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

176 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

Figure 7.4 Local Cartesian coordinates (x1, y1, z1) with axes x1 and z1 placed in wedge face 1 (ϕ = 0), where the axis y1 is normal to this face (yˆ1 = nˆ). Similar Cartesian coordinates (x2, y2, z2) are also introduced with axes x2 and z2 in face 2 (ϕ = α), and with yˆ2 = nˆ .

(

r2 = (x2 − ξ2)2 + h22 determine the distances between the observation and integration points (Fig. 7.4).

For the Green function of the free space, we use Equation (6.616-3) in Gradshteyn and Ryzhik (1994):

two branch cuts (−∞ ≤ Re( p) ≤ −k and k ≤ Re( p) ≤ ∞) in the complex plane ( p). The integration contour in Equation (7.21) is located on the upper side of the left cut and skirts over the branch point p = −k, then it follows along the real axis to the right cut and skirts under the branch point p = k. After that, the integration contour follows along the lower side of the right cut.

The normal derivatives of the Green function are defined by

Note that the differential operator ( ) performs the differentiation with respect to the coordinates of the integration (observation) point. The quantities y1,2 are expressed in terms of the spherical coordinates (R, ϑ , ϕ) as

and the quantities h1,2 are determined in terms of the coordinates (R, β1, β2) as

TEAM LinG

7.3 Triple Integrals for Elementary Edge Waves 177

After substituting the sources js,h(1) and the above expressions into the field formulas (7.19) and (7.20), one obtains

×e−ikξ1 sin2 γ0 cos η [w−1(η + ϕ0) − w−1(η − ϕ0)] sin η dη,

D

It may be seen that Equations (7.29) and (7.30) for the fields generated by strip 2 can be obtained from Equations (7.25) and (7.26) for the fields from strip 1 by the formal replacements

x1 −→ x2, h1 −→ h2, β1 −→ β2 · ϕ0 −→ α − ϕ0, ϕ −→ α − ϕ. (7.31)

For this reason, further calculations are carried out only for the fields du1(1), dv1(1), and the final expressions for the fields du2(1), dv2(1) will be obtained using the relationships (7.31).

The next section deals with the analytical work on the integrals Iu and Iv .

TEAM LinG

180 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

Because the quantity p2 is real and the argument of the logarithm in Equations (7.42)

and (7.43) is always positive, we take the regular arithmetic branch for this logarithm, where ln(0) = −∞, ln(1) = 0, and ln(∞) = +∞.

Now one can trace the position of the pole σ in the complex plane (η) as the function of the variable p in Equations (7.35) and (7.36). When this variable changes from p = −∞ to p = k cos(2γ0), the pole η3 = σ runs in the complex plane (η) along the imaginary axis Im(η) from η = +i∞ to η = 0. When the point p moves from p = k cos(2γ0) to p = k, the pole η3 = σ runs along the real axis Re(η) from η = 0 to η = π . When the point p moves from p = k to p = +∞, the pole η3 = σ runs down along the vertical line from η = π to η = π − i∞. The location of the pole η3 = σ in the complex plane (η) is shown in Figure 7.5 by the thick solid line. The dashed line shows the location of the pole η4 = −σ .

To calculate the integrals over the contour D, we connect its branches with the additional contours F+ and F− (Fig. 7.5), where Im(η) = A and Im(η) = −A, respectively. First, we place these contours at a large finite distance from the real axis (A 1) and apply the Cauchy residue theorem to the integrals over the closed contour C = D + F+ + F−. Then the results of this theorem are extended to the case when the contours F+ and F− are shifted to the infinite distance (A → ∞). This procedure is realized in the following for the integrals

The basic details of this procedure are the same for both integrals. That is why we demonstrate these details only for the integral Ju and then bring the final result for Jv .

Figure 7.5 Integration contours in the complex plane (η).

TEAM LinG