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

Problems 69

Consider these two examples:

(a) α = 270◦, ϕ0 = 45◦, 0 ≤ ϕ ≤ α, (b) α = 360◦, ϕ0 = 45◦, 0 ≤ ϕ ≤ α.

Prove analytically that F(π ± ϕ0, ϕ0, α) = 1. Formulate your conclusion regarding δf (0).

3.2 Analyze the accuracy of the PO approximation. Compute and plot the relative error

δg(0)(ϕ, ϕ0, α) = |G(ϕ, ϕ0, α) − 1| · 100%,

where

G(ϕ, ϕ0, α) = g(0)(ϕ, ϕ0)/g(ϕ, ϕ0, α). Consider these two examples:

(a) α = 270◦, ϕ0 = 45◦, 0 ≤ ϕ ≤ α, (b) α = 360◦, ϕ0 = 45◦, 0 ≤ ϕ ≤ α.

Prove analytically that G(π ± ϕ0, ϕ0, α) = 1. Formulate your conclusion regarding δg(0).

3.3The Sommerfeld function f (ϕ, ϕ0, α) satisfies the reciprocity principle, but its PO approximation f (0)(ϕ, ϕ0) does not. Analyze the PO deviations from the reciprocity principle. Compute the relative level of these deviations,

df (ϕ, ϕ0) = |F(ϕ, ϕ0, α) − F(ϕ0, ϕ, α)| · 100%,

where

F(ϕ, ϕ0, α) = f (0)(ϕ, ϕ0)/f (ϕ, ϕ0, α).

Investigate the case with α = 350◦, set ϕ = ϕ0 = 70◦. Prepare a (6 × 6) square table with numerical data for the deviations df . Formulate your conclusion.

3.4The Sommerfeld function g(ϕ, ϕ0, α) satisfies the reciprocity principle, but its PO approximation g(0)(ϕ, ϕ0) does not. Analyze the PO deviations from the reciprocity principle. Compute the relative level of these deviations,

dg(ϕ, ϕ0) = |G(ϕ, ϕ0, α) − G(ϕ0, ϕ, α)| · 100%,

where

G(ϕ, ϕ0, α) = g(0)(ϕ, ϕ0, α)/g(ϕ, ϕ0, α).

Investigate the case with α = 350◦, set ϕ = ϕ0 = 70◦. Prepare a (6 × 6) square table with numerical data for the deviations dg. Formulate your conclusion.

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72 Chapter 4 Radiation by the Nonuniform Component of Surface Sources

By summation with the incident wave (3.1), these equations determine the PO part of the total field:

where us,h(0)d is the diffracted part of the field, which is described by functions vs,h± . The geometrical optics part us,hgo of the PO field is the same quantity as that in Equation (4.1).

The field (4.1) is generated by total surface source js,h = js,h(0) + js,h(1), consisting of the uniform and nonuniform components, and the PO field (4.2) is radiated only by the uniform component js,h(0). Therefore, the field created by the nonuniform component is the difference

In the case 0 < ϕ0 < α − π , when only one face (ϕ = 0) is illuminated, this field is determined by

In the case α − π < ϕ0 < π , when both faces are illuminated, the field us,h(1) is determined as

us(1)/u0 = v(kr, ϕ − ϕ0) − v(kr, ϕ + ϕ0)

(4.7)

The function v(kr, ψ ) is defined in Equation (2.51) and the functions vs,h± (kr, ϕ, ϕ0) in Equations (3.39) and (3.40) by the integrals in the complex plane

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74 Chapter 4 Radiation by the Nonuniform Component of Surface Sources

Thus, the field generated by the nonuniform component js,h(1) represents by itself the cylindrical wave diverging from the edge of the wedge. This form of the field is a consequence of the fact that the nonuniform component js,h(1) concentrates near the edge. Just for this reason the quantity js,h(1) is sometimes called the fringe component. Approximations (4.12) and (4.13) reveal a ray structure of this part of the diffracted field and because of that they can be termed ray asymptotics.

The directivity patterns of the field (4.12) and (4.13) possess a wonderful property. In contrast to the functions f , g, f (0), g(0), which are singular at the geometrical optics boundaries, the functions f (1) and g(1) are finite there. It turns out that the singularities of functions f and g are totally cancelled by the singularities of functions f (0) and g(0), respectively. The following equations determine the finite values of functions f (1) and g(1) for these special directions.

For the direction ϕ = π − ϕ0 (which is the boundary of the plane wave reflected from the face ϕ = 0, Fig. 2.7), the functions f (1) and g(1) have the values

when 0 < ϕ0 < π . The direction ϕ = π + ϕ0 under the condition α − π < ϕ0 < π is inside the wedge and is not of interest.

In the case α − π < ϕ0 < π , when both faces of the wedge are illuminated (Fig. 2.8), the functions f (1) and g(1) have the following values at the direction ϕ = 2α − π − ϕ0 (which is the boundary of the plane waves reflected from the

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