Multiple scattering by random configurations of circular cylinders: Weak scattering without closure assumptions

Acoustic scattering by random collections of identical circular cylinders is considered. Each cylinder is penetrable, with a sound-speed that is close to that in the exterior: the scattering is said to be “weak”. Two classes of methods are used. The first is usually associated with the names of Foldy and Lax. Such methods require a “closure assumption”, in addition to the governing equations. The second class is based on iterative approximations to integral equations of Lippmann–Schwinger type. Such methods do not use a closure approximation. Our main result is that both approaches lead to exactly the same formulas for the effective wavenumber, correct to second-order in scattering strength and second-order in filling fraction. Approximations for the average wavefield are also derived and compared.