水底地形变化对摇板造波的影响_英文_

J. Marine Sci. Appl. (2011) 10: 7-16 DOI: 10.1007/s11804-011-1035-8 Effect of Bottom Undulation on the Waves Generated Due to Rolling of a Plate Puspendu Rakshit1 and Sudeshna Banerjea2* 1. Deshapran Birendranath Institution for Boys, 198-B, S. P. Mukherjee Road, Kolkata-700026, India 2. Department of Mathematics, Jadavpur University, Kolkata-700032, India Abstract: In the present paper, the effect of a small bottom undulation of the sea bed in the form of periodic bed form on the surface waves generated due to a rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of non uniform finite depth is investigated. A simplified perturbation technique involving a non dimensional parameter characterizing the smallness of the bottom deformation is applied to reduce the given boundary value problem to two independent boundary value problems upto first order. The first boundary value problem corresponds to the problem of water wave generation due to rolling oscillation of a vertical barrier either partially immersed or completely submerged in water of uniform finite depth. This is a well known problem whose solution is available in the literature. From the second boundary value problem, the first order correction to the wave amplitude at infinity is evaluated in terms of the shape function characterizing the bottom undulation, by employing Green's integral theorem. For a patch of sinusoidal ripples at the sea bottom, the first order correction to the wave amplitude at infinity for both the configuration of the barrier is then evaluated numerically and illustrated graphically for various values of the wave number. It is observed that resonant interaction of the wave generated, with the sinusoidal bottom undulation occurs when the ratio of twice the wavelength of the sinusoidal ripple to the wave length of waves generated, approaches unity. Also it is found that the resonance increases as the length of the barrier increases. Keywords: bottom undulation; rolling oscillation; partially immersed barrier; submerged plate Article ID: 1671-9433(2011)01-0007-10 1 Introduction1 The interaction of linear waves with a thin floating plate present in laterally unbounded sea can be used the depth of the sea is not constant. Due to this reason the effect of irregular bottom topography on waves are studied. The interaction of incident waves with irregular bottom topography of sea bed finds its importance in understanding the mechanism of wave induced mass transport that forms sand ripples of some wavelength. If this wavelength is half that of the incident wave train then these ripples produce a resonant behaviour in the reflected waves. A significant research both theoretical and experimental, has been carried out in this direction (Davies, 1982; Davies and Heathershaw, 1984; Fitz-Gerald, 1976; Roseau, 1976). In all above studies, the bottom undulation was the only obstacle in the propagation of waves. Later on, the study of the behaviour of surface waves due to irregular bottom topography was modified by including the effect of a thin floating plate. These problems were studied when the floating plate is in form of a very large floating structures (Wang and Meylan, 2002; Belibassakis and Athanassoulis, 2005; Watanabe et al., 2004). Received date: 2010-10-06. Foundation item: Supported by DST through the Research Project No. SR/SY/MS: 521/08. *Corresponding author Email: sbanerjee@math.jdvu.ac.in © Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2011 In recent past, Mandal and Gayen (2006) and Mandal and De (2007) studied the effect of a thin floating plate in form of a breakwater either partially immersed or submerged in sea with variable bottom topography, on the surface waves incident on it. They used a perturbation technique together with Green's integral theorem and Galerkin technique to obtain the reflection and transmission coefficients. In the present paper we have investigated the effect of irregular bottom topography on the wave generated by rolling oscillation of a vertical barrier either partially immersed or completely submerged in water. Problems concerning generation of waves due to rolling oscillation of a vertical barrier partially immersed in deep water was studied long back in 1948 by Ursell. He used singular integral equation formulation to obtain closed form solution of this problem. Later Evans (1970) and Banerjea and Mandal (1992) studied the problem of generation of waves due to rolling of a submerged vertical barrier in deep water. Evans (1970) used complex variable method while Banerjea and Mandal (1992) used integral equation formulation to obtain the closed form solution. These problems are among a limited number of problems which admit of closed form solution. Later Banerjea et al. (1996) and (1997) considered the effect of finite depth of water region on the waves due to rolling of a vertical barrier partially immersed or completely submerged in water of uniform finite depth. They used two different methods to obtain the amplitude of the waves Puspendu Rakshit, et al. Effect of Bottom Undulation on the Waves Generated Due to Rolling of a Plate 8 produced. The first method involves the eigen function expansion of the velocity potential describing the ensuing motion in water while the other method involves a hypersingular integral equation formulation. The two methods produce almost the same result for the amplitude at infinity of the radiated waves for both the configurations of the barrier. In the present paper we have applied a perturbation technique (Mandal and Gayen, 2006; Mandal and De, 2007) in terms of a parameter describing smallness of the bottom undulation to reduce the boundary value problem to two independent boundary value problems upto first order. The first boundary value problem corresponds to the known problem of wave generation by a vertical barrier either partially immersed (Banerjea et al., 1996) or completely submerged (Banerjea et al., 1997) in water of uniform depth h. From the second boundary value problem the first order correction to the wave amplitude at infinity for both the configurations of the barrier is obtained by applying suitably the Green's integral theorem. The numerical results for the wave amplitude at infinity is illustrated for different values of wave number when the sea bottom has a patch of sinusoidal ripples. It is observed that when the wavelength of the waves generated is equal to twice the wavelength of the sinusoidal ripples at the sea bed, a resonant exact is seen to occur. Also, this resonant exact is more pronounced as the length of the barrier increases. 2 Formulation of the boundary value problem We consider two dimensional motion due to rolling oscillation of a thin rigid plate described by 0,x y L= ∈ , present in water region with a small bottom undulation described by ( )y h c xε= + . Here h is the mean depth of water, ε is a small non dimensional parameter, ( )c x is a function of compact support describing the shape of the bottom. Here we choose rectangular cartesian coordinate system in which y axis is directed vertically downwards and x axis is along the mean free surface. The plate is hinged at a point (0, )s , s L∈ and is forced to perform simple harmonic oscillation about its mean vertical position with amplitude θ = i0Re{ e }σtθ − , σ being the frequency of oscillation and 0θ is small. We have considered the problem for two configurations of the barrier, namely, (i) partially immersed barrier for which [0, ]L a= and (ii) completely submerged plate for which [ , ]L a b= . Assuming the motion to be irrotational, it can be described by the velocity potential Re { }i( , )e tx y σφ − , where φ satisfies the following boundary value problem. 2φ∇ =0 in the water region (1) 0yΚφ φ+ = 0y = , 2 g σΚ = (2) g being the acceleration due to gravity, 0i ( )x y sφ θ= σ − 0, , ,x y L s L s y= ∈ ∈ < (3) 0nφ = ( )y h c xε= + (4) n denotes the outward drawn normal, 1 2r φ∇ is bounded as 1 2 2 2 1 1{( ) ( ) } 0r x c y d= − + − → (5) Here 1 1( , ) (0, )c d a≡ or ( , )a b according as the barrier is partially immersed or submerged. The far field conditions satisfied by φ are 0 0 ( , ), ( , ) ( , ), A x y x x y B x y x Φφ Φ → ∞⎧= ⎨ − → −∞⎩ (6) where A and B are unknowns which represent the amplitudes of radiated waves at infinity, and 0i0 ( , ) ( )e k xx y yΦ ζ= (7) 0 0 cosh ( )( ) cosh k h yy k h ζ −= (8) 0k is the unique real positive root of the equation tanhk kh K= (9) 3 Method of solution The bottom condition (4) can be expressed approximately as 2d { ( ) } ( ) 0 dy x c x x φ ε φ Ο ε− + = on y=h (10) The condition (10) implies that we can adopt the following expansion of ( , , ), (x y Aφ Βε ε), (ε) as 2 0 1 2 0 1 2 0 1 ( , , ) ( , ) ( , ) ( ) ( ) ( ) ( )= + ( ) x y x y x y A A A B B B φ ε φ εφ Ο ε ε ε Ο ε ε ε Ο ε ⎧ = + +⎪ = + +⎨⎪ +⎩ (11) Substituting (11) in the boundary value problem (1)–(3), (5), (6) and (10), we find after equating the coefficients of 0ε and ε from both sides, that 0φ and 1φ satisfy the following boundary value problems BVPI and BVPII. BVPI The function 0φ (x, y) satisfies 2 0 0φ∇ = in 0 y h≤ ≤ 0 0 0yKφ φ+ = on 0y = 0 0i ( )x y sφ θ= σ − on 0,x y L= ∈ 0 0yφ = on y h= 1 2 0r φ∇ is bounded as 0r → The far field conditions for 0φ are Journal of Marine Science and Application (2011) 10: 7-16 9 0 0 0 0 0 ( , ), ( , ) ( , ), A x y x x y B x y x Φφ Φ → ∞⎧⎨ − → −∞⎩ ∼ (12) BVPII 1( , )x yφ satisfies 2 1 0φ∇ = in 0 y h≤ ≤ 1 1 0yKφ φ+ = , on 0y = 1 0xφ = on 0,x y L= ∈ 1 0 d [ ( ) ( )] dy c x x x φ φ= on y h= 1 2 1r φ∇ is bounded as 0r → The far field conditions are 1 0 1 1 0 ( , ), ( , ), A x y x B x y x Φφ Φ → ∞⎧⎨ − → −∞⎩ ∼ (13) It may be noted here that BVPI corresponds to the known problem of generation of waves due to rolling oscillation of a vertical plate present in water of uniform finite depth h. For the problem involving a partially immersed barrier, L=[0, ]a and the solution of BVPI is given in Banerjea, Dolai and Mandal (1996). For the problem involving a completely submerged barrier, L=[ , ]a b and the solution is given in Banerjea, Dolai and Mandal (1997). For completeness we derive the expression for 0B for the two problems in Appendix. Now we will proceed to obtain A1, B1 from BVPII for (i) L=[0, ]a , (ii) L=[ , ]a b . Let us denote Re i( , )e }tx yψ − σ{ to be the velocity potential corresponding to the problem of scattering of an incoming wave by a vertical barrier either partially immersed or completely submerged in water of finite uniform depth h, where ( , )x yψ satisfies the following boundary value problem. 2 0ψ∇ = in 0 y h≤ ≤ 0yKψ ψ+ = on y=0 0xψ = on 0,x y L= ∈ 1 2r ψ∇ is bounded as 0r → 0 y ψ∂ =∂ y h= 0 0 0 i i 0 i 0 (e e ) ( ), ( , ) e ( ), k x k x k x R y x x y T y x ψψ ψ −⎧ + → −∞⎪= ⎨ → ∞⎪⎩ (14) where 1 2 0 0 0( ) cosh ( )y N k h yψ −= − (15) with 0 00 0 2 sinh 2 4 k h k hN k h += (16) and 0k being unique real positive root of Eq.(9). Also ( , )x yψ can be expressed as (Mandal and Gayen, 2006; Mandal and De, 2007) 0 0 0 i i 0 1 i 0 1 (e e ) ( ) e ( ), 0 ( , ) e ( ) e ( ), 0 n n k x k x k x n n n k x k x n n n R y C y x x y T y D y x ψ ψ ψ ψ ψ ∞− = ∞ − = ⎧ + + <⎪⎪= ⎨⎪ + >⎪⎩ ∑ ∑ (17) where i ( 1,2,....)nk n± = are the purely imaginary roots of Eq.(9), nC , nD ( 1,2,....)n = are unknown constants and 1 2( ) cos ( )n n ny N k h yψ −= − (18) with 2 sin 2 4 n n n n k h k hN k h += it can be shown that 1 n n R T C D = −⎧⎨ = −⎩ (19) where R and T are the unknown reflection and transmission coefficients. Now we proceed to evaluate A1, B1 when the barrier is partially immersed and completely submerged. Case I: Partially Immersed Barrier, L=[0, ]a To obtain A1 and B1, in this case we apply Green’s integral theorem to the functions 1( , )x yφ and ( , )x yψ in the region bounded by the lines 0,0 ; ,0 ; , ; ,0 ; 0, 0; 0 ,0 . y x X x X y h y h X x X x X y h y X x x y a = < ≤ = ≤ ≤ = − ≤ ≤ = − ≤ ≤ = − ≤ < = ± ≤ ≤ We obtain after making X to tend to infinity, 1 2 0 1 0 0 02i cosh ( ) ( , ) ( , )dx xk hB N k h c x x h x h xψ φ ∞− −∞ = ∫ (20) Similar application of Green’s integral theorem to ( , ) ( , )x y x yχ ψ= − and 1( , )x yφ in the same region and making X → ∞ , we get 1 2 0 1 0 0 02i cosh ( ) ( , ) ( , )dx xk hA N k h c x x h x h xψ φ ∞− −∞ = − −∫ (21) Case II: Completely Submerged Plate, L=[ , ]a b In this case, to obtain A1 and B1, we apply Green’s integral theorem to the functions 1( , )x yφ and ( , )x yψ in the following region by the lines Puspendu Rakshit, et al. Effect of Bottom Undulation on the Waves Generated Due to Rolling of a Plate 10 0, ; ,0 ; , ; 0 , . y X x X x X y h X x X y h x a y b = − ≤ ≤ = ± ≤ ≤ − ≤ ≤ = = ± ≤ ≤ Making X → ∞ , we obtain 1 2 0 1 0 0 02i cosh ( ) ( , ) ( , )dx xk hB N k h c x x h x h xψ φ ∞− −∞ = ∫ (22) Similar application of Green’s integral theorem in the region described above ( , ) ( , )x y x yχ ψ= − and 1( , )x yφ and making X → ∞ , we obtain 1 2 0 1 0 0 02i cosh ( ) ( , ) ( , )dx xk hA N k h c x x h x h xψ φ ∞− −∞ = − −∫ (23) In the expression for A1 and B1 given by Eq.(20)–(23), we need to know ( , )x yψ , 0 ( , )x yφ and ( )c x . A suitable representation for ( , )x yψ is given by (17). Also we use the representation of 0 ( , )x yφ as ( cf Banerjea, Dolai and Mandal, 1996, 1997) 0 0 1 0 0 0 1 ( , ) cos ( )e , 0 ( , ) ( , ) cos ( )e , 0 n n k x n n n k x n n n A x y P k h y x x y B x y Q k h y x Φ φ Φ ∞ − = ∞ = ⎧ + − >⎪⎪= ⎨⎪ − + − <⎪⎩ ∑ ∑ (24) where 0 0 n n A B P Q = −⎧⎨ = −⎩ (25) For completeness, we have obtained 0A , 0B , nP , nQ in Appendix and 0( , )x yΦ is given by Eq.(7). For describing bottom undulation at sea bed we take the shape function ( )c x as 0 π πsin , ( ) 0,otherwise m mc x x l x l c x λ λ λ ⎧ − ≤ ≤ ≡ − ≤ ≤⎪= ⎨⎪⎩ (26) where m is a positive integer. This represents a patch of m sinusoidal ripples with wave number λ . Substituting 0φ and ψ from Eq.(17) and Eq.(24) and using Eq.(19) and Eq.(25), we obtain from Eq.(20) and Eq.(23). 0 0 0 0 0 1 1 2 2 0 1 0 0 1 0 0 0 0 i i 0 0 i0 0 1 0 i 0 0 1 10 i0 0 0 2i 2i cosh cosh ( ){i (e e ) ( ) ie ( )}{ e cosh e }d ( ){(1 )i e ( ) e ( )} i{ e cosh m n m k x k x l k x k x m m m m l k x k x k x n n m m m n m k x n k hB N k hA N k h k h c x k R h k BC k h k h Q k x c x R k h C k h k B Q k k h ψ ψ ψ ψ − − ∞ − = ∞ ∞ − = = = − = − + − + + − + ⋅ − + ∫ ∑ ∑ ∑∫ 1 e }dnk xn n x ∞ − = ∑ Using the expression for ( )c x given by Eq.(26) and noting that ( )c x is an odd function of x we obtain after simplification 0 1 1 2 2 0 1 0 0 1 0 0 0 2 2i0 0 0 0 0 0 0 100 2i 2i cosh cosh ( )sin [ (e 1) 2 ( )sin e ]d cosh n l k x k x n n n k hBN k hAN k h k h k B hc x k h k x Q k x k h ψλ ψ ∞ − = = − = − − ∑∫ (27) Now on integration we obtain B1 as 1 2 0 0 0 0 02 0 1 0 1 0 02 0 2 0 0 0 0 0 12 2 0 0 0 2 0 0 0 0 0 0 0 1 2 2 1 02 0 cos( 2 ) cos( 2 )2i [ 2 22 sin( 2 ) sin( 2 ) 2 cos 1i i ] [ ] 2 2 4 cosh ( )sin( ) cos( )[{ ( ) ( n n n n n c B k k l k lk hBN k kN k l k l c B k l k k k N c k k h k k l k k lQ k k kN k λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ∞ = + −= − − ++ − − + −− + + −− + − − − − − −− + + ∑ 0 0 0 2 2 0 2 2 2 2 0 0 )sin( ) cos( ) }e ( ) ] ( ) ( ) nk ln n n n n n k l k k l k k k k k k k k λ λ λ λ λ −+ − + ++ + −− + + + (28) For partially immersed barrier i.e. for L= [0, ]a , the expression for 0B and nQ are substituted from (A20) to (A22) into (28) to get B1. Similarly for completely submerged barrier i.e. for [ , ]a b the expression for 0B and nQ are substituted from (A33) to (A35) into (28) to get B1. In the next section we evaluate 1| |B numerically and discuss the numerical results. 4 Numerical result For numerical evaluation of first order correction to wave amplitude 1| |B we need to know the value of 0| |B and the constants | |nQ in the expression for 0 ( , )x yφ given by Eq.(24). A multiterm Galerkin approximation is used to evaluate these quantities when the barrier is (i) partially immersed and (ii) submerged in water of uniform finite depth. In our numerical computation we have chosen the value of hλ to be unity and 0 0.1c h = . In Fig.1, the graph of 0| |B , the wave amplitude at infinity due to rolling oscillation of a partially immersed plate in water of finite depth h (in nondimensional form) is drawn for various values of the wave number Ka, for a h =0.04, 0.1 and s a =0.1. Here the results for 0| |B is compared with the value of | |B∞ , where B∞ is the wave amplitude due to Journal of Marine Science and Application (2011) 10: 7-16 11 rolling oscillation of a barrier partially immersed in deep water (Ursell, 1948). It is observed that the curve of 0| |B matches almost exactly with that of the curve for | |B∞ when a h =0.04, 0.1. Thus it is observed that the far field amplitude when the depth of water is constant for both the configurations of the barrier is consistent with the results when the depth of water is large. (a) a h =0.04 and s a =0.1 (b) a h =0.1 and s a =0.1 Fig.1 Partially immersed barrier A similar comparison between 0| |B and | |B∞ (Evans, 1970) is illustrated in Fig.2 when the barrier is submerged in water of uniform finite and infinite depth respectively for the values of the parameters a b =0.2, s a =0.4 and b h =0.06 and 0.1. As expected, a very good matching of these two results is observed from Fig.2 for these values of the parameters. (a) b h =0.06, a b =0.2 and s b =0.4 (b) b h =0.1, a b =0.2 and s b =0.4 Fig.2 Submerged plate Fig.3 b h =0.6, m=1, c h =0.1 for partially immersed barrier Puspendu Rakshit, et al. Effect of Bottom Undulation on the Waves Generated Due to Rolling of a Plate 12 Fig.4 b h =0.6, a b =0.1, m=1 and c h =0.1 for submerged plate Fig.3 depicts 1| |B against Ka for different values of s a when a h =0.6, m=1 in the case when barrier is partially immersed. Also in Fig.4, 1| |B is drawn against Kb for different values of s b , and a h =0.6, a b =0.1, m=1 when the barrier is submerged. From both the figures it is observed that 1| |B increases first, and then decreases for large values of the wave number for any value of s a or s b . Also it is observed in both cases that for a single ripple and for fixed length of the barrier, the lowering of the position of hinge point of the barrier causes decrease in the amplitude 1| |B . Fig.5 a h =0.4, s a =0.2 and c h =0.1 for partially immersed barrier Fig.6 b h =0.8, a b =0.2, s b =0.1 and c h =0.1 for submerged plate Fig.5 and Fig.6 depict 1| |B against various values of the wave number for different values of m. In Fig.5, 1| |B is plotted against Ka for a h =0.4, s a =0.2 and for m=1,3,5 when the barrier is partially immersed. In fig6, 1| |B is drawn against Kb f