Elastic Wave Transmission at an Abrupt Junction in a Thin Plate, with   Application to Heat Transport and Vibrations in Mesoscopic Systems
  The transmission coefficient for vibrational waves crossing an abrupt junction between two thin elastic plates of different widths is calculated. These calculations are relevant to ballistic phonon thermal transport at low temperatures in mesoscopic systems and the Q for vibrations in mesoscopic oscillators. Complete results are calculated in a simple scalar model of the elastic waves, and results for long wavelength modes are calculated using the full elasticity theory calculation. We suggest that thin plate elasticty theory provide a useful and tractable approximation to the full three dimensional geometry. 
M. C. Cross and Ron Lifshitz