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Recent experiments suggest that the superconducting order parameter ofSr$_2$RuO$_4$ has two components. A two-component order parameter has multipledegrees of freedom in the superconducting state that can result in low-energycollective modes or the formation of domain walls -- a possibility that wouldexplain a number of experimental observations including the smallness of thetime reversal symmetry breaking signal at T$_\mathrm{c}$ and telegraph noise incritical current experiments. We perform ultrasound attenuation measurementsacross the superconducting transition of Sr$_2$RuO$_4$ using resonantultrasound spectroscopy (RUS). We find that the attenuation for compressionalsound increases by a factor of seven immediately below T$_\mathrm{c}$, in sharpcontrast with what is found in both conventional ($s$-wave) andhigh-T$_\mathrm{c}$ ($d$-wave) superconductors. We find our observations to bemost consistent with the presence of domain walls between differentconfigurations of the superconducting state. The fact that we observe anincrease in sound attenuation for compressional strains, and not for shearstrains, suggests an inhomogeneous superconducting state formed of twodistinct, accidentally-degenerate superconducting order parameters that are notrelated to each other by symmetry. Whatever the mechanism, a factor of sevenincrease in sound attenuation is a singular characteristic with which anypotential theory of the superconductivity in Sr$_2$RuO$_4$ must be reconciled.

Strong Increase in Ultrasound Attenuation Below T$_\mathrm{c}$ in Sr$_2$RuO$_4$: Possible Evidence for Domains