The sound absorption coefficient is computed as

where is the sound pressure reflection coefficient, given by

Here, is the acoustic surface impedance, is the density of air, is the speed of sound in air, and is the characteristic impedance of air. The acoustic surface impedance is a quantity that depends on the sound frequency , and therefore, the sound pressure reflection coefficient and the sound absorption coefficient are quantities that also depend on the sound frequency . The objective is to calculate the relationship between the sound absorption coefficient and frequency for a given porous medium represented as a 3D pore-scale structure in GeoDict. Refer to Table 2 for a summary of the relevant quantities and their physical units.

The relationship between the acoustic surface impedance and the properties of the sound-absorbing specimen (porous medium) in the Kundt tube is dependent on whether the porous medium is a single homogeneous layer or if it is composed of multiple layers that are parallel to the sound-reflecting wall.

Single Layer

For a single layer, the acoustic surface impedance can be derived by

where is the thickness of the medium, is the characteristic impedance, and is the complex wave number. Those three parameters , , describe the acoustic properties of the porous layer.

Multilayer

Consider stacked porous layers with acoustic properties described by sets of parameters for , where the layer adjacent to the wall has the index 1. Then, the acoustic surface impedance for the stack consisting of layer 1 to is given by the impedance transmission theorem [Allard and Atalla, Eq 2.16]:

i.e., the acoustic surface impedance for the multilayer is

The absorption coefficient and the reflection coefficient for the multilayer are then computed by (35) and (36) , respectively.