ABSTRACT
As shown in Chapter 2, a diffraction problem in acoustic media may be reduced to integral equations obtained from the Kirchhoff–Helmholtz integral formula. The Dirichlet problem is described by an equation of the rst kind and the Neumann problem by an equation of the second kind, if we speak about the direct BIE method:∫
ψ(y)G(y0, y) dSy = F1(y0), y0 ∈ S, (9.1)
p(y0) – 2 ∫ S
p(y) ∂G(y0, y) ∂ny
dSy = F2(y0), y0 ∈ S. (9.2)
Here ψ(y) = ∂p/∂ny, p is the diffracted pressure on the boundary surface S; F1 and F2 are functions expressed in terms of boundary functions. To be more specic, we assume that the surface is closed and sufciently smooth; the normal ny is directed outwards the surface S. Equations (9.1) and (9.2) describe the diffraction problem both in the 2D and 3D cases. In such cases we have, respectively
G(y0, y) = i4H (1) 0 (kr), G(y0, y) =
1 4pi
eikr
r , r = |y – y0|. (9.3)
Note that equations (9.1), (9.2) may be written out for full wave eld too. This changes only the form of the right›hand sides F1 and F2. As we could see from the previous study, equations (9.1), (9.2) are investigated in detail, and a number of analytical and numerical methods have been developed to solve these equations. The exact analytical solutions are known only for canonical regions (circle, sphere, ellipse, segment, etc.). In the high›frequency range (ka 1, a is a characteristic size of the S›surface) the short› wave asymptotic methods are efcient for regions of complex shape. As regards direct numerical methods, we note once again that these methods become inefcient for extremely high frequencies. The main disadvantage of numerical methods is their isolation from the physical essence of phenomena. Besides, it is known that numerical methods lose their stability for large values of ka (see also the Preface). Here we propose a method that uses a combination of analytical and numerical techniques. The analytical solution obtained from the Kirchhoff theory is used as an initial›step approximation for a certain iterative method. It is effective for ka 1 only, and for lower values of ka this is improved by the iterative method, and we show that each step of the proposed technique improves the chosen analytical solution.
