Yadtürdeş dalga klavuzunda harmonik dalgalar
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Abstract
ÖZET Bu çalışmada, iki boyutlu dalga yayılışının karşı düzlem durumunda, yadtürdeş dalga kılavuzundaki harmonik dalgalar incelenmiştir. Birinci bölümde konu ile ilgili genel tanımlamalar ve açıklamalar yapılmıştır. Temel hareket denkleminin kararlı çözümümü elde etmek üzere Uz(x,y,t) = <f){x,y)e~ltJJt seçilip s/2ıp + k2s(y) = 0 biçiminde bir diferansiyel denklem elde edilmiştir. Bu denklemin çarpım tipi bir çözümü incelenmiştir. ikinci bölümde y koordinant değişkenine bağlı olan diferansiyeldenklemin genel çözümü hiper geometrik bir fonksiyon olduğu belirtilmiştir. İndir genmiş dalga denkleminin hiper geometrik fonksiyonlarına bağlı bulunan genel çözümü, hiper geometrik fonksiyon serilerinin yakınsama davranışlarındaki olumsuzluk nedeniyle kullanişlı değillerdir. Seri çözümüne 'ANALİTİK SÜREK YÖNTEMİ' uygulanarak,daha kullanışlı bir yol geliştirilmiştir. Üçüncü bölümde ise 'y' koordinant değişkeni'ne bağlı olan diferansiyel denklemin 'k' öz değerlerine karşılık - değeri hesaplanmıştır. Yadtürdeşlik üzerine herhangi bir kısıtlama konulmamıştır. Sayısal örneklerin sonuçları çizgeler (grafik) ile verilmiştir. Sonuçlar bölümümde, uygulanan yöntemin kullanışlığı tartışılmış ve yadtürdeşlik durum ile türdeş durumun sonuçları karşılaştırılmıştır. SUMMARY HARMONIC WAVES İN İNHOMOGENEOUS WAVEGUIDE In this study, two dimensionals antiplane harmonic waves have been examined At first, the basic concepts and the governing equations of the problem have been given.For the case of antiplane shear, the equation of motion can be written as follows d<rxs dayz.d2Uz dx dy dt* (1) Here o-xz,ayz,UX ve p(y) represent the x and y components of stress tensor, displacement component in the z drection and the mass density respec tively. We shall write our equation for a half space (fig..1) fig.. '.1: General inhomogeneous waveguide viThe constituve equation for the meterial have been given as follows = My) dUz dx (2) °v* = tiv) dU2 dy (3) Here fj,(y) represents the shear modulus which varies with y Substituting (2) and(3) into equation (1), one can obtain dx Kv) dUz dx d_ dy Kv) dUz dy _ = p{y) d2Uz dfi (4) Assuming steady-state case the solutions of equation (4) can be written as follows Uz(x,y,t) = <f>(x,y)e-^ (5) After substituting this expression into equation (4) we come to the equation (6) for function 4>(x,y) d_ dx Kv) df dx d_ dy tiv) d£ dy + p(y)u>2<f> = 0 (6) To obtain a simpler form, we introduce a i^(x,y) function as fol lows <Kx,y) i>{x,y) (v(y))* (7) After substituting this definition in equation (6) we obtain the follow ing differential equation for function ij)(x,y) V2V> + k2s(y)iP = 0 Here, V2 and ^l can ^e written as follows (8) V dx2 dy vnkl(y) = pjy)^ dy 1 (dn{y)/ 4/i(v) / dy J dfi(y) dy (9) We considers an inhomogeneous layer given in fig.2 to be the region of our problem. fig.2:Examined inhomogeneous waveguide In this region the boundary conditions are defined as follows for y = 0 then ( ^dU* n for y = H then < /dU* n (10.1) (10.2) At first (J,(y) is assumed to be an arbitrary function of y. But this function will be restricted to be an analytical function of y. After this definition,the solution is assumed in the following form: vmi>(x,y) = X(x).Y(y) (11) Substituting (11) in (8),and we come -id2x ia2r.,2 X dx2 Y dy2 K(y,`,k) (12) Both sides of this equation must have been equal to the same constant k2 for the existance of the equation (12). Then we come the equation (13) and (14) for X and Y functions with a new unknown constant k2 £ + **-<> (13) İL+(k',(y,u,)-l<?)Y = 0 (14) The solution of equation (13) can be written easily as follows X{x) = Atikx + Be~Lk-x (15) Here, A, and B are integration constants. d^r+(k2(y,u))Y = k2 (16) Eq.(14) is a lineer differential equation with variable coefficents. Some special cases of k2(y,uj) the solutions of this equation are given in [3]. Another solution of this equations have been given to be hypergeometric functions in [4]. Let 'L' be the operator denned belows L = -^ + k2s(y^k) (17) Using this definition equation (16) can be represented by LY = XY (18) ixHere, An = h/ (n = 1,..., N) represent the eigenvalues of this equation. Using boundary conditions Aı, A2,..., are obtained. This A values are distinct. For our problem we choose the mass density p(y) and shear modulus fi(y) as follows p(y) = Po(l + hiy) (19) H{y) = Mo(l + h2y) (20) Under these definitions, the solution of equation (19), (20) satisfying the boundary conditions (10.1), (10. 2) are obtained as follows Y{y) = A0Yi{y) + A1Y2{y) (21) In this equations still we have two integration constants A0 and A/ to be determined. To determine these constants, we shall go to the concept analytical continuation. Using this concept,we obtain a series for An eigenvalues Using this series,we examined the scattering of harmonic waves de pending on to the variation of inhomogeneity in the layer. According to the choosen direction of propagation, the integration constant B must vanish. Using the expressions (5), (7), (11), (15) and (21) we obtain the Uz(x,y,t) function as follows Uz(x,y) = -^-rY(y)eik-*-«t (22) Here any restriction on in homogenety has not been put Finally, we compare the results of homogeneous and in homogeneous cases.
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