Kenneth E. Gilbert

A fast Green’s function method for one‐way sound propagation in the atmosphere

A Green’s function method is used to derive a fast, general algorithm for one‐way wave propagation. The algorithm is applied to outdoor sound propagation. The general method is not limited to atmospheric sound propagation, however, and can be applied to other problems, such as sound propagation in the ocean and electromagnetic wave propagation. The new algorithm, called ‘‘GF‐PE’’ (Green’s function method for the parabolic equation), reduces to the well‐known Fourier split‐step algorithm for the parabolic equation (PE) when no boundary conditions are imposed (e.g., at a ground surface). With the GF‐PE, range steps many wavelengths long are possible, while with a PE algorithm based on a finite‐difference range step, such as the Crank–Nicolson method, the range steps are typically limited to a fraction of a wavelength. Because of its longer range step, the new algorithm is 40–450 times faster than PE algorithms that use the Crank–Nicolson method. For outdoor sound propagation over a locally reacting ground surface, the computed GF‐PE field is the sum of three terms: a direct wave, a specularly reflected wave, and a surface wave. With the new method, the air–ground impedance condition is treated exactly and results in an analytic expression for the surface wave contribution. Numerical results from the GF‐PE model are presented and compared to exact calculations, fast‐field program (FFP) calculations, and PE results computed with the Crank–Nicolson method. The GF‐PE algorithm is shown to be accurate and approximately two orders of magnitude faster than a PE based on the Crank–Nicolson method. Hence, the new algorithm opens the door to some useful new computational capabilities such as real‐time predictions on desktop computers, fast pulse calculations, and practical three‐dimensional calculations.

A tutorial on the parabolic equation (PE) model used for long range sound propagation in the atmosphere

Abstract The theory of the parabolic equation (PE) method and the manner in which a finite difference numerical solution scheme can be constructed are presented. The treatment deals with both narrow and wide angle cases. The implementation of a second-order accurate ground boundary condition and an upper boundary condition, designed to minimise reflections into the computational region, are also described.

Application of the parabolic equation to sound propagation in a refracting atmosphere

A wide‐angle parabolic equation (PE) model is presented that is applicable to sound propagation in a steady (nonturbulent) atmosphere overlying a flat, locally reacting ground surface. The numerical accuracy of the PE model is shown by comparing PE calculations to calculations from a ‘‘fast‐field program’’ (FFP). For upward refraction, the PE and FFP solutions agree to within 1 dB out to ranges where the sound‐pressure levels drop below the accuracy limits of both models. For downward refraction, the PE and FFP agree to within 1 dB except at deep interference minima. Parabolic equation calculations are also compared to measured values of excess attenuation for 15 different combinations of frequencies and ranges. In general, the PE model gives good agreement with the average experimental values. For upward refraction at the highest frequency (630 Hz), however, the PE predicts a strong shadow zone that is not observed in the data.

The radiation of atmospheric microbaroms by ocean waves

A two-fluid model, air over seawater, is used to investigate the radiation of infrasound by ocean waves. The acoustic radiation which results from the motion of the air/water interface is known to be a nonlinear effect. The second-order nonlinear contribution to the acoustic radiation is computed and the statistical properties of the received microbarom signals are related to the statistical properties of the ocean wave system. The physical mechanisms and source strengths for radiation into the atmosphere and ocean are compared. The observed ratio of atmospheric to oceanic microbarom peak pressure levels (approximately 1 to 1000) is explained.

Calculation of turbulence effects in an upward refracting atmosphere

In an upward refracting atmosphere, measured values of excess attenuation (50–500 Hz) seldom exceed 20 to 30 dB at a range of 1 km. Calculations of excess attenuation for a steady (nonturbulent) atmosphere predict a deep shadow zone with much higher excess attenuation. This paper investigates the contribution of atmospheric turbulence to decreasing the predicted excess attenuation. Since no convenient analytical method presently exists which can simultaneously account for turbulence, upward refraction, and a finite‐impedance ground surface, a parabolic equation method is used to numerically simulate sound propagation. As an initial test, the calculation is compared to Daigles theoretical and experimental results for a homogeneous atmosphere [J. Acoust. Soc. Am. 65, 45–49 (1979)]. For a test with upward refraction, the calculation is compared to the experimental results of Wiener and Keast [J. Acoust. Soc. Am. 31, 724–733 (1959)].

ACOUSTIC SCATTERING AND THE SPECTRUM OF ATMOSPHERIC TURBULENCE

Some issues regarding atmospheric turbulence modeling and its role in acoustic scattering calculations are discussed. Discrepancies between turbulence spectral models appearing in the acoustical and in the atmospheric sciences literature are noted, and it is argued that these discrepancies can be understood by recognizing that the acoustic wavelength and scattering geometry combine to act as an “acoustic filter” which selects a specific part of the turbulence spectrum. A particular model spectrum can yield satisfactory acoustic scattering predictions if it fits the actual spectrum well at the acoustically filtered turbulence scales, even if the model is a poor overall representation of the turbulence spectrum. Proper interpretation of length scales determined by fitting two-point correlation functions, and the importance of averaging times in estimating variances, are also discussed in relation to the action of the acoustic filter.

An exact Laplace transform formulation for a point source above a ground surface

An exact analysis is given for a point source in air above a ground surface. By representing the plane‐wave reflection coefficient as the Laplace transform of an image source distribution, a well‐behaved image integral, instead of the usual Sommerfeld integral, is obtained. The approach is valid for both locally and extended reacting surfaces. For a locally reacting ground surface, the image integral is an especially simple, rapidly convergent integral. The integral for local reaction is investigated analytically for a number of limiting cases. The resulting analytic solutions are compared with analytic solutions obtained from more standard approaches. Finally, the image integral for local reaction is analyzed numerically, and an upper limit on the numerical integration is given. It is shown that with realistic values of ground impedance, the prescribed integration limit allows the image integral to be easily and accurately computed numerically.

A theoretical treatment of the long range propagation of impulsive signals under strongly ducted nocturnal conditions

On clear nights, over flat land, a sound duct develops in which sound can carry to great distances. As is the case with all ducted propagation, there is strong dispersion so that a broadband signal undergoes severe distortion as it propagates. The signal received at long ranges from an impulsive source is a wave train, of much greater duration than the initial impulse, consisting of a series of arrivals. The long range ground to ground propagation of an impulsive signal in a typical nocturnal duct is studied and the natures of the various arrivals are explained. A direct connection is drawn between the meteorological and ground conditions and the structure of the propagated signal.

A stochastic model for scattering from the near‐surface oceanic bubble layer.

A stochastic scattering model has been developed that allows the backscatter from a subsurface bubble layer to be written as the product of a geometric factor times the horizontal wave‐number (‘‘power’’) spectrum of the bubble layer. By dividing the measured backscatter versus frequency data by the geometric factor, one can ‘‘invert’’ the backscatter data and directly infer the horizontal wave‐number spectrum of the bubble layer. Three different data sets give power‐law wave‐number spectra: P(K)≊A‖K‖−β, where β≊4 for two of the data sets and β≊3 for the third data set. The factor A is a constant that is different for each data set. When the inferred wave‐number spectrum is used to predict backscatter versus angle, good agreement is obtained with the data at low frequencies and low grazing angles. The consistency in the inferred wave‐number spectrum strongly suggests that a systematic power‐law spectrum exists for the near‐surface oceanic bubble layer. A time‐dependent bubble plume model is discussed that ...

Application of the parabolic equation to the outdoor propagation of sound

Abstract A wide-angle parabolic equation has been developed for atmospheric sound propagation above locally reacting surfaces. The method considers only outward-going lateral waves in computation of the acoustic field. It is applied to a propagation problem over flat, open, finite impedance ground and to realistic atmospheric profiles. The comparisons with data exhibit good agreement (1–2 dB) in most downward-refracting cases and all low-frequency cases examined. The high-frequency, upward-refracting predictions contained dramatic shadow zones that were not as pronounced in the data.