The interaction of harmonic plane and spherical pressure waves with an absorbing solid elastic sphere is modeled. The analytic, boundary‐element‐method (BEM) and finite‐element‐method (FEM) solutions for the internal displacement, strain, and stress fields are evaluated numerically for 50‐mm‐diameter solid spheres of poly(methyl methacrylate) (PMMA) and high‐density polyethylene (HDPE) in an immersion medium of mass density 1000 kg/m3 and sound speed 1500 m/s. The mass density and longitudinal and transverse sound speeds of the PMMA sphere are 1191 kg/m3, 2690 m/s, and 1340 m/s, respectively. The corresponding properties of the HDPE sphere are 957 kg/m3, 2430 m/s, and 950 m/s. The two materials are assumed to have a hysteresis‐type absorption, hence with constant product of absorption coefficient and wavelength. The respective values of this product for longitudinal and shear waves for PMMA are assumed to be 0.19 and 0.29 dB, and for HDPE, 0.40 and 1.20 dB. Each of two frequencies is considered, 10 and 100 kHz, for which the wavenumber‐radius product is π/3 and 10π/3, respectively. Results for the three solution methods are compared. [Work partly supported by NOPP through ONR Award No. N000140710992.]