Similar to electromagnetics, sound waves that are incident upon a medium with different properties will experience reflection, transmission/refraction or absorption depending on multiple factors. The analysis of the transmission and reflection of sound is greatly simplified when the boundary between media and the incident wave are planar. The amount of transmission and reflection depends on each material’s acoustic or characteristic impedance (r = p*c) and the angle that the incident wave makes with the boundary.

Much like in electromagnetics, transmission and reflection and reflection coefficients can be defined as shown.

The figure shows that the transmission coefficient is the ratio of the transmitted pressure wave to the incident pressure wave. The reflection coefficient is the ratio of the reflected pressure wave to the incident pressure wave. This is similar to the voltage reflection coefficient from transmission line theory.

The angle of the sound wave with the boundary can either be “normal” (at a 90 degree angle) or “oblique”. For normal incidence, the problems are greatly simplified, as there is no refracted wave, only a reflected component and a transmitted component. These are described by plane wave equations:

The propagation vector (k) is dependent on the material. The transmitted wave has a different propagation vector value because it has surpassed the boundary into the second medium. This is due to the fact that a material’s characteristic impedance is dependent on the speed of sound.

In order for boundary conditions to be satisfied, both the normal component of velocity and pressure must be continuous. This means the acoustic pressure on both sides of the boundary must be equal, leaving no net force on the planar boundary separating the fluids. The fluids must also remain in contact, meaning the normal component of the velocity must be continuous.

These equations can be used to derive the reflection and transmission coefficients.

It is important to note that the reflection coefficient is always real. When the second medium (the medium the transmitted wave propagates into) has a greater acoustic impedance than the first, the reflection coefficient is positive. This makes sense because when a sound pressure wave comes in contact with a rigid boundary, sound echoing occurs. When r2 is much greater than r1, this defines the rigid boundary condition. When r1 is much greater than r2, the boundary is termed “pressure release” and there is an 180 degree phase shift between incident and reflected wave.

Having an incident angle other than 90 degrees complicates the solution process a bit. This is termed “oblique incidence”. The pressure equations become a bit more complex.

The angles made by an obliquely incident wave are shown below.

Applying the same continuity of pressure boundary condition from before leads to Snell’s law.

The critical angle can be defined as

This angle, as well as the comparison of speeds of the materials determines the bending of the refracted wave.