On formulation of the high-order boundary conditions for stratified media

Abstract

Boundary conditions containing spatial derivatives of high orders, Â(∂x,∂y)(∂U/∂z)+ikB̂(∂x, ∂y)U‖z=0=0, where z is the coordinate orthogonal to a plane surface of a layered medium, Â, B̂ are the differential operators with respect to the tangent coordinates, U is a scalar function representing acoustic pressure in the region above the surface (z≥0), k is the wave number, are widely used to describe the complicated process of interaction of sound waves with thin elastic plates or stratified slabs. The high-order boundary conditions are usually obtained from the equations of motion of the elastic structures by expanding the wave fields into power series according to the slab thickness. This paper describes a new approach to formulation of the high-order boundary conditions for isotropic stratified media. Via Fourier transform the problem is rigorously reduced the approximation of certain entire functions by polynomials in the complex plane. Choosing an appropriate method of approximation such boundary conditions are constructed that are specially adjusted to model the structure of interest. The technique proposed is regular; it does not require the separation of symmetrical and antisymmetrical processes, and it includes the slim layer theories as a special case. For structures placed between two fluid half-spaces, the scalar boundary conditions must be replaced by matrix ones combining the values of U and its derivatives on the opposite sides of the slab.