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

48 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

This expression represents the slightly modified first term in the Pauli asymptotic expansion. The next term is of order (kr)−1/2 near the boundaries ϕ = π ± ϕ0 and it is of order (kr)−3/2 away from them.

The upper limit of the Fresnel integral in Equation (2.75) should be read as sgn(cos ψ/2)∞. It always equals infinity but changes its sign when the observation point intersects the geometrical optics boundaries (ψ = ϕ ± ϕ0 = π ). Here, the function v(kr, ψ ) undergoes the discontinuity and in this way it ensures the continuity of the function u(kr, ψ ) and therefore the continuity of the total field. Indeed, by using the formula

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50 Chapter 2 Wedge Diffraction: Exact Solution and Asymptotics

which can be converted to the Fresnel integral. To do this, let us separate the contour D0 (see Fig. 2.6) into two parts at the point ζ = 0. Summation of the integrals over these parts of the integration contour leads to the expression

With the help of Equation (2.73) this expression transforms to Equation (2.82). The latter, together with Equations (2.39) to (2.41) and (2.52) to (2.54) provides the exact solution to the half-plane diffraction problem.

Thus, the Pauli asymptotics (2.75) possesses valuable properties. It is simple. It provides the exact solution to the half-plane diffraction problem. It describes both the transverse diffusion of the wave field near the geometrical optics boundaries and the diffracted rays away from these boundaries. However, it is not free from certain drawbacks. These drawbacks are as follows:

•The total field us,h determined with the Pauli asymptotics (2.75) exactly satisfies the boundary conditions (2.2) and (2.3) on the face ϕ = 0. However,

on face√ ϕ = α, these boundary conditions are satisfied only asymptotically, |cos ψ/2| 1 and the Pauli asymptotics converts to the Sommerfeld

expression (2.60).

• The Pauli asymptotics (2.75) provides correct values for the wave field in the direction of the shadow boundary (ϕ = π + ϕ0) and in the direction ϕ = π − ϕ0 of the plane wave reflected from the face ϕ = 0. However, it fails at the direction ϕ = 2α − π − ϕ0 of the plane wave reflected from the face ϕ = α (Fig. 2.8). It predicts a wrong infinite value for the field in this direction.

One can suggest the following remedy to diminish these drawbacks, to some extent. The asymptotics (2.75) should be used only in the region 0 ≤ ϕ ≤ α/2. In order to calculate the field in the rest of the region α/2 ≤ ϕ ≤ α, it is necessary to introduce new polar coordinates with the angle ϕ measured from the face ϕ = α and then to apply the expression (2.75) in the region 0 < ϕ ≤ α/2. In this way, one can obtain correct values for the field at the boundary of the plane wave reflected from

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2.6 Uniform Asymptotics: Extension of the Pauli Technique 51

the face ϕ = α and satisfy the boundary conditions on this face, but at the expense of the field discontinuity in the direction ϕ = α/2.

The mentioned discontinuity of the field at ϕ = α/2 is manifestation of the fact that asymptotics (2.75) does not satisfy the fundamental physical principle. It is not invariant with respect to choice of the coordinate system. Indeed, if we choose the polar coordinates ϕ and ϕ0 measured from the face ϕ = α, the Pauli asymptotics leads to the relationships

The last inequality indicates that, strictly speaking, the Pauli asymptotics does not satisfy the invariance principle. However, it satisfies this principle approximately, when it transforms into the Sommerfeld ray asymptotics.

In the next section, we derive new asymptotics applicable in all scattering directions (0 < ϕ < α).

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 the concave wedges/horns (π/2 < α < π ), but only for certain directions of the incident wave. However, the theory developed below can be easily extended for any narrow horns (0 < α < π/2) with multiple reflections.

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 aside 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

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