The geometry of the problem is shown in Figure 13.1. A solid circular cylinder with flat bases is illuminated by the incident plane wave
(13.1)
Figure 13.1 Cross-section of the cylinder by the y0z-plane. The dots at 1, 2, and 3 are the stationary-phase points visible in the region π/2 < ϑ < π, φ = 3π/2.
The total length of the cylinder and its diameter are denoted as L = 2l and d = 2a, respectively. The scattered field is evaluated for the backscattering direction ϑ = π − γ, φ = 3π/2 in the far zone (R ≫ ka2, R ≫ kl2).
13.1.1 PO Approximation
According to Equation (1.37), the PO far fields backscattered by acoustically hard and soft objects differ from each other only in sign. Hence, it is sufficient to exhibit the PO calculations only for the case of scattering at a hard cylinder.
First we calculate the far field scattered by the left base/disk of the hard cylinder. In this case, use of Equation (1.37) leads to the expression
(13.2)
where ϑ = π − γ, , x′ = r′cos ψ, and y′ = r′sin ψ. In view of Equation (6.55), ...
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, x′ = r′cos ψ, and y′ = r′sin ψ. In view of Equation (6.55), ...