Sound velocity and absorption in a coarsening foam.

We present experimental measurements of sound velocity and absorption in a commercial shaving foam. We observe that both quantities evolve with time as the foam coarsens increasing its mean bubble radius <R>. By varying the acoustic frequency we probe the foam from the large wavelength regime, lambda approximately 1500<R>, down to the scale lambda approximately 20<R>. Sound absorption alpha varies significantly with both the foam age and the excitation frequency. After an initial transition time of 20 min, the attenuation per wavelength, alphalambda, varies linearly with the foam age. In addition, for evolution times smaller than approximately 90 min, we observe that alphalambda scales linearly with both foam age and frequency. From these scalings we show that the thermal dissipation mechanism is the dominant one. Sound velocity c is initially frequency independent but the medium becomes slightly dispersive as the foam coarsens. We observe that sound velocity depends on the evolution of the structure of the foam, even in the large wavelength regime. After 2 h of foam coarsening, c decreases at least by a factor of 20%, due to the softening of the foam. These facts are explained by considering the liquid matrix elasticity, due to the presence of surfactant molecules. A simple model of foam structure, combined with results of Biot's theory for porous media, gives both good qualitative and quantitative agreement with our experimental results in the low frequency regime.