Exact analytical solution of linear acoustic scattering by a large-amplitude pulsating sphere

An exact analytical solution is developed for the linear acoustic scattering of a plane sound wave by a sphere undergoing large–amplitude radial pulsations. The analytical solution is expressed in the form of a multi–harmonic modal expansion, where the scattered acoustic field is represented as a series of Legendre polynomials and spherical Hankel functions. Analytical models available in the literature successfully address scattering by rigid or weakly deforming bodies of simple shape, but no exact solution exists for obstacles undergoing large dynamic deformations. The present work addresses this gap by deriving an analytical solution within linear acoustic theory, rigorously enforcing the impermeability boundary condition on the time-varying surface of the pulsating sphere without relying on perturbation expansions. Although the governing wave equation remains linear, large–amplitude boundary motion introduces strong kinematic nonlinearities that lead to frequency mixing and the generation of multi–harmonic scattered fields even under purely tonal incident excitation. Fluid nonlinearities are intentionally neglected to isolate the effects associated solely with boundary kinematics. Validation is performed through comparison with a multi‑harmonic Boundary Element Method solver previously developed and validated for arbitrarily deforming bodies. Excellent agreement is achieved across a broad range of pulsation amplitudes, reaching 80% of the undeformed radius, and for observers located in both the near and far fields. The results further demonstrate the limitations of existing perturbation-based analytical approaches beyond moderate deformation regimes. The present solution provides a fundamental reference for the verification and calibration of numerical acoustic solvers and a basis for future extensions toward nonlinear scattering models.