Spectral and Scattering Theory for Acoustic Operators in Non-Homogeneous Fluids. Continuous and Discrete Models

Abstract

In this paper we formulate the results on the spectral property and scattering theory for the linearized Navier-Stokes system, discribing the process of sound propagation in the non-homogeneous halfspace $\mathbb{R}_ + ^3 = \left\{ {x = \left( {x_1 ,x_2 ,x_3 } \right):x_3 > 0} \right\}$

$$ \begin{array}{*{20}c} {\frac{{\partial \vec \upsilon }} {{\partial t}} = - \rho ^{ - 1} \nabla p + \rho ^{ - 1} \vec f,} \\ {\frac{{\partial p}} {{\partial t}} = - \rho C^2 \nabla \cdot\vec \upsilon + c^2 \rho a.{\text{ }}} \\ \end{array}$$

(1)

Here ∇ denotes the gradient operator and ∇· denotes the divergence operator, ρ denotes the fluid density, c is the sound speed, v⃗ = (v ₁, v ₂, v ₃) is the vector of vibrating volocity, p is the acoustic pressure, f⃗ = (f ₁, f ₂, f ₃) and a are solid densities corresponding the source of force and solid velocity respectively.