Properties of a periodically stratified acoustic half-space and its relation to a Biot fluid

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Abstract
The problem of the reflection of acoustic waves from a periodically layered acoustic half‐space is solved exactly at all frequencies. Pass and stop bands and the associated complex slowness surfaces of propagation through the periodic medium are found. In the low‐frequency limit, when the wavelength normal to the layering is much greater than one period, it is shown that there always exists a homogeneous ideal fluid with the same reflection properties as the layered medium. However, the effective acoustic medium that models transmission as well as reflection at low frequencies has a bulk modulusK eff given by 〈K − 1〉− 1 where the brackets denote a volume weighted average, and, for the inertial density appearing in the equations of motion, the effective medium has a t r a n s v e r s e i s o t r o p i c d e n s i t y t e n s o r with ρ∥, the density component parallel to the layering, given by 〈1/ρ〉− 1, and ρ⊥, the density component perpendicular to the layering given by 〈ρ〉. The range of wavespeeds speeds as a function of angle is shown to parallel the range of speeds in a Biot fluid as a function of the Biot coupling parameter. Normal propagation corresponds to full locking of the two phases in a Biot material and parallel propagation corresponds to the fully uncoupled case. The relation between the angle of propagation θ measured from the normal to the layering, and the corresponding Biot coupling parameter α is that α−1 is proportional to cot2 θ.

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