Nonlinear Fresnel diffraction of weak shock waves

Abstract

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.

Nonlinear focusing of acoustic shock waves at a caustic cusp

“…Finally, we indicate that a modified version of the present code has already been used to confirm theoretical predictions in a study about the nonlinear Fresnel diffraction of weak shock waves ͑Coulouvrat and Marchiano, 2003͒. In that previous study, the numerical code was similar to the code presented here except for the boundary conditions. The agreement between theoretical predictions and numerical results on self-similar solutions of the nonlinear KZ equation also permitted us to validate quantitatively the numerical method.…”

“…Once the solution of the linear KZ is computed, it is used to initialize the calculation of the solution of the Burgers equation for potential on an elementary step ⌬z. That solution is calculated semianalytically with the Hayes graphical method ͑1969͒, which was recently adapted for the numerical procedure by Marchiano ͑2003͒ andMarchiano et al ͑2003a͒. Finally, on a ⌬z step, the solution takes into account the diffraction and the nonlinear effects.…”

Bulk Probing of Shock Wave Spatial Distribution in Opaque Solids by Ultrasonic Interaction

“…(b)). The resulting scattering profile (Fig.3(b)) ideally reproduces the experimental V shape, including the strongly attenuated lateral oscillations likely due to the Fresnel diffraction of the probe on the shock circumference [29]. However, a huge difference (3 decades) in amplitude remains.…”

Numerical analysis of weak acoustic shocks in aperiodic array of rigid scatterers

“…Previous studies of nonlinear propagation in periodic media have focused on time harmonic signals propagating in one-dimensional waveguides [1]. Nonlinear scattering of a two-dimensional plane wave from a single semi-infinite rigid barrier was studied under a paraxial approximation using similarity theory [2]. The purpose of this paper is to provide a description of pulse propagation in a periodic twodimensional system of scattering objects based on numerical simulations.…”

“…These simulations are, as with Ref. [2], based on a paraxial approximation known as the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation [3].…”