  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.
