Sound waves incident on a thin elastic rod whose radius is smaller than the wavelength of the incident sound induce flexural and uniform compressional oscillations in the rod. These elastic oscillations, in turn, radiate sound waves into the fluid medium and affect the scattered waves. This paper deals with an analytic study on sound scattering by, and acoustoelastic vibrations of, a thin elastic unbound rod in a viscous fluid. The shear viscosity of the fluid is considered in the solutions to boundary value problems concerning the sound scattering and the elastic response of the rod. Results show that the scattered compressional waves consist of the rigid‐rod scattering of compressional waves, monopolar waves due to the uniform pulsating of the rod, and dipolar waves due to the flexural vibration of the rod. The scattered viscous waves consist of the rigid‐rod scattering of viscous waves and dipolar waves due to the flexural vibration of the rod. Acoustic resonances occur when the effective inertia force of the rod balances the stiffness force of the rod. The fluid viscosity and the scattering of sound give rise to damping effects for the rod vibrations and signficantly affect the acoustic resonances.