Branched flow and caustics in nonlinear waves

Abstract

Rogue waves, i.e.high amplitude fluctuations in random wave fields, have been studied in several contexts, ranging from optics via acoustics to the propagation of ocean waves. Scattering by disorder, like current fields and wind fluctuations in the ocean, as well as nonlinearities in the wave equations provide widely studied mechanisms for their creation. However, the interaction of these mechanisms is largely unexplored. Hence, we study wave propagation under the concurrent influence of geometrical (disorder) and nonlinear focusing in the (current-modified) nonlinear Schrödinger equation. We show how nonlinearity shifts the onset distance of geometrical (disorder) focusing and alters the peak intensities of the fluctuations. We find an intricate interplay of both mechanisms that is reflected in the observation of optimal ratios of nonlinearity and disorder strength for the generation of rogue waves.

Branched Flow

“…The branching, while somewhat smoothed off by a 10% or 20% dispersion in initial wave direction, actually survives in quite high contrast and surprisingly develops more contrast and even finer structure downstream (figure 6). The excess freak wave probability was accounted for without involving nonlinear effects (not denying their crucial importance in the "end game" of the formation of a very large wave 22 ) by nonuniform Gaussian sampling over the energy density of branches of the branches surviving the fuzzy manifold. It is thus possible that the ignition step of freak waves is still in the linear regime.…”

Branched flows of flexural elastic waves in non-uniform cylindrical shells

“…The assumption of slow spatial variation of properties enables the simplification of the wave elastodynamics to a set of ray equations using the eikonal/ WKB approximations [26,27]. They are further simplified using the paraxial approximation [28], permitted by the weak scattering nature of the problem, which asserts a predominantly axial direction of the wave vector. These ray equations are then used to analytically derive the scaling law (Eq 1) relating the position of focusing and the severity of the non-homogeneity.…”

Branched flows of flexural waves in non-uniform elastic plates

“…The assumption of slow spatial variation of properties enables the simplification of the wave elastodynamics to a set of ray equations using the eikonal/WKB approximations 22,23 . They are further simplified using the paraxial approximation 24 , permitted by the weak scattering nature of the problem, which asserts a predominantly axial direction of the wave vector. These ray equations are then used to analytically derive the scaling law (Equation 1) relating the position of focusing and the severity of the non-homogeneity.…”