Transparency Property of One Dimensional Acoustic Wave Equations

摘要

This thesis proposes a new proof of the acoustic transparency theorem for material with a bounded variation. The theorem states that if the material properties (density, bulk modulus) is of bounded variation, the net power transmitted through the point z = 0 over a time interval [−T,T] is greater than some constant times the energy at the time zero over a spatial interval [0,Z], provided that T equals the time of travel of a wave from 0 to Z. This means the reflected energy of an input into the earth will be received. Otherwise, the reflections may not arrive at the surface. A proof gives a lower bound for material properties (density, bulk modulus) with bounded variation using sideways energy estimate. A different lower bound that works only for piecewise constant coefficients is also given. It gives a lower bound by analyzing reflections and transmissions of the waves at the jumps of the material properties. This thesis also gives an example to illustrate that the bounded variation assumption may not be necessary for the medium to be transparent. This thesis also discusses relations between the transparency property and the data of an inverse problem.