Nonlinear focusing of acoustic shock waves at a caustic cusp

Abstract

The present study investigates the focusing of acoustical weak shock waves incoming on a cusped caustic. The theoretical model is based on the Khokhlov-Zabolotskaya equation and its specific boundary conditions. Based on the so-called Guiraud's similitude law for a step shock, a new explanation about the wavefront unfolding due to nonlinear self-refraction is proposed. This effect is shown to be associated not only to nonlinearities, as expected by previous authors, but also to the nonlocal geometry of the wavefront. Numerical simulations confirm the sensitivity of the process to wavefront geometry. Theoretical modeling and numerical simulations are substantiated by an original experiment. This one is carried out in two steps. First, the canonical Pearcey function is synthesized in linear regime by the inverse filter technique. In the second step, the same wavefront is emitted but with a high amplitude to generate shock waves during the propagation. The experimental results are compared with remarkable agreement to the numerical ones. Finally, applications to sonic boom are briefly discussed.

Evidence of wave front folding of sonic booms by a laboratory-scale deterministic experiment of shock waves in a heterogeneous medium

“…Note, however, that this wave focusing does not correspond to caustics as described by the high-frequency asymptotic theory of catastrophes ͑Thom, 1972; Berry, 1976͒. Indeed, that theory is valid only if the curved wave front aperture is large compared to the wavelength ͑the difference between ideal infinite incident wave front satisfying catastrophe theory and a finite one has been outlined by Marchiano et al, 2005͒, an assumption that is obviously not satisfied here because of the small size of the heterogeneity. Moreover, if the incoming wave presents shocks, the analytical solutions of the linear problem at the fold ͑Guiraud, 1965͒ or the cusp caustics ͑Coulouvrat, 2000͒ lead to an unphysical infinite amplitude that can only be reduced by introducing local nonlinear effects.…”

Nonlinear and diffraction effects in propagation of N-waves in randomly inhomogeneous moving media

“…[8][9][10] Linear scattering of sound waves on random scalar inhomogeneities have also been considered using Born approximation. 11 Focusing of high amplitude and wide band acoustic signals has been considered either in homogeneous medium giving the initial distortion of the wavefront [12][13][14] or in media with deterministic inhomogeneity. 15,16 The diffraction of nonlinear acoustic pulses from random structures has only been reported for media with scalar type inhomogeneities.…”

Random focusing of nonlinear acoustic N-waves in fully developed turbulence: Laboratory scale experiment

“…In outdoor sound propagation, the importance of wind in acoustics has been known since the end of the 19th century, when initial studies on effective sound speed and formation of shadow zones were performed based on observations of various sources of intense sound like volcanic eruptions, thunder, explosions, and artillery fire. 1 While propagating in turbulent air, acoustic waves are distorted by the combined effects of diffraction and scattering by atmospheric inhomogeneities, [2][3][4][5] nonlinear dissipation, 6,7 and linear absorption/relaxation. 8,9 At sufficiently long propagation distances multiple focusing of acoustic waves is predicted.…”