Two-dimensional (2-D) interfacial waves on flowing films are unstable with respect to both two-and three-dimensional instabilities. In this paper, several distinct three-dimensional instabilities that occur in different regions of the parameter space defined by the Reynolds number R and the frequency f of forced two-dimensional waves are discussed in detail. (a) A synchronous 3-D instability, in which spanwise deformations of adjacent wave fronts have the same transverse phase, appears over a wide range of frequency. These transverse modulations occur mainly along the troughs of the primary waves and eventually develop into sharp and nearly isolated depressions. The instability involves many higher harmonics of the fundamental 2-D waves. (b) A3-D surbharmonic instability occurs for frequencies close to the neutral curve f,(R). In this case, the transverse modulations are out of phase for successive wave fronts, and herringbone patterns result. It is shown that this weakly nonlinear instability is due to the resonant excitation of a triad of waves consisting of the fundamental two-dimensional wave and two oblique waves. The evolution of wavy films after the onset of either of these 3-D instabilities is complex. However, sufficiently far downstream, large-amplitude solitary waves absorb the smaller waves and become dominant. Q I995 American Institute of Ph.ysics.

A method is presented for application of the perfectly matched layer ͑PML͒ absorbing boundary condition ͑ABC͒ to the P-SV velocity-stress finite-difference method. The PML consists of a nonphysical material, containing both passive loss and dependent sources, that provides ''active'' absorption of fields. It has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies. In this work, numerical simulations are used to compare the PML and an ''optimal'' second-order elastic ABC ͓Peng and Toksöz, J. Acoust. Soc. Am. 95, 733-745 ͑1994͔͒. Reflection factors are used to compare angular performance for continuous wave illumination; snapshots of potentials are used to compare performance for broadband illumination. These comparisons clearly demonstrate the superiority of the PML formulation. Within the PML there is a 60% increase in the number of unknowns per grid cell relative to the velocity-stress formulation. However, the high quality of the PML ABC allows the use of a smaller grid, which can result in a lower overall computational cost.

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A Monte-Carlo finite-difference time-domain (FDTD) technique is developed for wave scattering from randomly rough, one-dimensional surfaces satisfying the Dirichlet boundary condition. Both single-scale Gaussian and multiscale Pierson-Moskowitz surface roughness spectra are considered. Bistatic radar cross sections are calculated as a function of scattering angle for incident angles of 0, 45, 70, and 80 degrees measured from the vertical. The contour path FDTD method is shown to improve accuracy for incident angles greater than 45 degrees. Results compare well with those obtained using a Monte-Carlo integral equation technique.