Fundamentals of the Physical Theory of Diffraction, 2nd Edition

2.6 Uniform Asymptotics: Extension of the Pauli Technique

Here we derive asymptotic expressions under the condition that the incident wave does not undergo double and higher-order multiple reflections at faces of the wedge. This condition is always realized for convex wedges (π < α ⩽ 2π) and also for concave wedges and horns (π/2 < α < π) but only for certain directions of the incident wave. However, the theory developed below can easily be extended for any narrow horns (0 < α < π/2) with multiple reflections.

Now we return to Equation (2.66) and we observe that two poles,

(2.89)

can approach the saddle point s = 0 when ψ = φ ± φ₀ → π or ψ = φ + φ₀ → 2α − π. The pole s₁ approaches the saddle point when the direction of observation φ tends to the shadow boundary φ = π + φ₀ or to the boundary φ = π − φ₀ of the wave reflected from the face φ = 0 (Fig. 2.7). The pole s₂ approaches the saddle point when the direction φ tends to the boundary φ = 2α − π − φ₀ of the wave reflected from the face φ = α (Fig. 2.8). All other poles in (2.66) can be ignored, as they are outside the integration contour and never reach the saddle point in the absence of multiple reflections.

Taking these observations into account, we multiply and divide the integrand in (2.66) by the factor

(2.90)

and obtain ...