{'fraction_non_alphanumeric': 0.015081206496519721, 'fraction_numerical': 0.0, 'mean_word_length': 5.537878787878788, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Phase-shift derivation of expansions for material and frequency dependence of progressive-wave radiation forces and backscattering by spheres.

When considering the scattering of sound and radiation forces for spheres, it has historically been helpful to understand situations lacking dissipation. In that case the scattering is characterized by real partial-wave phase shifts. At low frequencies expansions show the dependence of each phase shift on material properties and on frequency. Those expansions are used here to describe the frequency and material dependence of scattering and radiation forces beyond the usual Rayleigh-scattering approximation. Results for radiation forces on spheres in standing waves are extended to plane progressive waves. The expansion coefficients use algebraic functions. Results for movable and fixed rigid spheres are shown.