  We study scattering of waves by impurities in strongly precompressed granular
chains. We explore the linear scattering of plane waves and identify a
closed-form expression for the reflection and transmission coefficients for the
scattering of the waves from both a single impurity and a double impurity. For
single-impurity chains, we show that, within the transmission band of the host
granular chain, high-frequency waves are strongly attenuated (such that the
transmission coefficient vanishes as the wavenumber $k\rightarrow \pm\pi$),
whereas low-frequency waves are well-transmitted through the impurity. For
double-impurity chains, we identify a resonance --- enabling full transmission
at a particular frequency --- in a manner that is analogous to the
Ramsauer--Townsend (RT) resonance from quantum physics. We also demonstrate
that one can tune the frequency of the RT resonance to any value in the pass
band of the host chain. We corroborate our theoretical predictions both
numerically and experimentally, and we directly observe complete transmission
for frequencies close to the RT resonance frequency. Finally, we show how this
RT resonance can lead to the existence of reflectionless modes even in granular
chains (including disordered ones) with multiple double impurities.
