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

222 Chapter 8 Ray and Caustics Asymptotics for Edge Diffracted Waves

For the observation points behind the caustic, no stationary points exist on the edge and therefore no diffracted rays come here. This is a shadow region for diffracted rays. Below we develop the asymptotic approximation for the field in the illuminated region, in front of the caustic, including the points on the caustic itself.

We proceed with the general integral expression for the edge diffracted field of the Equations (8.3), (8.4) type,

In front of the caustic, the integrand in Equation (8.44) has two stationary points ζ1 and ζ2. As the observation point P approaches the caustic, the stationary points move toward each other, and merge when P reaches the caustic. In this case, (ζ1,2) → 0 and the ray asymptotics (8.41) become invalid. One can define a smooth caustic as a surface where both (ζ ) = 0 and (ζ ) = 0.

A main contribution to the integral (8.44) is provided by the stationary points. Real edges usually have ends. To extract the caustic effect in a pure form, we extend the integration limits in Equation (8.44) to infinity (−∞ ≤ ζ ≤ ∞) and in this way remove the contributions of the ends to the field, which are usually less in magnitude than those of Equation (8.41).

Uniform asymptotics valid in the whole illuminated region, including the caustic, can be found by using the stationary-phase method, extended for the case of two merging stationary points (Chester, et al. 1957). In the following, we provide the basic details of this technique and derive the caustic asymptotics:

•As the first and second derivatives of the phase function (ζ ) equal zero at the caustic, it is expedient to represent this function as a cubic polynomial

under the condition τ 2 = μ = 0 for the observation point on the caustic.

•The function τ (ζ ) shapes a three-sheeted Riemann surface. The regular branch of this function is selected by setting

The quantities τ (ζ1,2) are the stationary points of the function [ζ (τ )]. Parameters μ and ψ are found from Equation (8.45), setting ζ = ζ1 and ζ = ζ2:

•According to Equation (8.41) and in agreement with Figure 8.3, the function(ζ ) possesses the following properties:

◦(ζ1) ≥ 0 and (ζ2) ≤ 0. This means that (ζ2) ≥ (ζ1) and μ ≥ 0,

μ1/2 ≥ 0, μ3/2 ≥ 0.

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(8.55)

•The further substitution of Equation (8.54) into (8.51) leads to the asymptotic expression

u k−1/3eikψ [ pAi(−k2/3μ) − ik−1/3qAi (−k2/3μ)], with k → ∞, (8.56)

The asymptotic approximation (8.56) is uniformly valid in the whole illuminated region, including the caustic. In particular, on the caustic itself,

where, according to Equation (10.4.4) in (Abramowitz and Stegun, 1972),

In order to specify the final asymptotics (8.56) for the fields us,h(1), we obtain the explicit expressions for coefficients p and q:

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where the quantity μ is defined in Equation (8.47). After replacement of functions f (1), g(1) by f , g (or by f (0), g(0)), the expressions (8.62) and (8.63)

determine the coefficients p(s,ht) , qs,h(t) (or p(s,h0), qs,h(0)) related to the fields us,h(t) (or to us,h(0)).

For large real arguments (t 1), the Airy function and its derivative are determined by the asymptotic expressions (Abramowitz and Stegun, 1972):

(k2/3μ 1), the general asymptotics (8.56) transform into the ray asymptotics (8.41).

8.2.2Electromagnetic Waves

For electromagnetic waves the caustic asymptotics are derived in the same way (Ufimtsev, 1991) and can be written as

Here,

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The subscript m = 1, 2 indicates that a quantity with this subscript relates to the stationary point ζ1 or ζ2 at the edge.

(1) (1)

Formulas like Equations (8.67) and (8.68) define the vectors ph and qh . It is

(t) (t) (0) (0)

(8.62), (8.63) determine the coefficients pe,h, qe,h (or pe,h , qe,h ) related to the fields

(t) (t) (0) (0)

E , H (or to E , H ).

(1)

(1) (1) (1) (1)

The derived asymptotics for the field us,h , E , H (with functions f , g )

(t) (t)

have an important advantage compared to the asymptotics for the total field us,h, E ,

(t)

H . They remain finite at the boundaries of ordinary incident and reflected rays

(ϕ = π ± ϕ0, ϕ = 2α − π − ϕ0), where functions f , g become singular.

It follows from Equations (8.56) and (8.66) that the structure of the caustic field is the same for acoustic and electromagnetic waves. The difference is only in the coefficients p and q, which are scalar quantities for acoustic waves and vectors for electromagnetic waves.

Notice also that asymptotics (8.56) and (8.66) are valid for the field calculation only in the illuminated region, in front of the caustic. When the observation point moves across the caustic into the shadow region, the diffracted field continuously changes and exponentially attenuates, because the elementary edge waves asymptotically cancel each other there. We do not consider this topic here. Details regarding the wave field in the vicinity of arbitrary caustics can be found in the review paper by Kravtsov and Orlov (1983).

PROBLEMS

8.1Use the theory of EEWs from Section 7.5 and derive the edge wave scattered by an infinite straight edge of a wedge. The incident acoustic wave is given by

uinc = u0 e−ikz cos γ0 e−ikr sin γ0 cos(ϕ−ϕ0).

Consider the scattering by both a soft and a hard wedge. Integrate the EEWs over the whole edge. Apply the stationary-phase method and obtain the asymptotics (4.48) and (4.49).

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