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Various parabolic equations for advected acoustic waves have been derived based on the assumptions of small Mach number and narrow propagation angles, which are of limited validity in atmospheric acoustics. A parabolic equation solution that does not require these assumptions is derived in the weak shear limit, which is appropriate for frequencies of about 0.1 Hz and above for atmospheric acoustics. When the variables are scaled appropriately in this limit, terms involving derivatives of the sound speed, density, and wind speed are small but can have significant cumulative effects. To obtain a solution that is valid at large distances from the source, it is necessary to account for linear terms in the first derivatives of these quantities [A. D. Pierce, J. Acoust. Soc. Am. 87, 2292-2299 (1990)]. This approach is used to obtain a scalar wave equation for advected waves. Since this equation contains two depth operators that do not commute with each other, it does not readily factor into outgoing and incoming solutions. An approximate factorization is obtained that is correct to first order in the commutator of the depth operators.

A mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested. At each range, the medium is rigidly translated vertically so that a sloping interface becomes horizontal. To simplify the approach, the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected. The effects of these terms can be approximately accounted for by applying a leading-order correction to the phase. The mapping introduces variations in topography, which are relatively easy to handle for the case of a pressure-release boundary condition. The accuracy of the approach is demonstrated for problems involving fluid sediments. The approach should also be accurate for problems involving elastic sediments and should be useful for solving three-dimensional problems involving variable topography.

Atom scattering from disordered surfaces: Randomly corrugated hard walls and the sudden approximation Scattering from disordered surfaces in the sudden approximation J. Chem. Phys. 78, 4277 (1983); 10.1063/1.445105 Rotationally inelastic molecule-surface scattering in the sudden approximationIt was recently proposed that the sudden approximation should be a powerful tool for the calculation of the angular intensity distribution in high-energy atom scattering from disordered surfaces. In the present study the sudden approximation is applied to scattering from one-and two-dimensional models of: (1) Isolated adsorbed impurities on crystalline surfaces (Ar on eu) ;(2) Mixed overlayers on an underlying surface (Xe + Ar mixtures on a smooth surface). The results are tested against numerically exact quantum-mechanical wave packet calculations. Except for very low collision energies, the sudden approximation gives results of excellent quantitative accuracy for both types of noncrystalline surfaces. At low energies, several features ofthe intensity distribution are not produced correctly by the sudden: These are found to be due mainly to double collision effects. The accuracy and validity range of the method are discussed in the light of the results obtained in the test calculations.

On-surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles. An analytic representation of the Dirichletto-Neumann map for a circle is derived and used to construct a higher-order on-surface radiation condition for a generally convex perfectly conducting body in two dimensions. This approach is based on a Hankel function in which a tangential operator appears in the index. In the high-frequency limit, this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems. Treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters, the representation is fit to a rational function to provide an accurate and efficient on-surface radiation condition that is tested for various examples.