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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 $k_0$) scattering from an isolated circular domain of radius $a$ with surface tension $σ_{1}$ which is imbedded in an otherwise infinite interface of surface tension $σ_{0}$. 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 $σ_{0}/σ_{1}$ and incident wavenumber $k_0 a$. The analogous problem of a discontinuous change in bending rigidity is also considered and the solution to the complete viscous problem is outlined in an appendix. Experimental implications of these results are discussed.