Nonlinear multiple scattering of acoustic waves by a layer of bubbles

We present a theoretical and experimental study of the acoustic second-harmonic generation by a single layer of bubbles. This simple system allows us to investigate the subtle interplay between nonlinear effects and multiple scattering. A perturbative model is shown to give an excellent agreement with the experimental measurements, and we demonstrate the existence of an optimal concentration of bubbles, for which the harmonic generation is maximum. The potential of bubble screens as efficient subwavelength acoustic nonlinear sources is discussed. Introduction. – Wave transport in a multiple scattering environment has been a subject of intense research, demonstrating a large variety of behaviors, from the dispersive propagation of a coherent wave [1, 2] to the existence of a diffusive regime, sometimes leading to localization [3, 4]. On the other hand, the nonlinear propagation of waves also comes with many intriguing phenomena, such as self-induced transparency [5] or secondharmonic generation [6], for instance. The question arises of how waves propagate when both strong multiple scattering and nonlinearities are present. For mechanical waves, this question has been addressed in granular media [7, 8], with the complication that both scattering and nonlinearities are strongly dependent on the contact between the grains. Gas bubbles appear as perfect candidates for looking at nonlinear acoustic propagation in a multiple scattering regime: they are efficient acoustic scatterers, as well as strong nonlinear sources. Bubbly liquids have already been shown to exhibit substantial acoustic nonlinearities [9–12]. But an important limitation of the previous studies was the lack of quantitative comparison between the theoretical predictions and the experimental measurements, often due to a weak knowledge of the structure of the bubbly liquids used for the experiments. In this Letter, we use stable and well-characterized bubbly media to carefully study the interplay between nonlinearities and multiple scattering. Our experimental system is a single layer of bubbles, organized on a square lattice (see inset of Fig. 1). We focus on a particular nonlinear mechanism: harmonic generation, i.e. how a pressure wave at frequency f generates, due to the presence of bubbles, a wave at frequency 2f . Theory. – We limit ourselves to the long wavelength regime, which means that the wavelength is much larger than the typical distances involved in the system, namely the radius R of the bubbles, and the distance d between two neighboring bubbles. Linear regime. Excited by a monochromatic pressure with complex amplitude P , P exp[−iωt], a bubble oscillates and generates at distance r a spherical pressure field p(r, t) = V̈ (t−r/c)/(4πr), where V (t) is the instant bubble volume and c the sound speed in the liquid. With a linear development of Rayleigh-Plesset equation [13], this pressure field can be written p(r, t) = fsP exp[iω(r/c − t)]/r, where the scattering function fs(ω) = R ( ω0 ω )2 − 1− i(kR+ δ) (1) has been introduced. In Eq. (1), ω0 is the resonance angular frequency of the bubble and kR+ δ the total damping constant (k = ω/c is the wavenumber in the liquid). The damping of the bubble is due to radiative losses (the kR term) and dissipation (the δ term), the latter being divided into a viscous and a thermal contribution. As they are strong scatterers, bubbles couple efficiently one to each other: if several bubbles are present in a liquid, the total pressure field experienced by one bubble is