub p0)(z)/f at the observer and the foregoing factor inside the path integral. The PWE and GO predictions for are identical and involve equivalent assumptions. Previous GO and PWE results are recovered in the limits that f(sub p0)(z prime)/f is constant or zero. The new PWE and GO results will permit more accurate calculation of angular broadening for solar system and astrophysical sources. Moreover and importantly, due to the PWE and GO results for being identical, previous GO analyses of in solar system contexts are essentially correct, except for the neglect of or minor deficiencies in the treatment of nonzero, spatially varying f(sub p0)/f effects. The identical GO and PWE results for and the form of the PWE equation for the correlation function raise questions as to whether diffraction is unimportant for angular broadening (under the usual PWE conditions). Future direct comparisons of the PWE predictions with angular spectra calculated using existing GO ray-tracing codes should answer these questions. Diffraction effects are probably important when the medium and turbulence are not sufficiently homogeneous transverse to the central ray path. The implications are also discussed for studies of the 2-3.5 kHz radiation observed in the outer heliosphere.

Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces

This paper consists of two parts. In the first part, the solution of the Helmholtz equation under forward-scattering or propagation conditions is sought as a uniform asymptotic perturbation expansion using the method of multiple scales. It is then shown that the parabolic wave equation (PWE) solution is the zeroth-order term in this expansion. In the second part, the electric-field integral equation and the magnetic-field integral equation, derived under the PWE approximation, are solved for surface currents induced on a sinusoidal surface. The scattered fields produced by these currents are then calculated using the appropriate radiation integrals. Results are compared to those obtained using the method of ordered multiple interactions developed by Kapp and Brown.

Pulse propagation in random media

Wave propagation in randomly fluctuating media is a problem of continuing interest in ocean acoustics. Usually the work done in this field is carried out in the frequency domain since there are powerful techniques to study wave scattering in the frequency domain, while time-domain formulations are notoriously more difficult to use and obtain results. Nevertheless, our intuition about wave phenomena is much better when discussing such phenomena in a space-time formulation than in a space-frequency formulation. In this presentation wave propagation in a weakly and randomly fluctuating medium is discussed entirely within the space-time formulation. McDonald and Kuperman derived from the wave equation an approximate time-domain progressive wave equation which is the equivalent in the space-time formulation of the well-known parabolic wave equation in the space-frequency formulation. A path integral representation for the solution of this equation is used here to study wave propagation in random media. The weakly inhomogeneity assumption yields an approximate evaluation of the path integral for the Green’s function thus allowing an analysis of the time-dependent statistics of acoustic fields propagating through random media. [Work supported by ONR.]

Nonlinear wave propagation from a discrete annular array: Theory

Theoretical investigations of finite amplitude wave propagation from bounded beam sources has been considered by Lee and Hamilton [J. Acoust. Soc. Am. 97, 906–917 (1995)] as well as others. These researchers have restricted their analyses to a source geometry of either an unfocused circular piston or a weakly, spherically focused circular piston. When the peak pressure does not exceed approximately 150 MPa, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation yields numerical solutions in agreement with experiments. However, little research has been reported on finite amplitude wave propagation from an array of discrete sources. Time domain simulations, based on the KZK equation, will be discussed for an eight element discrete annular array immersed in fresh and sea water. The source signal is typically a short sine wave tone burst with a center frequency of 500 kHz. The transitions from linear acoustics to nonlinear acoustics will be discussed, and comparisons between a spherically focused circular source and a properly phased array will be presented. While some data from recent experiments may be shown, the scope of this talk will be restricted to the theoretical treatment and the numerical simulations for the discrete annular array. [Work supported by ONR.] by Edson

Nonlinear wave propagation from a discrete annular array: Experiments

To date, efforts to model nonlinear wave propagation from a bounded beam source have chiefly been concerned with unfocused and weakly focused circular piston sources [Lee and Hamilton, J. Acoust. Soc. Am. 97, 906–917 (1995)]. When the peak pressure does not exceed approximately 150 MPa, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation yields numerical solutions in agreement with experiments. However, little research has been reported on nonlinear wave propagation from an array of discrete sources. Theoretical and experimental results for an eight-element annular array will be presented. The source signal is typically a ten-cycle sine wave tone burst with a center frequency of 500 kHz. The outer diameter of the array is 0.14 m, and the surface areas of the elements are equal. While theoretical predictions will be shown, the experimental aspects of this research will be emphasized. [Work supported by ONR.] by Kargl

Tissue harmonic imaging in cardiology and radiology applications

Harmonic imaging, the formation of ultrasound images from the harmonic components of the sonification signal, has been used in diagnostic ultrasound applications that utilize microbubble contrast agents. In the presence of microbubbles the harmonic signals are mainly due to nonlinear bubble oscillations. Recently, harmonic imaging has been applied without contrast agents, the harmonic signals arising from nonlinear propagation. Near-field artifacts and aberrations are reduced, and tissue borders are enhanced. The harmonic signals are at a lower amplitude and suffer more attenuation than the fundamental. However, the wide dynamic range, digital architecture, and signal processing capabilities of modern diagnostic ultrasound systems make it possible to utilize this tissue-generated harmonic energy for image formation. The nonlinear parabolic wave equation is used to model tissue propagation. A time domain numerical solution for a nonaxisymmetric source function is presented. Theory and experiments demonstrate harmonic generation in tissues. In vitro animal tissue experiments and in vivo collected radio frequency data are shown. A novel broadband pulsing scheme that uses two pulses with a phase difference of 180° is introduced, which results in cancellation of the propagated fundamental component and doubling of the second harmonic. Cardiac and abdominal harmonic images are shown.

Acoustical helicoidal waves and Laguerre–Gaussian beams: Applications to scattering and to angular momentum transport

Traveling waves having helicoidal wavefronts are cylindrically symmetric except for an azimuthal angular dependence of exp(imφ−iωt), where m is an integer. The amplitude vanishes at the screw-phase dislocation [J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165–190 (1974)] on the z propagation axis. Examples of paraxial helicoidal waves are the Laguerre–Gaussian beam solutions of the parabolic wave equation. Some potentially useful properties of acoustical helicoidal waves are analyzed. Just as for linearly polarized optical Laguerre–Gaussian beams [L. Allen et al., Phys. Rev. A 45, 8185–8189 (1992)], the ratio of the axial angular momentum flux to the acoustical beam power is found to be m/ω so that axial radiation torques are generated when the acoustical energy is absorbed. Helicoidal waves are predicted to backscatter in a way which reveals the axisymmetry of the scatterer when combined with a helicoidal mode-selective detector. Some strategies for acoustical helicoidal wave generation and detection will also be described. [Work supported by the Office of Naval Research.]

On energy-conserving boundary conditions for parabolic wave equations

Realistic description of sound interaction with the ocean bottom requires environmental models having piecewise-continuous dependence of their parameters on position. For a boundary-value problem to be well posed for a parabolic equation (PE), unlike the wave equation, type, and even the number of boundary conditions (BCs) to be imposed at an interface, depends on the interface geometry and the PE itself. In this paper, the boundary conditions at staircase and sloping interfaces are considered for a class of PEs for which the operator square root is approximated by a rational function. Boundary conditions are derived from the PEs themselves thereby ensuring consistency between the equation and the BCs. In particular, energy-conserving PEs produce energy-conserving BCs. Physically, this procedure can be viewed as the substitution of an interface by an infinitesimally thin transition layer. The existence and uniqueness of solutions for the boundary-value problem for a wide-angle PE in the presence of interfaces are both illustrated by considering plane-wave reflection and transmission. Moreover, this fundamental problem provides insight into asymptotic accuracy of the parabolic approximation in media with piecewise-continuous parameters. [Work supported by NSERC.]

Average received signal power after two‐way propagation through ionized turbulence

In this paper we consider propagation through a structured ionosphere separating a monostatic or bistatic radar and its target. A new analytic calculation of the correlation function of the received, demodulated signal is presented. This result is a function of the locations of the transmit and receive apertures and of time during the duration of the received pulse. Strong scattering is assumed where the one‐way scintillation index is unity; under this assumption the parabolic wave equation can be solved using the quadratic approximation for the phase structure‐function; this facilitates tractable expressions for the mutual coherence function (MCF) for both one‐ and two‐way propagation. A new expression is given for the two‐position, two‐time, two‐frequency MCF for spherical wave propagation that allows simultaneous description of all effects of the ionospheric propagation channel (or transmission medium) on a propagating radar signal. To obtain useful and tractable results, the transmitted signal is assumed to be a chirp (linear FM) pulse, a waveform used by most long‐range defense radars. The result for the correlation function of the received signal is a function of the basic parameters that describe wave propagation through the ionospheric structure, including the channel decorrelation distance, decorrelation time, and coherence bandwidth. Both the two‐way enhancement in average power and aperture averaging effects are explicitly included in this derivation, and new results are presented that relate these effects to the characteristics of the propagation channel. The results are useful for propagation under conditions of saturated scintillation, for example HF through UHF radar in the natural equatorial environment and HF through gigahertz‐band radars that are required to operate in a disturbed propagation environment (i.e., in chemical releases or nuclear detonations).

Numerical Computations of Elastic Wave Equations

An improved numerical scheme is introduced for solving a set of three-dimensional parabolic elastic wave equations. The new scheme is based on the combination of an ordinary differential equation method and a finite difference scheme. A stability analysis of the numerical method is presented along with a numerical example in which computational accuracy is verified by comparison with an exact three-dimensional analytic solution.

Nonparaxial equation for linear and nonlinear optical propagation

The formalism of coupled-mode theory, specialized to the continuum of radiation modes, allows us to extend the standard parabolic wave equation to include nonparaxial terms and vectorial effects, and, in particular, to generalize the nonlinear Schrödinger equation that describes propagation in the presence of an intensity-dependent refractive index.