**Audrey V. Osipov
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*Radiophys. Dept., Inst. of Phys., St. Petersburg State Univ., Ulianovskaya
1-1, 198904 St. Petersburg, Russia
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Boundary conditions containing spatial derivatives of high orders,
A((cursive beta)[sub x],(cursive beta)[sub y])((cursive beta)U/(cursive
beta)z)+ikB((cursive beta)[sub x],(cursive beta)[sub y])U|[sub z=0]=0, where z
is the coordinate orthogonal to a plane surface of a layered medium, A, B 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(greater than or equal to)0), k is the wave number, are widely used in the
modern acoustics 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 [R. G. Mindl, J. Appl. Mech. 18, 31--8 (1951); G. I. Petrashen and L.
A. Molotkov, Vestn. Leningr. Univ. Fiz. & Khim. (USSR) (4), 137--56 (1958), in
Russian; S. A. Rybak and B. D. Tartakovskii, Akustich. Zh. 9, 66--71 (1963), in
Russian; P. S. Dubbelday and A. J. Rudgers, J. Acoust. Soc. Am. 70, 603--14
(1981); B. L. Wooley, J. Acoust. Soc. Am. 70, 771--81 (1981); J. Acoust. Soc.
Am. 72, 859--69 (1982)] which automatically leads to the theory of slim layers.
Some other approaches have been also presented in the literature but the most
of them are associated with significant simplifications in physical models of
the processes (the classical theory of elastic plates is a good example).
Moreover, many theories treat symmetrical and antisymmetrical processes
separately, despite the fact that there are a great deal of situations when
these processes are not separable (e.g., structures with nonsymmetrical
stratification). 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 to the classical one of
approximation of certain entire functions by polynomials in the complex plane.
Choosing an appropriate method of the approximation it is possible to construct
such boundary conditions that are specially adjusted to model the most
important properties of 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. The formulation of the matrix boundary
conditions is also discussed in this paper. To illustrate the basis principles
of the theory the boundary conditions for a thick elastic homogeneous slab
surrounded by an acoustic medium are considered in detail.

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