Elektromagnetik Dalgaların Dikdörtgen Kesitli Bir Silindirden Saçılması

Abstract

Bu çalışmada yüzeyleri empedans özelliği gösteren, dikdörtgen kesitli bir sonsuz uzun silindirden elektromagnetik dalgaların kırınımı incelenmiştir. Gelen dalganın "BİF çizgisel kaynak tarafından uyarıldığı var sayılarak problem, önce, birbirine bağlı üçüncü tipten iki Wiener-Hopf denkleminin oluşturduğu bir sisteme indirgenmiş; sonra da, bir takım dönüşümlerle ikinci tipten bir Predholm integral denklem sistemine indirgenerek iteratif bir yöntemle çözülmüştür. Bulunan çözüme dayanılarak, alanın, değişik bölgelerde gözlenen(gelen, yansıyan, kırman v.b.) bileşenlerinin açık ifadeleri çıkarılmıştır. Cismin boyutlarının ve yüzey empedansının değişik terimlere etki sini daha açık bir biçimde gözleyebilmek amacıyla, sonuçlar bazı sayısal örneklere uygulanmış ve ayrıntılı bir biçimde tartışılmıştır.

It goes without saying that the scattering of electromagnetic waves by rectangular cylindrical bodies, which plays extremely important roles on the performance of actual communication systems constitutes a chal lenging problem in the diffraction theory. In this work this phenomenon is analyzed rigorously by assuming that the boundary of the cylinder can be modelled by impedance type conditions. The basic procedure consists of reducing the problem to a pair of simultaneus modified Wiener-Hopf equations(MWHE) of the third kind. By means of some elementary trans formations, these coupled equations are reduced into a system of Fredholm integral equations of the second kind involving also infinitely many un known constants and then solved by iterations. The unknown constants are determined numerically by solving certain linear algebraic equations. In order to show clearly the effects of various parameters on diffracted field, the results were applied to some illustrative examples. 2. Formulation of the Problem Let a rectangular cylinder S be illuminated by an electrical line source I parallel to the cylinder located through P(xo,yo,0) (see Fig.2.1). The source is assumed to be time harmonic with time factor exp(-iu:t) while the horizontal and vertical walls of the cylinder are characterized by differ- ent(homogeneous) surface impedances Z\ = 7}\Zq and Zi - 772-2/0, respec tively. Here Zq represents the free-space impedance, i.e.: Zq = y/^Ao- Then the total electric field outside the cylinder becomes parallel to the cylinder and independent of the z- coordinate, namely: E = u(x,y)ez. (1) The problem consists in finding an explicit expression of u(x, y). For the sake of analytical convenience, we will consider the following vi; regions Bn separately: #1 = {(*,y) #2 = {(x,y) #3 =?{(*, y) #4 = {(*,y) y > yo,x e (-co,oo)} y e (6, y0),x e (-co, c»)},, y6(-M),W>o} lZj y < -h,x G (-00,00)}. The expression of u inside £?" will be denoted by un. They satisfy the Helmholtz equation Aa" + k2un = 0, (a:, y) G £", n = 1,2, 3, 4. (3) Here A: denotes the wave number of the space, which is assumed to have a small positive imaginary part. The method we use is based on a spectral representation of un. In what follows we give the essential steps of this representation briefly. 2.1. Spectral Represantatioîis of un(x, y) The function u\(x,y) which satisfies also the radiation condition can be written as follows: 1 f+°° (x,y) = - / A(a)eiKM*e-iaxda. (4a) 27r J-00 Here A(a) is an unknown coefficient to be determined through the bound ary conditions while A'(a) is given by K(a) = y/k2-a2. (46) The square-root function is defined in the complex «-plane cut as shown in Fig.2.2 such that K{6) - k. The known asymptotic behaviors of ui(x,y) valid for x - > ±00 show that the function A(a)exp{iK(a)y} appearing in (4a) is regular in the strip Im(-k) < Im(a) < Im(k). For the function 1*2(2:, y), which is defined in the region yo < y < 6, one can write 1 f+0° u2(x,y) = ~ / [B(a)eiK<="" style="margin: 0px; padding: 0px; outline: 0px; color: rgb(34, 34, 34); font-family: Verdana, Arial, sans-serif; font-size: 10px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">aH*-b) + C{a)e-iK^a)^-h) = F-(a,y)e-iaa + Fx(a,y) + F+(a,y)eiaa, (6) viii where and iaaF-(a,y)= [ u-2(x,y)eiaxdx, (7a) J-oo /.OO iaaF+(a,y) = / u2{x,y)eiaxdx, (76) J a F1{a,y)= f u2(x,y)eiaxdx. (7c) J - a Owing to the well-known analytical properties of Fourier integrals, i*+(a) and F-(a) are regular functions in the upper(Jm(a) > Im(-k)) and the lower (Im(a) < Im(k)) halves of the complex a- plane, respectively, while Fi (a, y) is an entire function. It can also be shown that one has eiaaF+(a,y) -> 0 as \a\ -? oo, Ima > Irn(-k) (8a) eiaaFi(a,y) -» 0 as \a\ -* oo,Ima > Im(-k) (86) and e~lQaF_(a,y) -> 0 as \a\ -> oo, Ima < Im(k) (8c) e~iaaFi(a,y) -^ 0 as \a\ -> oo, Ima < Im(k). (8d) As to the function u3(x,?/) which satisfies the equation (3) in the ranges x ? (- oo, - a) and x G (a, oo), by assuming u$(x,y) = 0 in the complementary region \x\ < a we can write d2 d 8 [- + K2(a)]Û3(a, y) = (- - ia)u3(a, y)e>Qa - (- - ia)u3(-a, y)e~iaa (9) with /oo u3(x,y)eiaxdx := G-{a,y)t~iaa + G+(a,y)eiaa. (10) -oo The functions G-(a) and G+(a) appearing in (10) are regular functions in the upper(/m(«) > Im( - k)) and the lower(Jm(o/) < Im(k)) halves of the complex cv-plane. Let us consider finally the function «4(2, y) which is defined in the re gion y < -6. By repeating the same procedure outlined above for «2(2, y), we write /oo D(a)e-iK yo) - u2(x,y0) = 0, x e (-00,00) (12a) d - [uı(x,yQ) - u2(x,y0)] = ikIZ0S(x - x0),x ? (-00,00) (126) Tit d (1 + ^-ğ-M*, b) = 0, \x\ < a (12c) (l-^^-M*,-&)=0, |x| a (12g) ti3(a;, - 6) - «4(^5- fc) = 0, |ic | > a (12h) - [v.2(x,b) -u3(x,b)] - 0 |a:| > a (12i) oy - [u3(x,-b)-u4(x,-b)} = 0 \x\>a. (12 j) ay By taking into account (4a) and (6), C(a) can be solved directly from (12a) and (126) to give piatxo C(a) = klZoi- - eiK(°>)(yo-V. (13) 2A(«) v ' The determination, of the other coefficients yields a rather tough problem. In what follows we will recapitulate some basic steps in the method we follow for this purpose. From (126 - j) one gets rather easily (14a) + (1 - a^)g*me-iaa] + W1(a) + 2ikC(a)X(a) and fc Here M (a) and iV(ft) are known functions, namely: + (1 - aai)g°ie-ia*] + W2(a) - 2ikC(a)X(a). (146) M (a) = cosKb - K^-sinKb (15a) IK N(a) = sinKb + K~cosKb (156) ik while a*;0 and K^° are known constants defined by M(±aem) = 0, JV(± 0 (16) and Klf = K{±aX). (17) The remaining functions and constants appearing in (14a, 6), i.e.: P±(ft), Q±{cn), Wi,2(ar), /,'"." and g*>° are unknowns. The meanings of the subindices (- ), (+) are already known. Thus (14a, 6) constitute two modified Wiener-Hopf equations which permit us to solve the functions P±(ft), and Q±(ft) in terms of the unknown parameters f*i° and g^f. An approximate solution to these equations will be given in the next chapter. Here we confine ourselves to notice that after having determined the func tions P±(ft) and Q±(&), the coefficients A,B,D and G±{a) are written as follows: A(a) = 2.1\. e~iKya (18a) B(a) = R-{a)'-iaa +H+(a)eiaa - (1 - mK/k)C(a) 1 } {1 + mK/k) { } XI g.(g)e-'"« + S+(")c*" D{a) (i + mK/k) (18c) R±( Im(-k) and lower Im(a) < Im(k) half- planes respectively. Therefore their right hand sides should also be regular in their respective regions of regularity. This requirement is satisfied if the following relations hold: Q+KJ = [1 " C^fWh + |*m2/0]^^(l + <"f="" )c="" (20c)="" p-(- <="[1" -(^)p+="" |-^,="" style="margin: 0px; padding: 0px; outline: 0px;">]^d +0-.S) + Ae2K") + Ae1(-<tfyia b and y < -b one should evaluate the following integrals: «aOr, y) = ~- / [B(a)eiKiy-k) + C{a)e-iKiy-b)]e-iaxda (29a) i y00 u4(x,y) = - D(a)e-*K{y+b)e-iaxda. (296) 2tt J_oo Here S(o), C(a) and D(a) are known functions given in (186), (13) and (18c), respectively. Evaluation of these integrals was made by using the well-known "saddle-point" technique. The results were applied to some numerical examples which permit us to grasp the effect of various param eters on the diffraction phenomena.