Reflection from and transmission through a periodically stratified ideal fluid half-space
Michael Schoenberg, Pabitra N. Sen · The Journal of the Acoustical Society of America · 1982
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 for 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. The effective acoustic medium that models transmission as well as reflection at low frequencies has a bulk modulus, Keff, given by 〈K−1〉−1, where the brackets denote a volume weighted average. However, for the inertial density appearing in the equations of motion, the effective medium has an anisotropic density tensor 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 slowness surface for such an anisotropic acoustic medium is the usual ellipse with a speed that is always greater parallel to the layering than normal to the layering.