Capillary wave scattering from a surfactant domain

The study of capillary wave scattering by a circular region with different interfacial properties from the rest of an otherwise homogeneous interface is motivated by experiments on wave attenuation at a monolayer-covered air-water interface where domains of one surface phase are dispersed in a second surface phase. Here the scattering function is calculated for an incident wave of frequency ω (wavevector k₀) scattering from an isolated circular domain of radius a with surface tension σ₁ which is imbedded in an otherwise infinite interface of surface tension σ₀. The underlying fluid is treated as irrotational and the three-dimensional flow problem coupling the heterogeneous surface to the underlying liquid is reduced to a set of dual integral equations, which are solved numerically. With this solution the scattering amplitudes and the total scattering cross sections are calculated as a function of the surface tension ratio σ0/σ₁ and incident wavenumber k₀a. The analogous problem of a discontinuous change in bending rigidity is also considered and the solution to the complete viscous problem is outlined in the Appendix. Experimental implications of these results are discussed.