Файл: Ufimtsev P. Fundamentals of the physical theory of diffraction (Wiley 2007)(348s) PEo .pdf

118 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

In view of the above comments, the scattered field (1.16–1.17) can be represented in this form:

0

The first integrals in the brackets of Equations (6.8) and (6.9) relate to the illuminated surface (z < 0), and the second integrals relate to the shadowed surface (z > 0). These integrals are calculated over the interval 0 ≤ ξ ≤ ξeff , as the nonuniform sources Js,h(1)(kξ ) decrease with increasing ξ and can be neglected at a certain distance ξ > ξeff from the edge. In the next sections we present the asymptotic estimates for the scattered field.

6.1.2Ray Asymptotics

First we investigate the field in the observation points, which are visible from all edge points (0 ≤ ψ ≤ 2π ). This happens for the two intervals of the observation directions: 0 ≤ ϑ ≤ and π − ω ≤ ϑ ≤ π . We assume that

TEAM LinG

6.1 Diffraction at a Canonical Conic Surface 119

and apply the stationary phase technique (Copson, 1965; Murray, 1984) to the integrals over the variable ψ . The stationary points ψ1,2 are found from the condition

In accordance with this asymptotic method, the function cos(ψ − ψ ) contained in± is approximated by

The initial integral over the variable ψ asymptotically equals the sum of two integrals calculated in the vicinity of each stationary point. The intervals of integration in these integrals are extended from −∞ to +∞. These standard manipulations lead to the asymptotic expression

TEAM LinG

120 Chapter 6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

Equation (6.17) allows one to reduce Equations (6.8) and (6.9) to single integrals over

the variable ξ . These integrals will contain the factors √a − ξ sin ω and √a − ξ sin ,

√

which can be approximated to a under the condition a ξeff .

Before we present the resulting expressions for Equations (6.8) and (6.9), we introduce the local polar coordinates ϕ1 and ϕ2 at the stationary points ψ1 and ψ2. In Figure 6.2 these points are denoted as 1 and 2. The local coordinates are shown in Figure 6.3 for point 1 and in Figures 6.4 and 6.5 for point 2.

Figure 6.3 Local polar coordinates at the stationary point 1 (ψst = ψ ).

Figure 6.4 Local coordinates at the stationary point 2 (ψst = π + ψ ) for the observation directions in the interval π − ω ≤ ϑ ≤ π .

Figure 6.5 Local coordinates at the stationary point 2 (ψst = π + ψ ) for the observation directions in the interval 0 ≤ ϑ ≤ .

TEAM LinG

6.1 Diffraction at a Canonical Conic Surface 121

Considering the above relationships between coordinates ϕ1, ϕ2, and ϑ , and utilizing Equation (6.17), one can obtain the following approximations for Equations (6.8) and (6.9):

Under the condition ka 1, the scattering sources Js,h(1) near the edge of the conic surface are asymptotically equivalent to those near the edge of the tangential wedge. Hence, the expressions inside the brackets in Equations (6.19) and (6.20) are also asymptotically equivalent to the similar expressions in Equations (4.29) and (4.30), which relate to the wedge diffraction problem. Utilizing this observation, one can rewrite Equations (6.19) and (6.20) as

These expressions can be interpreted as the ray asymptotics for the field us,h(1).

TEAM LinG