Wave propagation in alternating solid and viscous fluid layers: Size effects in attenuation and dispersion of fast and slow waves

Abstract

In a system of alternating parallel elastic solid layers (plates) and viscous fluid layers (channels), two types of waves propagate parallel to the layers in the low-frequency limit. They are related to fast and slow waves in Biot’s theory of wave propagation in porous media. The slownesses s0 fast and s0 slow obtained previously for the case of inviscid fluid layers between elastic plates are modified to reflect dispersion and attenuation due to fluid viscosity. Two important dimensionless dynamic parameters are L=φH/δ, where H is the period width, φ is the relative width of the fluid channel, i.e., the ‘‘porosity,’’ and δ=(2ν/ω)1/2 is the viscous skin depth, and Ω=ωφH/α f which is channel width times the fluid wave number. Here ν is the kinematic viscosity of the fluid, α f is the fluid sound velocity, and ω is the radial frequency. In the frequency range for which L is very large L≫1 and Ω is very small Ω≪1, the attenuation and dispersion of both slow and fast waves exhibit similar dependences on L. In particular, both wave velocities are decreased by an amount proportional to L−1(∼ω−1/2) and both have a quality factor Q proportional to L(∼ω1/2). To leading order in L−1 the wave slownesses have the form s2j=s20j[1+(1+i)Mj/L], j=fast or slow, where Mj are combinations of material and geometric parameters. The Q≡Re(s2)/Im(s2) factors are, for the fast wave, Qfast≊L/Mfast≊L/{[φρ f/(1−φ)ρ−ε]} and, for the slow wave, Qslow≊L/Mslow ≊L/(1+ε). ρ f and ρ are fluid and solid densities, respectively. ε is a usually small parameter given by 2(1−2γ)(ρ f/ρ)/ [α2pl/α2 f+(1−φ)ρ f/φρ−1], where γ is the square of the ratio of shear velocity to longitudinal velocity in the bulk solid and αpl is the velocity of long wavelength extensional plate waves.