Nonlinear Fresnel diffraction of weak shock waves.

Fresnel diffraction at a straight edge is revisited for nonlinear acoustics. Considering the penumbra region as a diffraction boundary layer governed by the KZ equation and its associated jump relations for shocks, similarity laws are established for the diffraction of a step shock, an "N" wave, or a periodic sawtooth wave. Compared to the linear case described by the well-known Fresnel functions, it is shown that weak shock waves penetrate more deeply into the shadow zone than linear waves. The thickness of the penumbra increases as a power of the propagation distance, power 1 for a step shock, or 3/4 for an N wave, as opposed to power 1/2 for a periodic sawtooth wave or a linear wave. This is explained considering the frequency spectrum of the waveform and its nonlinear evolution along the propagation, and is confirmed by direct numerical simulations of the KZ equation. New formulas for the Rayleigh/Fresnel distance in the case of nonlinear diffraction of weak shock waves by a large, finite aperture are deduced from the present study.