Heat transport across interfaces in nanostructures is an important open problem that arises into play in the design and operation of electronic and thermoelectric devices. In particular, lattice mismatch across the interface and the interface profile strongly influence thermal transport. In this work, we study elastic-wave scattering at the junction between two different materials. We solve the elastic-wave equations with the finite-difference time domain (FDTD) technique. We investigate the effects of mode conversion and energy trapping at the interface and how they are affected by interface roughness and lattice mismatch. The results provide a unique outlook on the interfacial heat transport and its effect on the operation of nanodevices.