Two-dimensional linear and nonlinear wave propagation in a half-space

We examine a staggered pseudospectral method to solve a two-dimensional wav e propagation problem with arbitrary nonlinear constitutive equations, and evaluate a general image method to simulate the traction-free boundary cond ition at the surface. This implementation employs a stress-velocity formula tion and satisfies the free surface condition by explicitly setting surface shear stress to zero and making the normal stress antisymmetric about the free surface. Satisfactory agreement with analytical solutions to Lamb's pr oblem is achieved for both vertical point force and explosion sources, and with perturbation solutions for nonlinearly elastic wave propagation within the domain of validity of such solutions. The Rayleigh wave, however, suff ers much more severe numerical dispersion than do body waves. At four grids per wavelength, the relative error in the Rayleigh-wave phase velocity is 25 times greater than the corresponding error in the body-wave phase veloci ty. Thus for the Rayleigh wave, the pseudospectral method performs no bette r than a low-order finite difference method. A substantial merit of the image approach is that it does not assume any pa rticular rheology, the method is readily applicable even when stresses are not analytically related to kinematic variables, as is the case for most no nlinear models. We use this scheme to investigate the response of a nonline ar half-space with endochronic rheology, which has been fit to quasi-static and dynamic observations. We find that harmonics of a monochromatic source are generated and evolve with epicentral range, and energy is transferred from low to higher frequencies for a broadband source. This energy redistri bution characteristic of the propagation is strain-amplitude dependent, con sistent with laboratory experiments. Compared with the linear response, the nonlinear response of an endochronic layer near the surface shows a deampl ification effect in the intermediate-frequency band and an amplification ef fect in the higher-frequency band. The computational method, with modificat ions to accommodate realistic nonlinear soil characteristics, could be appl ied to estimate earthquake strong ground motions and path effects.