Audrey V. Osipov
Radiophys. Dept., Inst. of Phys., St. Petersburg State Univ., Ulianovskaya 1-1, 198904 St. Petersburg, Russia
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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