Since elastic surface waves are examples of guided waves, nonlinear effects are significant only between linearized modes which have good matching of both phase and group velocities. Within homogeneous half-spaces, linearized modes travelling across the surface in any direction are completely non-dispersive. The phase speed can depend upon direction, but not upon frequency (or wavelength). Consequently, the standard weakly nonlinear theory equates the derivative of the (complex) Fourier transform of the surface displacement to an integral of convolution type, with a kernel which involves various elastic moduli and which takes account of the depth-dependence of the displacement fields within interacting pairs of modes having any two distinct wavenumbers.The direct formulation of the equation governing the evolution of surface slope involves a quadratically nonlinear, nonlocal operator, incorporating the fact that waveform evolution is influenced by quadratically nonlinear contributions to the stress at all depths. This kernel splits naturally into one entirely local part, a nonlocal part allowing wave profiles to preserve symmetry and one necessarily causing asymmetry. Details are determined for elastic materials of arbitrary anisotropy.
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- Nonlinearity in Elastic Surface Waves Acts Nonlocally
D. F. Parker
- Springer Netherlands
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