Fringe integral equations for the 2-D wedges with soft and hard boundaries. r Fringe Wave Integral Equation

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1 RESEARCH ARTICLE Key Point: An alternative approac baed on te integral euation derived directly for tefringecurrentipreented A MoM-baed algoritm i developed for direct modeling of fringe wave around oft and ard wedge for diffraction problem Correpondence to: G. Apaydin, g.apaydin@gmail.com Fringe integral euation for te 2-D wedge wit oft and ard boundarie Gokan Apaydin 1, Levent Sevgi 2, and Pyotr Ya. Ufimtev 3 1 Zirve Univerity, Gaziantep, Turkey, 2 Department of Electrical and Electronic Engineering, Okan Univerity, Itanbul, Turkey, 3 EM Conulting, Lo Angele, California, USA Abtract Novel, fringe wave integral euation tat account for te diffraction from nonpenetrable wedge wit bot oft and ard boundarie are derived. Metod of moment imulation of fringe wave generated by a plane wave tat excite te wedge i performed uing ti fringe wave integral euation. Te reult are compared wit te exact pyical teory of diffraction fringe wave. Citation: Apaydin, G., L. Sevgi, and P. Y. Ufimtev (216), Fringe integral euation for te 2-D wedge wit oft and ard boundarie, Radio Sci., 51, , doi:. Received 7 JUN 216 Accepted 8 SEP 216 Accepted article online 13 SEP 216 Publied online 26 SEP Introduction Fringe wave i a fundamental component of diffracted field widely ued in antenna and cattering problem. It wa introduced in Ufimtev [1957, 27, 213, 214] a te diffracted field generated by te nonuniform part of te ource-induced urface current. A number of related reference can be found in Ufimtev [23, 27, 29, 213, 214; Hacivelioglu et al., 211a, 213a, 213b, 213c] were it wa invetigated analytically and numerically. In addition to finite difference time domain and finite element metod, metod of moment (MoM) a alo been ued in diffraction modeling [ee, e.g., Harrington, 1993; Arva and Sevgi, 212; Apaydin and Sevgi, 212, 214a]. MoM-baed fringe wave a been recently calculated in Apaydin et al. [214b, 216]. In toe paper, MoM wa applied twice: firt, for te total current on te cattering object and ten, for te pyical optic (PO) approximation. Te fringe wave wa ten found a te difference between te total field and it PO approximation. Ti approac i called 2-Step MoM [Apaydin et al., 214b]. In ti paper, we preent an alternative approac baed on te integral euation derived directly for te fringe current. Te reult are compared wit te data found troug te exact analytical expreion [Ufimtev, 27, 214]. Advantage of ti approac i dicued. Furter analytical and numerical diffraction modeling and imulation tudie may be found in Apaydin et al. [216], Balani et al. [213], Hacivelioglu et al. [211b], Cakir et al. [212], Ulu and Sevgi [212], Ulu et al. [214], Ozgun and Sevgi [215], and Wu et al. [212, 215]. We conider ere te fringe wave induced on wedge wit oft and ard boundary condition (acoutic formulation). Ti formulation i euivalent to te electromagnetic (EM) formulation of te diffraction at a perfectly conducting wedge excited by te incident EM wave wit E z /tranvere magnetic (TM) or H z /tranvere electric (TE) polarization (Figure 1). Te total field u(r,φ) outide te wedge atifie te euation ( 2 r r r + 1 ) 2 r 2 φ + 2 k2 u = I r δ(r r )δ(φ φ ) (1) and te boundary condition (BC) on φ = and φ = α E z = for oft wedge (2) H z = for ard wedge under te line ource illumination at (r,φ ). Here k i te wave number, I i te line current amplitude, and δ i te Dirac delta function. Te ource become plane wave wen r and I. Plane wave illumination and exp( iωt) time dependence are aumed in ti paper American Geopyical Union. All Rigt Reerved. 2. Fringe Wave Integral Euation A novel fringe wave integral euation i derived in ti ection. Firt, oft wedge i conidered. Te incident plane wave i given a u inc = e ikr co(φ φ ) = e ik(x co φ +y in φ ). (3) APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 157

2 Figure 1. Geometry of te wedge problem. Te total field around te oft wedge i decribed a u(p) =u inc (P) i u(x, y ) H (1) (kr)dl (4) 4 L were r = (x x ) 2 +(y y ) 2, n i te outward normal, L 1,2 denote te wedge face ( L = L 1 + L 2 ), and u(x, y ) i te total field on te wedge. Placing P on L, one obtain te integral euation u inc (P) i 4 p.v. u(x, y ) H (1) (kr)dl = (5) L in accordance wit te oft boundary condition. Here p.v. mean te Caucy principal value of te integral. Ti i te integral euation of te firt order wit repect to te unknown uantity u = J and can be interpreted a a urface current. According to PTD [Ufimtev, 1957, 27, 214], ti current conit of te uniform ( J ()) and nonuniform ( J (1)) component: u = u() + u(1). (6) Here te firt term i determined by te PO, and te econd term i te diffracted current, o-called te fringe component. Euation (6) can be rewritten a were u = upo + ufr, (7) { u PO 2 u = inc on te illuminated face on te adowed face. (8) Depending on te angle of incidence of te plane wave, tree poible cenario occur in ti diffraction problem: Cae I: ( <φ <α π ) Only te face L 1 i illuminated, Cae II: ( α π<φ <π ) Bot face, L 1 and L 2, are illuminated, and Cae III: ( π<φ <α ) Only te face L 2 i illuminated. Firt, let u conider Cae I. On te face L 1, te urface current i defined by (6), wile on te adowed face L 2,by u = ufr. (9) APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1571

3 Now (5) take te form u inc (x, ) i 4 p.v. i 4 p.v. u fr (x, ) k x x dx i u fr (r co α, r in α) kr21 4 dr u PO (x, ) k x x dx = (1) were r 21 = (x x ) 2 + y 2 = r 2 + x 2 2r x co α and u inc (x, ) =e ikx co φ (11) u PO (x, ) = 2ikin φ e ikx co φ. (12) Te lat term in (1) i a particular cae of te integral I (x, y) = k in φ 2 wit y =+ and according to (3.1) and (3.37) in Ufimtev [27, 214] eual Subtitution of (13) into (1) reult in e ikx co φ H (k ) (1) (x x ) 2 + y 2 dx (13) I (x, +) = v () kx,,φ e ikx co φ = v () kx,,φ u inc (x, ). (14) u fr (x, ) p.v. k x x dx u fr (r co α, r in α) + kr21 dr = 4iv () kx,,φ wit P L 1. (15) Te function v () kr,φ,φ i defined in (3.39) of Ufimtev [214] a v () i in φ kr,φ,φ = 2π π 2 i π 2 +i e ikr co ζ dζ co φ + co (ζ + φ). (16) It can be tranformed into te fat convergent form (3.49) of Ufimtev [214]. Togeter wit te MATLAB code given in Table 4 ofhacivelioglu et al. [213a], it accurate numerical computation i poible. Now conider te ituation wen te obervation point P i on te lower face L 2. Euation (5) take te form e ikr co(α φ ) i 4 u fr (x, ) kr12 dx i 4 p.v. k in φ 2 e ikx co φ H (1) ( k u fr (r co α, r in α) k r r dr ) (x x ) 2 + y 2 dx = (17) were x = r co α, y = r in α, and r 12 = r 2 + x 2 2rx co α. (18) According to (3.1) and (3.37) of Ufimtev [214], te lat integral in (17) eual Taking ti obervation into account, one can rewrite (16) a u fr (x, ) kr12 dx + p.v. v () kr,α,φ e ikr co(α φ ). (19) = 4iv () u fr (r co α, r in α) k r r dr kr,α,φ wit P L 2. (2) APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1572

4 Expreion (15) and (2) provide te integral euation for te fringe current. In contrat to te integral euation (5) for te total field, it a a ignificant advantage: all it integrand in te left and contain only te decreaing fringe current. Te function v () kr,φ,φ i alo a decreaing uantity. Let u denote te left-and ide of (15) and (2) a LS P L 1 and LS P L2, repectively. One can ow tat te imilar procedure a above, being applied to te integral euation (5), lead to te following fringe euation in Cae II and III. In Cae II, wen bot face L 1 and L 2 are illuminated, LS ( P L 1 ) = 4iv () kx,,φ 4iv () kx,α,α φ (21) LS ( P L 2 ) = 4iv () kr,α,φ 4iv () kr,,α φ. (22) In Cae III, wen only te lower face L 2 i illuminated, LS P L 1 = 4iv () kx,α,α φ (23) LS P L 2 = 4iv () kr,,α φ. (24) Te fringe current ( u fr ) are ten ued to calculate te fringe field around te oft wedge a u fr (r,φ) = i u fr (x, y ) H (1) (kr) dl (25) 4 L wit r = (x x ) 2 + (y y ) 2. Note tat (5) i ued for te egment on bot face of te wedge, wile (15) and (2) are ued for top and bottom face egment, repectively (remember: a egment interact wit all oter no matter on wic face tey are). Now conider te ard wedge. Te total field around te ard wedge i decribed a u(p) =u inc (P)+ i u(x, y ) H (1) (kr)dl (26) 4 L and placing P on L, te integral euation of te econd order for ard wedge i obtained a u inc (P)+ i 4 p.v. u(x, y ) H (1) u(p) (kr)dl = L 2. (27) Here te unknown uantity u(x, y ) i available intead of u(x, y ) in (6) (8). Conidering te ame principle of oft wedge, wen only te upper face L 1 i illuminated (Cae I), te integral in (27) reduce to u fr (x, ) i 2 kx in α for te obervation point P L 1 and to u ( fr r co α, r in α ) dr kr21 = (28) 1 r 21 u fr (r co α, r in α) i 2 kr in α u ( fr x, ) dx kr12 = 2v () kr,α,φ 1 r 12 (29) for te obervation point P L 2. Te function v () kr,φ,φ i defined in (3.4) of Ufimtev [214] a v () 1 kr,φ,φ = 2πi π 2 i e ikr co ζ in (ζ + φ) dζ π 2 +i co φ + co (ζ + φ). (3) It can be tranformed into te fat convergent form (3.5) of Ufimtev [214]. Togeter wit te MATLAB code given in Table 4 of Hacivelioglu et al. [213a], it accurate numerical computation i poible. Let u denote te APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1573

5 Figure 2. A flow cart of te fringe MoM algoritm. left-and ide of (28) and (29) a LH P L 1 and LH P L2, repectively. In Cae II, wen bot ard face L1 and L 2 are illuminated, LH P L 1 = 2v () kx,α,α φ (31) LH P L 2 = 2v () kr,α,φ. (32) APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1574

6 Figure 3. MoM modeling of fringe wave for wedge wit repect to te illumination of face (wit imple pule bai function). Upper and lower face are illuminated only for Cae I and Cae III, repectively. Bot face are illuminated for Cae II. Figure 4. (top) Wedge urface current total and uniform (PO) current. (bottom) Only nonuniform (fringe) current (r = correpond to te tip of te wedge, TE/HBC, α = 27. Reproduced from Apaydin et al. [214b]. Figure 5. Fringe wave around te tip of te wedge, α = 24, r = 2λ, d = λ 2, f=3 MHz, TM/SBC ((top) Cae I, φ = 45 and (bottom) Cae II, φ = 12 ). APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1575

7 Figure 6. Fringe wave around te tip of te wedge, Cae III, r = 2λ, d = λ 2, f=3 MHz, TM/SBC ((left) α = 24, φ = 2 and (rigt) α = 27, φ = 225 ). In Cae III, wen only te lower ard face L 2 i illuminated, LH P L 1 = 2v () kx,α,α φ (33) LH ( P L 2 ) =. (34) Te fringe current ( u fr) are ten ued to calculate te fringe field around te ard wedge a u fr (r,φ) = i u ( fr x, y ) H (1) (kr) dl. (35) 4 L Te derived fringe integral euation can be olved by te claic MoM procedure. A related flow cart i given in Figure 2. An appropriate dicretization i own in Figure 3. To demontrate te typical current ditribution, we bring Figure 4 taken from Apaydin et al. [214b]. Tee current are generated by te ource wit freuency f = 3 MHz located at te line r = 2λ, φ = 7. Here te left and rigt ide belong to te lower and upper face of te wedge from te tip up to 1 wavelengt ditance. 3. Numerical Example and Comparion MoM-baed fringe wave are computed uing te novel fringe integral euation derived in ection 2. Te reult are compared wit te PTD for a nonpenetrable wedge wit variou cenario under different exterior angle and plane wave illumination angle. Te reult at 3 MHz (i.e., λ=1 m) are preented in Figure 5 7. Te receiver are located outide te wedge between φ = and φ = α around te tip wit 2λ radiu (kr = 4π and.5 angular reolution). Since infinite wedge face mut be truncated for te MoM procedure, tey are coen a 1λ, and a total of 4 egment are ued to atify λ/2 egment lengt. In Figure 5 7, fringe-mom and PTD reult for Cae I, II, and III are own. A oberved, very good agreement i obtained between fringe-mom and PTD model for oft boundary condition (SBC) and ard boundary condition (HBC) wedge. Numerical tet ow tat wedge face mut be truncated between 8λ and 12λ, and egment lengt wit λ/1 λ/3 would be enoug to approac an appropriate MoM accuracy. Te fringe integral euation ave a ignificant advantage compared to te MoM for total current. Tey do not contain te incident wave free term tat i not zero everywere on te edge urface up to infinite ditance from te edge. In ti way, ti term generate te MoM matrix witout decreaing term and make a numeric olution difficult for infinite wedge face. Elimination of te free incident term in te integral euation i a ignificant novelty of ti tudy. Fringe current concentrate in vicinity of te edge; terefore, te integration area in te fringe integral euation are alway finite and allow one to find te olution wit a good preciion. Notice alo a fine feature of te 2-tep MoM. It error for eac tep partially cancel eac oter in te difference for fringe current and enure an accurate olution if te integration area are large enoug. A computer time in bot tecniue depend on ize of integration area. In our calculation te 2-tep MoM reuired le time tan te fringe euation wic involve calculation of function v () in (16) and (3), repectively. ( kr,φ,φ ) and v () ( kr,φ,φ ), given APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1576

8 Figure 7. Fringe wave around te tip of te wedge for variou incidence angle. (left column) α = 27, (rigt column) α = 3, (top row) Cae I, (middle row) Cae II, and (bottom row) Cae III, (r = 2λ, d = λ 2, f=3 MHz, TE/HBC). 4. Concluion Novel fringe wave integral euation are derived, and a metod of moment (MoM)-baed algoritm i developed for te direct modeling and imulation of fringe wave around perfectly reflecting oft and ard wedge under a plane wave illumination. Te tet and comparion are performed among PTD-extracted fringe wave and fringe MoM. A perfect agreement i found. Fringe integral euation of te derived type can be ueful for numeric olution of diffraction problem for complex object. Acknowledgment For te data, pleae end your reuet via to g.apaydin@gmail.com or l@leventevgi.net. Reference Apaydin, G., and L. Sevgi (212), A canonical tet problem for computational electromagnetic (CEM): Propagation in a parallel-plate waveguide, IEEE Antenna Propag. Mag., 54(4), Apaydin, G., and L. Sevgi (214a), Metod of moment (MoM) modeling of wave propagation inide a wedge waveguide, Appl. Comput. Electromagnetic Soc. J., 29(8), Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtev (214b), Wedge diffracted wave excited by a line ource: Metod of moment (MoM) modeling of fringe wave, IEEE Tran. Antenna Propag., 62(8), APAYDIN ET AL. FRINGE IE FOR 2-D SOFT AND HARD WEDGES 1577

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