Nonlineer su dalgaları

Abstract

Bu çalışmada, küçük fakat sonlu genlikli su dalgalarının yayılması ile ilgili sınır değer problemlerinin asimptotik çözümleri incelenmiştir. Çalışma beş bölümden oluşmaktadır. İlk bölümde, önce sürekli ortamlarda dalga yayılması problemlerinin, daha sonra da su dalgalan ile ilgili problemlerin tarihi gelişimi kısaca özetlenmiştir. Su dalgalan incelenirken akışkanın yoğunluğunun hareket esnasında değişmediği ve akımın irrotasyonel (çevrisiz) olduğu kabul edilir. İkinci bölümde, bu varsayımlar dikkate alarak viskoz olmayan sıkışmaz bir akışkanın çevrisiz hareketini yöneten temel denklemler verilmiştir. Bu denklemler birinci mertebe bir kuazilineer denklem sistemi teşkil ederler. Bu bölümde, denklem sistemine eşlik eden akışkanın serbest yüzeyindeki kinematik ve dinamik sınır koşullan ve akışkanı sınırlayan bir ara yüzeydeki sınır koşulu da türetilmiştir. Serbest yüzeydeki dinamik sınır koşulunda yüzey geriliminin (kapilariterin) de etkisi gözönüne alınmıştır. Bu koşullardan ilk ikisi nonlineerdir. Bilindiği gibi, nonlineer problemlerde çözümlerin süperpozisyonu ilkesi geçerli olmadığı için, böyle problemlerin çözümlerini inşa etmek için lineer analizin yöntemleri direkt olarak uygulanamazlar. Bu nedenle, bu tip problemler için değişik yöntemler türetilmiştir. Bunlardan önemli bir grubu asimptotik yöntemler oluşturmaktadır. Üçüncü bölümde, lineer harmonik dalgaların yayılması ele alınmış ve su dalgalarının nasıl sınırlandırıldığı özetlenmiştir. Dördüncü bölümde küçük fakat sonlu genlikli su dalgalarının yayılması ele alınmıştır. Bu problemin yaklaşık çözümü daha önce dissipatif veya dispersif nonlineer ortamlarda dalga yayılması problemleri için geliştirilen bir asimptotik yöntem kullanılarak inşa edilmiştir. Beşinci bölümde ise küçük fakat sonlu genlikli su dalgalarının modülasyonu incelenmiştir. Burada, kapilariterin de etkisi gözönüne alınmış ve daha önce tabakalı elastik ortamlarda uygulanan bir asimptotik teknik kullanılarak problemin çözümü inşa edilmiştir. Nonlineer dalga modülasyonunu asimptotik olarak karakterize eden nonlineer Schrödinger (NLS) denklemi elde edilmiş ve bu denklemin katsayılarının çeşitli limitleri alınarak, limit problem için daha önceki çalışmalarda elde edilen NLS denklemlerinin katsayılarının bulunabileceği gözlemlenmiştir,

In this work, we have considered the propagation of small but finite amplitude (weakly nonlinear) water waves on a liquid layer of uniform depth. When investigat ing the water waves, the fluid is assumed to be inviscid and incompressible and that the motion is irrotational. The two dimensional motion of such a fluid may be described by the Laplace equation for the velocity potential <|>(x,y,t) ; - f + - f- = 0 for -oo<x<y, Ti = hri, c = -Jgh (5) c where L is a characteristic wave length. In terms of the non-dimensional quantities (1),(2),(3) and (4) are rewritten as 8»£t + £t = 0 (6) ox2 öy2 K ' 52 - 1+ - L- l + ti- - - - =- - + t\ - r +- ti - t at y = 0 (7) lot dxdx 'cxSxSyJ dy 'öy2 2 ' dy3 J K ' db 52dyj dy dy f =0 at y = -1 (9) dy Here the boundary conditions (2) and (3) at y = Ti(x,t) are expanded around the undisturbed free surface y = 0 to obtain (7) and (8) respectively. Note that, the bar "-" over the non-dimensional quantities is omitted in (6),(7),(8) and (9). Also the quantity 5 is defined as 8 = h/L (10) which measures the ratio of vertical to horizontal length scales. Since the propagation of weakly nonlinear waves are considered t| and § are expanded the following asymptotic power series in a small parameter e>0 which measures the degree of nonlinearity ; îl = İ>X(x,t), ? = !>"?"(***) (11) n=1 n=1 Here it is also assumed that 82 = 0(s) as e ->? 0 i.e. e = 82 (12) Then substituting the series (11) into (6-9) and equating terms with the same powers of s, a hierarchy of equation and boundary conditions are obtained from which it is possible to obtain §n and t\B successively. Up to third order in e, these are given as follows : (13a) on y= 0 (13b) on y = -l (13c) (14a) VI The solution of the first order problem is found to be T,=A,(*t), ^ = ~^ (16> where A,(x,t) is an arbitrary function to be determined in higher order problems. By employing the first order solution (16) into the second order problem, the second order solutions are obtained as ?.-f(f»)^M.,-HI'-«-«-' where A2(x,t) is an arbitrary function to be determined also in higher order problems. At this order, the boundary condition (14b) on y = 0 yields ^F "âF-0 (18) For the waves propagating along the positive x - axis, the solution of (18) is taken as A,=F(x-t) (19) where F is an arbitrary function. Then (17) becomes fyJ İS2F., A öA2 lföFV /om K2 J 8£,2 iy ',2 a 2 as If we employ <|>2 in (15a), and then integrating the resulting equation with respect to y once, and using the boundary condition (15d), we get a« or t->», i.e. A, increases without bound. Therefore the second order solutions $2 and t\2 indicate secular behaviour. That is, when x=0(e_1) or t = 0(e~') the first order solutions.j), and ti, and the second order solutions §2 and tj2 become the same order and then the formal asymptotic expansions given in (11) cease to be uniformly valid. Let us now assume that the waves propagating along the positive x-axis is developed from an initial state prescribed at t=0. In that case, for a fixed x when t = 0(e') the second order solutions <|>2 and ti2 indicate secular behaviour and therefore the asymptotic expansions (11) become non-uniform. The secularity may be removed by the method of multiple scales [16]. To do this, we introduce new time variables t"=t, t,= et (24) instead oft, where t, characterizes the slow variation with time ; and assume that <(> and ti are functions of x, t0,t, and y. Then utilizing the asymptotic expansion n = İeX(^t0,t1), = Zen«l>n(xy.t.,t1) (25) in (6-9) instead of those given in (11) a new hierarchy of problems are obtained to determine 4>", Tin successively. The solutions of the new first and second order problems are found to be V 4,t, t.), li=-^i (26> and "..?$+tyst""'k ?'.') ^-S7*-^-|<|*>' <27) where rf7 is an arbitrary function and /^n satisfies the equation JH-£*=°. (28) The solution of this equation for the waves propagating along the positive x-axis is of the form *,=*,(*. t.), S=x-t0 (29) VUl Note that at this order, the function /#, could not be determined completely. This result implies that /*, remains constant in a frame of reference moving with the unit velocity. The dependence of /*, on ğ and t, is determined at the next order. At the third order, it is found that /^2 must satisfy the equation ; âf^-i?^ z***2********* (30) Note that the solution of this equation will exhibit a secular behaviour like the solution of (22). But this can be eliminated by equating the terms on the right hand side of (30) to zero : * * 3 9. d1. 15' _ /(11v dL& 2 3£ 'SÇ2 ' 6SÇ4 By doing this not only the uniformity of the asymptotic expansions (25) are maintained but also a differential equation for the function /^,, which was left undetermined in the previous steps, is deduced. Under the condition (31) the solution of (30) for the waves propagating along the positive x - axis is of the form *3= *,(5, t.) (32) Note that at this order, it is only revealed that rf2 remains constant in the frame of reference moving with the unit velocity. The dependence of /t2 on \ and t, can be determined in higher order problems. But, since this work is centered around the small but finite amplitude waves, it is aimed to obtain just the uniformly valid zeroth order solution. Therefore, to obtain this solution the determination of si, as a solution of (31) will be sufficient. Note that *-?£*, (33) then (31) can be written for t|, as ön. 3 ön. 1 8 tl,~.^ - -+- "H,- + -= 0 (34) St, 2- 3Ç 6 SÇ* This is the equation known as the Korteweg-deVries equation. The coefficients of this equation is different from the one given in [14]. This difference stems from the use of different non-dimensional quantities. In the last part of the work, the nonlinear self modulation of capillary-gravity waves on a liquid of uniform depth is considered. The harmonic resonance phonemone is excluded in the analysis. The amplitude of the wave is assumed to be small but finite. Therefore, the problem is investigated by employing the method of multiple scales. Following the usual procedure of the method 4 and r\ are expanded in the following asymptotic power series in a small parameter e > 0 which measures the degree of nonlinearity and, at the same time, the narrowness of the side band width of the carrier wave number centered around a specific wave number ; <l»^zeb"(Xo.x..x2,y,t0.t1,t2) îl = ZenTi"(x".xI,x21t0.t1,t2) (35) where IX x, = e'x, t, = s't (36) are the multiple scales introduced to specify the slow variations of the amplitude compared with the phase of the carrier waves. Hence, writing first the equations and boundary conditions (1-4), in terms of the new independent variables |x0, x,, Xj, t0, t,, t2, yj, then employing the expansions (35) and collecting the terms of like powers in e yield a hierarchy of problems from which it is possible to determine ", ti" successively. Up to third order in e, these are given as follows ; SXo ay ±t-fL=0 and a0 dy St. + gn,- T 8\ pdx20 dy = 0 on on o(e'): fÜi+£Üı=_2Ü^- w ax;; ay2 d^dx, at" a^ ay £n1_£n1a^1_ a <[>, at, ex0 ax0 1 ay2 ak Tati2 a.)), - +gn2- ---=- - at0 p ax. at. Til a2*, at0ay 2 af..(*).+iI-£a~o P dx0dx. = 0 o(e>) : ay 92: aM», a2*: ax2 ay2 dq» &?,_ at0 ay s22 Sn2 ^h at, at2 ax" ax0 ax0 ax0 on y=0 on y=-h an, az4>, 1,a>, an, a*, ax0 ax0ay 2 ay ax0 ax, a^-öf, at, at2 -.n. at0ay Til, a2.),, a2*, - Til >,*i T ay ay at,ay ax0 ex, P o g2n2 a2^, a2n, ax2 ax"ax, ax"ax2j (38c) (38d) (39a) on y=0 (39b) on y=0 (39c) %=0 on y=-h (39d) 5y Note that, as usual in these types of asymptotic analysis, the problems at each step are linear. Moreover the first order problem is simply the classic linear wave problem. Accordingly we may take the first order solutions as ; 11=İA?>(x1Ix2,tI,t2)e»+c.c 1.1 . = |(BrMJ.t1.t2)e"<' +Ci»(x"x2,tI,t2)e-»*)e'» +c.c +DI(x"x2>tIIt2) (40) where e = kx0-<üt0 (41) and k is the wave number, ca is the angular frequency, c = co / k is the phase velocity and c.c. denotes the complex conjugate to the preceeding terms. Then the substutition of (40) into the boundary conditions (37b-d) yields W U(l> = 0 (42) - I -1 where LtJa',". B», eT)' (43) and f -fol -kl kl ' g + k2l2y - iool -icol (44) 0 kle-"" -kle1*, W,= Note that detW = 0 gives the dispersion relation of the linear waves, i.e. , can be constructed by (49). In the limit of no capillarity, (73) recovers the result for the gravity waves given in [17]. The coefficients r and A are responsible for the modulational instability of a nonlinear plane wave solution of (73). The plane wave train is stable or unstable according as TA<0 or TA>0. In the limit of infinite depth, it is seen that the capillary-gravity waves on the deep liquid are modulationally unstable for k>(g/2y)V2 or for k 1.363) TA>0 while for the shallow water waves (kh< 1.363) TA<0. Hence the deep gravity waves are modulationally unstable whereas the shallow water waves are stable. For purely capillary waves, TA>0. Therefore they are modulationally unstable for all wave numbers. These results agree with the existing ones [17,18]. </l»^zeb</x