
Propagation of a Sound Wave Through an Interface BarrierWhat happens to a sound wave when it interacts with some type of discontinuity? Each material has some particular acoustic impedance. This acoustic impedance is given by the following equation:
Where is the density and is the speed of sound. Consider a sound wave propagating between two different materials. The first material has acoustic impedance and the second . For simplicity let's assume the wave arrives perpendicular to the surface. Some of the sound wave will propagate through the boundary and some will be reflected. A sound wave that reaches an interface between two materials will generate a reflected wave and a transmitted wave. The Transmission coefficient measures the fraction of energy that passes through the barrier and the reflection coefficient measures the fraction of energy reflected back. Physically we might expect that the difference between the acoustic impedance values of the two materials might have something to do with how much sound is transmitted and how much is reflected. Suppose that we could adjust to be any value we wanted. If we make then there should be no reflection at all and the transmitted wave should be exactly the same as the incident wave. If is much larger or smaller than then most of the energy would be reflected at the boundary. It's also reasonable to expect that the total amplitude of in the Incident, Reflected and Transmitted waves must remain constant. Since all the waves meet at a thin layer, regardless of what happens on either side, the sound waves must smoothly join together. In other words, we expect the following to hold
It turns out that the reflection coefficient and the transmission coefficient can be written down as follows
Notice that if we set , then and . If we have a wave that reaches a boundary and the wave has an amplitude then the amplitudes of the reflected and transmitted waves are given by the following equations
Note that the amplitude of the reflected wave can be positive or negative. A negative value corresponds to a 180 degree phase change. In other words, the wave will appear upside down with respect to the incident wave. The "coupling efficiency" between a transducer and a material is actually the transmission coefficient. In order to calculate this coupling efficiency it's necessary to know the acoustic impedance of the transducer and the material. If the acoustic impedance of the material is not known, then we can't calculate the transmission coefficient. 
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