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7.10 Some References Related to Elementary Edge Waves 209

7.10 SOME REFERENCES RELATED TO ELEMENTARY EDGE WAVES

The investigation of EEWs has a long history. In Kirchhoff’s approach, the EEWs were first discovered by Maggi (Maggi, 1888; Bakker and Copson, 1950). The same result was rediscovered by Rubinowicz (1917). A similar approach to electromagnetic EEWs was developed by Kottler (1923).

Attempts to define EEWs more strictly (on the basis of the Sommerfeld (1896, 1935) exact solution of the wedge diffraction problem) were first undertaken by Bateman (1955) and Rubinowicz (1965). However, their expressions for EEWs satisfy the Dirichlet or Neuman boundary conditions everywhere at the faces of the canonical tangent wedge. For this reason, these EEWs predict incorrect values for the diffracted field at those parts of the virtual tangent wedge that are extended outside the real scattering object (as shown in Fig. 7.10 by dotted lines). Infact, according to this definition of EEWs, the infinite plane areas of free space (outside the scattering object) formally become perfectly reflecting.

The same drawback exists in another theory of EEWs suggested in by Tiberio et al. (1994, 1995, 2004). In PTD a similar shortcoming occurs only at the extensions of infinitely narrow elementary strips (Fig. 7.3). As PTD is a source-based theory, this shortcoming can be completely removed by the truncation of elementary strips (Johansen, 1996).

The directivity patterns of electromagnetic EEWs can be interpreted as equivalent edge currents (EECs). The EECs introduced in the work of Knott and Senior (1973) and Knott (1985) are based on GTD and are valid only for the directions of the diffracted rays. The EECs based on PTD are applicable for arbitrary scattering directions (Michaeli, 1986, 1987; Breinbjerg, 1992; Johansen, 1996). Notice that the untruncated EECs developed by Michaeli (1986) are in complete agreement with the EEWs derived in Section 7.8. However, his truncated EECs (Michaeli, 1987) are not free from some shortcomings, which were overcome by Johansen (1996). The paper by Johansen (1996) also contains additional references related to the EEC concept.

Another interpretation of the directivity pattern of EEWs is the so-called incremental length diffraction coefficient (ILDC). The term ILDC was introduced by Mitzner (1974), who determined the ILDCs on the basis of PTD. The ILDC concept was further developed in the work of Tiberio et al. (2004).

Figure 7.10 Perfectly reflecting solid prism of finite size (solid and dashed lines) and infinite faces of the virtual tangent wedge (dotted lines).

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210 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

Notice also the asymptotic theory for plane screens (Wolf, 1967), which is similar to PTD and the method of matched asymptotic expansions (Tran Van Nhieu, 1995, 1996), which lead to the PTD ray asymptotics.

PROBLEMS

7.1Start with Equations (7.3) and (7.4) and derive Equations (7.12) and (7.13) for the scattering surface sources js and jh on strip 1.

7.2Start with Equations (7.3) and (7.4) and derive Equations (7.16) and (7.17) for the scattering surface sources js and jh on strip 2.

7.3Start with Equations (7.19) and (7.20), use Equations (7.12) and (7.13), and verify the field expressions (7.25) and (7.26).

7.4Start with Equations (7.19) and (7.20), use Equations (7.16) and (7.17), and verify the field expressions (7.29) and (7.30).

7.5Start with Equation (7.37) for the pole σ ( p), verify its form (7.42), (7.43), and retrace

the trajectory of this pole in the complex plane (η), when the argument p changes from

−∞ to +∞.

7.6Apply the Cauchy residue theorem to the integral (7.35) and verify its transformation into the form (7.64).

7.7Apply the Cauchy residue theorem to the integral (7.36) and verify its transformation into the form (7.65).

7.8Apply the stationary phase technique to the integrals (7.64) and (7.65) and prove the asymptotics (7.89), (7.90) for EEWs.

7.9Verify that the directivity patterns Fs,h(1) of EEWs are always real functions, although their arguments σ1,2 can be complex quantities.

7.10Explain why the function Fs(1) equals zero for the grazing incidence (ϕ0 = 0 or ϕ0 = α).

7.11Functions Ut (σ1, ϕ0), Vt (σ1, ϕ0), as well as functions U0(β1, ϕ0), V0(β1, ϕ0), are singular at the point σ1 = ϕ0. Show that their differences, functions U = Ut − U0 and V = Vt − V0 remain finite there. Prove Equations (7.106) and (7.107).

7.12Show that for the scattering directions ϑ = π − γ0, 0 ≤ ϕ ≤ α (which belong to the diffraction cone, outside the wedge), functions Fs(1), Fh(1) transform into functions f (1), g(1), respectively. Prove Equations (7.115).

Ezinc = E0ze−ikz cos γ0 e− ikr sin γ0 cos(ϕ−ϕ0),

Hzinc = H0ze−ikz cos γ0 e−ikr sin γ0 cos(ϕ−ϕ0)

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Problems 211

generates on elementary strip 1 (Fig. 7.3) the nonuniform currents

• Represent the field excited by the incident wave around the wedge in the form

•Use the Maxwell equations and express components Er,ϕ (x, y), Hr,ϕ as functions of components Ez(x, y), Hz(x, y). See Equation (5.4) in Ufimtsev (1962).

•According to Chapter 4,

Ez = E0ze−ikz cos γ0 [v(k1r, ϕ − ϕ0) − v(k1r, ϕ + ϕ0)],

Hz = H0ze−ikz cos γ0 [v(k1r, ϕ − ϕ0) + v(k1r, ϕ + ϕ0)].

(1)

7.15 Use the same manipulations as those in Problem 7.14 and find the current j on elementary strip 2 (Fig. 7.3). Show that its components are determined by the equations:

j(1) = − 1 H e−ikz cos γ0 x 2α 0z

×e−ik1x cos η [w−1(η + α − ϕ0) + w−1(η − α + ϕ0)]dη,

The x -axis is shown in Figure 7.3.

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212 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

7.16Find the vector-potential dA(x1,z) generated by the nonuniform current jx(1,z) induced on elementary strip 1 (Fig. 7.3). Follow the procedure shown below:

•Start with Equation (1.89). Substitute there the current jx(1,z) found in Problem 7.14.

•Use Equation (7.21) for the Green function.

•Calculate the integral over variable ξ1 (along the strip) in closed form.

•Apply the Cauchy theorem to the integral over variable η.

•Apply the asymptotic procedure (7.69) to integrals of the type of Equation (7.68).

•Represent the vector-potential in the form of a spherical wave diverging from the stationary point.

When you have this result, use Equations (1.92) and (1.93) and obtain the asymptotic expression for the wave generated by elementary strip 1. Having this, use the replacements

H0z → −H0z, β1 → β2, σ1 → σ2, ϕ0 → α − ϕ0, ϕ → α − ϕ,

and obtain the wave generated by strip 2. The sum of these waves is the EEW shown in Equation (7.130).

7.17Section 7.9.1 develops the asymptotics of acoustic EEWs free from the grazing sin-

gularity. Show that for the directions of the diffraction cone (ϑ = π − γ0), functions

Fh,s(1) take the form of Equations (7.167) and (7.168). Then prove Equations (7.173) and (7.174) for the specular direction ϕ = π − ϕ0 and verify Equations (7.177) and (7.178) for the grazing incidence (ϕ0 = π ) and the grazing scattering (ϕ = 0).

7.18Section 7.9.2 develops the asymptotics of electromagnetic EEWs free from the grazing singularity. Show that for the directions of the diffraction cone (ϑ = π − γ0), functions

(7.193) and (7.194) between acoustic and electromagnetic waves.

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