Geri Dön

Düzlemsel dalgaların farklı sınır koşullarına sahip paralel iki yarım düzlemden kırınımı

Plane wave diffraction by a pour of soft and hard complementary half planes

  1. Tez No: 39134
  2. Yazar: ERDEM TOPSAKAL
  3. Danışmanlar: PROF.DR. MİTHAT İDEMEN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 34

Özet

ÖZET Bu çalışmada, Dirichlet ve Neumann sınır koşullan ile karakterize edilen, birbirine paralel, iki komplementer yarım düzlemden düzlemsel dal gaların kırınımı incelenmiştir. Ortaya çıkan sınır-değer problemi bir ma tris Wiener-Hopf denklemi olarak formüle edilmiş ve“zayıf faktorizas- yon”yöntemi ile sonsuz bilinmeyenli bir lineer cebirsel denklem sistemi nin çözümüne indirgenmiştir. Bu denklem sisteminin çözümü sayısal olarak yapılmış ve kırman alanın asimptotik ifadesi elde edilmiştir. III

Özet (Çeviri)

SUMMARY PLANE WAVE DIFFRACTION BY A PAIR OF SOFT and HARD COMPLEMENTARY HALF-PLANES 1. Introduction The diffraction of waves by arrays of parallel half-planes is an important topic in scattering theory from both mathematical and engineering points of view. The mathematical problems related to this type of configurations are generally formulated as matrix Wiener- Hopf equations whose solutions require the splitting of a matrix(Kernel matrix) into a product of two ma trices such that these matrices as well as their inverses are regular and of algebraic growth in certain overlapping halves of the complex plane. Except for a restricted class of matrices, no general method exist to achieve such a splitting process in a compact form. However the“weak factorization”method proposed first by Idemen, which allows the determinants of the split matrices to be entire functions which may have zeros in their respective re gions of regularity, can sometimes be effective in solving such problems. The aim of the present work is to analyse the diffraction of scalar plane waves by the configuration shown in Fig.l by using the“weak factorization”method in the case where the half-planes Si and 52 are characterized by Dirichlet(soft) and Neumann(hard) boundary conditions, respectively. Af ter solving the related matrix Wiener- Hopf equation through this method, asymptotic expressions for the scattered field are obtained and some scat tering patterns are presented. A time factor exp(-iu)t) is assumed and suppresed throughout this work. 2. Formulation of The Problem Let a scalar plane wave Ui(x, y) - exp[-ik(xcos(f>o + ysina)\ (1) illuminate the parallel complementary half planes shown in Fig.l. In (1) o is the angle of incidence while k denotes the wave number which is temporar ily allowed to have a positive imaginary part. The boundary conditions on the left and right half-planes are of hard and soft type conditions Respec tively. Our aim is to obtain an exact asymptotic expression of the scattered field. To this end, it is convenient to express the total field as follows: IVr^(x,y) + ^D)(x,y) + C/1(a:,y), y>0 U(x,y) = I U2(x,y), -d < y < 0.U3{x,y), y0 (5a) Ui(x,0)-U2(x,0) = 0,x e (-oo,+oo) (56) dUx{x,G) dU2(x,0) dy dy = 2iksino),x < 0 (5c) dU3(x,-d) dy = 0,x 0 (5c) (5/) In order to obtain the unique solution of the above-mentioned boundary- value problem from (1) - (5/) it is also necessary to take into account the following edge conditions: *7(x,0) = O(v^),x->-0 dU{x, -d) dy = 0(1),x-»0 (6a) (66) By replacing (4a- c) in (5a- f) and inverting the resulting integral equations one gets: AM = $;>) (7«) A1(u)-A2(u)-A3(u) = 0 (76) K(v){Ax(v)-A2(v) + A3(v)}^ + 2$+(i/), 7r(l/ - kcOS(f>o) Qmu < S?m(kcoso) (7c) K(u)A,(u)e~K^d = *+(!/) (Id) A2(u)eK^d - [A3(u) - A4(u)}e~K^d = 0 (7e) A2(u)eK^d + [Az(v) - A4(u)}e~K^d = 2$2“» (7/) The functions $ft2(t/) and $\$(v) appearing in (7a - /) are yet un known functions of u which are regular in the half-planes ^mu > ^m(-k) and 9my < ^m(kcos4>0), respectively. By using (6a, 6), it can be easily shown that: *r» = 0(”~f) (8a) and *i» = 0(i) (86) as \u\ - ¥ oo in their respective regions of regularity. The elimination of Aj(v)(j = 1,2,3,4) between (7a - /) leads to the following matrix Wiener-Hopf equation written in the strip Qm(-k) < *3mu < Qm(kcoso): K(v) 0 1 [-I !(u)\ #~(t/) + -7(1/) 1 1 0 $+(i/) = x -j(u) 1 ksinfo 2tt(u - kcos4>o) (9a) VIwith and 7(1/) = exp[- K{v)d\ *+(v) = $“ = *r(”) $2-» (9ft) (9c) 3.Solution of The Matrix Wiener-Hopf Equation Now let us introduce the following matrix: (10) where 7+(^) and 7 (1/) are the split functions regular in ^mu > ^m(-k) and SSmv < ^mk,respectively, resulting from the Wiener-Hopf factorization of (96) : 7(iz) = 7+(i/)7 {u). The explicit expresions for 7+(v) and f~{v) are known [4] as: +/ \ f K(v)d v\ 7 (v) = ea;p{arccos-}, 7T K (11a) 7 (i/) =7+(_i/). (116) (lie) Here the function axccos(u/k) is defined in the i/-plane cut as shown in Fig.2 with the condition arccos 1 = 0. The asymptotic behavior of ~f+(v) as u -* 00 in the upper half-plane is as follows : 7 \u) ~ exP -i( - j [log \1vjk\ + i(arg u - arg &)] The multiplication from left of both sides of (9a) by (10) gives G-(u)&-(v) + G+(*/)$+(*/) = F(v) with 1/2 G~(u) = (u-k) G+(i/) = (u + k)~ll2 7 (1/) -7 (i/)L(x/) 1 0 0 -1 7+(l/)L(t/) -7» (lid) (12a) (126) (12c) VIIF(o) y/u + k Here we put L{u) = 7(1/) + - - = 2cosh[K{u)d] l\v) (12d) (13) Note that the determinants of G+(i/) and G~(v) involve the same entire function L{u). The zeros of L{v) lying in the upper and lower half-planes are denoted by vn and (- un),respectively, with un = ik{[(2n - l)Tr/2dk}2 - 1}1/2, n = 1,2,... (14) The application of the standard Wiener- Hopf procedure to (9a) yields: *+(i0 = G+(i>) [F+(v) + -P{v)] IL{V) (15a) *-(!/) = G-(^)[F-(^ - P(i/)] lL{u), (156) where P(i^) denotes an entire column vector to be determined, whilst F±(i/) and Gq{v) are defined by F(u)=F+(u) + F-(u) (16a) (166) P(i/) appearing in (15a, 6) must be specified such that the right hand sides of (15a) and (156) have no poles at v = vn and v = - vn respectively, and their asymptotic behaviors for v - > 00 are consistent with the predicted behaviors of $±(i/) given in (8a, 6). Now consider the Wiener-Hopf decomposition in (16a) through the following well-known formulas: f±m=±_l/ mdT. 2X1 Jc± T - V (17) The position of the integration lines £* are indicated on Fig.2. From (12d) it can be checked easily that F{y) contains an exponentially growing term and consequently the integrals in (17) do not converge. This difficulty can be circumvented by introducing a diagonal matrix L+(v) 0 0 L-(u) satisfying F±(I/) = B(u)Fİ(u). X±(i/) appearing in (18) are defined by L(u) = L+(u)L~(u) ; L+(u) = L~(-u) (18) (19) (20a) VIIIso that L+(-un) = L-{vn) = 0, n = 1, 2,.... (206) The explicit expression of L+(u) having the above properties can be ob tained as : £+(“) = y/2 J] [(1 - k2c2n) - iucn] eivCne^u) (20c) n=l with d cn = n (n - 1/2)tt In (20c) x(^) is an arbitrary entire faction which will be chosen such that 7±(i/)i±(i/) have algebraic behavior at infinity. By choosing x(^) as one obtains 7+(I/)I+(I/) = 0(1) (21) (22) as v - * oo in the upper half-plane. In (21) C(=0.5771...) is the Euler constant. The use of (18a) reduces (17) to: nw=±7h 1 ksinQ 2-Kİ 2?r / 7+(r)L+(r) dr i± \A + r (r - fccos^oXr - v) ? (23) By virtue of (22), the integrals in (23) converge and give: *t(y) = 1 ksin(j>o 2x {y - kcoso) y/k + U,y+(kcos^>o)L+(kcos4>o) y/k(l + coso) (24a) F?(”) 1 ksin(j)Q y+(kcos4>o)L+(kcos(J)o) 2n {y ~ kcosfa) \Jk{l + cosfo) By substituting now (19) into (15a,b), one obtains *+(!/) = G+(u)D(u) [Fİ(u) + T>-\u)F(u)} /L(u) *» = G^(u)B(u)[F-(u) - D-\u)P(v)]/L(v) It follows from the choice of the matrix D(i/) as in (18) and from (8a, 6) that (246) (25a) (256) IXD_1(l/)P(l/) = O(l), U-K3Q. (26) Since the simple poles of the matrix valued meromorphic function D-1(i/)P(z/) occur at v - ±V{, it can be determined by its Mittag-Leffler's expansion. Thus, by taking also into account (26), one may write V ' V ' f-» V - V{ 4-^ V + Ui t=l 1=1 The constant column matrices ai and bi = 6 (Di 6! (2) (27) are to be determined with the aid of the regularity conditions of 3>+(i/) at u = i/,- and of &~(v) at v = -V{ (i = 1,2,...), namely: dn lim, dm hm - - v-*-vn dvm 2*+/ (»/-i/“r*+(i/) (ı/ + ı/n)2$ (u) = 0 m = 0, 1 = 0 m = 0,1 A straightforward calculation yields for n=l,2,... = 6?> = 0 n = l,2....^Wa(2) = _f^Jİ_ n = 12 [£+K)7+K)]2 n hUi + Un (28a) (286) (29a) (296) or L'(vn),(2) ^ l^nj b(l) _ _ y> «i [£+Kb+K)]2 ”£r *< + *» 1 ksinQ j+(kcoS(f>o)L+(kcos(l)o) 2tt (i/n + kcoso) ^k(l + cos0) L\vn) P+Kb+K)]' T+b\l) a(2), 6(i)l = _ y- ai +öi 1 ksin0 ?j+(kcos(j)Q)L+(kcos(j)o) 2?r (i/w + kcosfo) y/k{\ + cos(J)q) (29c) (30a) ^ a MM L(2) _ 6(d] = y- ai -fti [£+K)7+K)]2 [n n J £i »i + »» 1 ksin4>Q 7+(kcoso)L+(kcos(j>o) 2tt (t/“ + kcos(f>0) y/k(l + cos^o) (306)It follows from (30a - b) that the solution of the equation (9a) is given by *i» = y/u - k 1 ksinQ ~f+(kcoso)L+(kcos0) 1-K (v - kcOS(f>o) y/k(l + COS(j>o) 00 J2) 1 y^ ai (31a) (315) *f (”) = + L(u) ~f+(u)L+(u) 1 ksin< 1-k [y - kcos o /j+(u)L+(u) f+(kcoso) 00 J2) 1 (31c) oo 6(D *+(!/) = v^Ti7+(^i+(^E 4 «=i V + V{ (31d) 4. Asymptotic Analysis of The Diffracted Field and Some Numerical Results The diffracted field in the region y > 0 can be obtained by replacing A\{y) determined from (31a, 6) and (7a) into (4a) and evaluating the result ing integral through the saddle-point technique. For \un\ > \k\ the result is as follows: A) Whence 6(0,tt/2): C/(r,^) = Z7i(r^) + ^(r^) + J7oD(r^) if ? (0,tt - 0) and U{r,) = Ui{r,(j>) + Uf\r^) + UoD{r^) if ? (tt - 0,7c) B) When^o ?(7r/2,7r): U{r,4>) = Ui{r,) + U(D\r,) + UoD{rA) if ) = Ui(r,cf>) + Ui.N\r,) + UoD(r,) if 6 (tt - 0,ir) XIHere Ur denotes the wave reflected by the left half-plane 52, i.e. Jj(N)(r A) - e2ikdsinif>ae-ikrcoa{4>+4>a) while UoD(r,(i>) stands for the wave diffracted at the edge 0, namely: ? eikr UoD{r,) ~eiw/4v^^=7+(fcco^)L+(fcco^)v/l - cos(f> y/kr (v/1 - coso)L+(kcos(f>o) ^> a\ fsfk \ 2ir cos + cosfo 4^ cos(j> + (fi/k)J Some illustrative numerical applications show the variation of the edge- diffracted field as a function of different parameters. I XII

Benzer Tezler

  1. Tez No

    14172

    Local intrinsic modes: a new method to solve non-separable wave problems

    Başlık çevirisi yok

    LEVENT SEVGİ

    Doktora

    İngilizce

    İngilizce

    1990

    Elektrik ve Elektronik Mühendisliğiİstanbul Teknik Üniversitesi

    PROF.DR. ERCAN TOPUZ

  2. Tez No

    421157

    Nonlinear dynamic behaviour of tapered sandwich plates with multi-layered faces subjected to air blast loading

    Çok katmanlı yüzeylere sahip kalınlıkça sivrilen sandviç plakların anlık basınç yüklemesi altındaki lineer olmayan dinamik davranışı

    SEDAT SÜSLER

    Doktora

    İngilizce

    İngilizce

    2015

    Havacılık Mühendisliğiİstanbul Teknik Üniversitesi

    Uçak ve Uzay Mühendisliği Ana Bilim Dalı

    PROF. DR. HALİT SÜLEYMAN TÜRKMEN

  3. Tez No

    866227

    Faz modları kullanılarak silindirik dizi antenlerinin hızlı tasarımı ve analizi

    Fast analysis and design of cylindrical array antennas via phase modes

    HASAN AYDIN

    Yüksek Lisans

    Türkçe

    Türkçe

    2023

    Elektrik ve Elektronik Mühendisliğiİstanbul Teknik Üniversitesi

    Elektronik ve Haberleşme Mühendisliği Ana Bilim Dalı

    DOÇ. DR. ÖZGÜR ÖZDEMİR

  4. Tez No

    537872

    Tabakalı bazı ortamlarda nonlineer dalga yayılması probleminin asimptotik analiz

    Asymptotic analysis of nonlinear waves in certain layered media

    EKİN DELİKTAŞ

    Doktora

    Türkçe

    Türkçe

    2018

    Matematikİstanbul Teknik Üniversitesi

    Matematik Mühendisliği Ana Bilim Dalı

    PROF. DR. MEVLÜT TEYMÜR

  5. Tez No

    863979

    Sound radiation in a coaxial duct with a step discontinuity on the inner wall

    İç duvarında adım süreksizliğine sahip olan bir koaksiyel dalga kılavuzunda ses dalgasının yayılımı

    ADİLE ULUCA

    Yüksek Lisans

    İngilizce

    İngilizce

    2024

    MatematikGebze Teknik Üniversitesi

    Matematik Ana Bilim Dalı

    DOÇ. DR. HÜLYA ÖZTÜRK

Geri Dön