Note on Acoustic Pseudopotentials: Energy Viewpoint, Real Scatterers and Linear Arrays

A study of scalar acoustic pseudopotentials and their connection to actual spherical scatterers is presented. The pseudopotential concept and the governing scattering equations are established through an alternative energy viewpoint from which the operator-based definition follows. Limiting formulas are derived for spherical acoustic scatterers with subwavelength size and it is shown that pseudopotentials can approximate only certain scattering types: impedance or soft impenetrable ones or penetrable ones having equal density but different compressibility from the surrounding medium. The accuracy of the approximation is investigated numerically and mapped versus the scatterer size and distance of the observation point. As an application, the infinite linear periodic array of small spheres is considered and it is shown that the frequency dependence of a scatterer's responsivity to the incident field forbids the existence of traveling waves except in the cases of positive impedance scatterers or penetrable scatterers having the same density but higher compressibility than the surrounding medium.