Superresonant systems of scatterers. I

Abstract

Single compact scatterers insonified at frequencies near their intrinsic radial resonance ω0 act primarily as monopole sources, amplifying the local pressure field by a factor of order (kRa)−1, a being the scatterer radius and kR the wavenumber at ω0 in the surrounding material (kRa≂1.37×10−2 for air bubbles in water). This paper deals with systems of such scatterers (bubbles, inflated balloons, thin shells in water) which, through coupling via multiple scatter, may themselves exhibit true resonant modes at frequencies close to ω0, amplifying the local pressure field by a further factor of order (kRa)−1, leading to net amplifications in excess of 103 for bubbles or balloons, i.e., of over 120 dB for the intensity. This phenomenon, referred to as superresonance, is predicted by formulas for the equivalent source strength B of each scatterer of the system, proper account being taken of multiple scatter between all elements, using a self-consistent scheme. It has long been known that for a pair of scatterers (doublet), or in periodic lattices, B exhibits maxima for certain values of x=kl (l being the distance between scatterers), which had been called resonances [Twersky, J. Opt. Soc. Am. 52, 145–171 (1962)], but are in fact only partial (quasi) resonances. It is demonstrated here that, under suitable conditions, true resonances exist; i.e., in the absence of attenuation, B exhibits real poles (the infinities are removed by introduction of acoustic damping and finite radii). The existence of such superresonances is furthered by the presence of elastic boundaries in the fluid near the system (e.g., a thin plate or a solid half-space). Multiple scatter coupling then occurs by means of guided flexural or surface waves, which offer a more efficient energy transport mechanism between scatterers than the acoustic modes (volume waves) of a homogeneous fluid fullspace. In the latter case, a doublet of resonant bubbles, or shells, exhibits no superresonances (poles), only partial ones (quasiresonances); it takes at least three scatterers to create a true superresonant system in a fullspace, and even so there is only one resonant spacing (kl=1.8955, ω=1.0022 ω0 for an equilateral triangle of bubbles in water). But near a thin plate, both doublet and triplet configurations develop spectra of resonant spacings xn and frequencies ωn. Near an elastic solid half-space, Stoneley (Scholte) wave coupling creates similar effects. The role of small interface roughness on this phenomenon is also examined.