Generation of solitary waves by forward- and backward-step bottom forcing. Citation Journal of Fluid Mechanics, 2001, v. 432 n. 1, p.

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1 Title Generation of solitary waves by forward- and backward-step bottom forcing Autor(s) Zang, D; Cwang, ATY Citation Journal of Fluid Mecanics, 21, v. 432 n. 1, p Issued Date 21 URL ttp://dl.andle.net/1722/42135 Rigts Tis work is licensed under a Creative Commons Attribution- NonCommercial-NoDerivatives 4. International License.; Journal of Fluid Mecanics. Copyrigt Cambridge University Press.

2 J. Fluid Mec. (21), vol. 432, pp Printed in te United Kingdom c 21 Cambridge University Press 341 Generation of solitary waves by forward- and backward-step bottom forcing By DAO-HUA ZHANG AND ALLEN T. CHWANG Department of Mecanical Engineering, Te University of Hong Kong, Pokfulam Road, Hong Kong (Received 1 February 2 and in revised form 18 September 2) A finite difference metod based on te Euler equations is developed for computing waves and wave resistance due to different bottom topograpies moving steadily at te critical velocity in sallow water. A two-dimensional symmetric and slowly varying bottom topograpy, as a forcing for wave generation, can be viewed as a combination of fore and aft parts. For a positive topograpy (a bump), te fore part is a forward-step forcing, wic contributes to te generation of upstream-advancing solitary waves, wereas te aft part is a backward-step forcing to wic a depressed water surface region and a trailing wavetrain are attributed. Tese two wave systems respectively radiate upstream and downstream witout mutual interaction. For a negative topograpy (a ollow), te fore part is a backward step and te aft part is a forward step. Te downstream-radiating waves generated by te backwardstep forcing at te fore part will interact wit te upstream-running waves generated by te forward-step forcing at te aft. Terefore, te wave system generated by a negative topograpy is quite different from tat by a positive topograpy. Te generation period of solitary waves is sligtly longer and te instantaneous drag fluctuation is skewed for a negative topograpy. Wen te lengt of te negative topograpy increases, te oscillation of te wave-resistance coefficient wit time does not coincide wit te period of solitary wave emission. 1. Introduction Te study of solitary waves generated by moving objects or bottom topograpies as received muc attention since te initial eperiments of Huang et al. (1982) and te remarkable numerical findings of Wu & Wu (1982). Wu & Wu (1982) studied forced waves in a transcritical state using te generalized Boussinesq (g-b) model proposed by Wu (1981). Tey identified numerically te eistence of a train of solitions of equal size moving aead of a bump. Immediately beind te bump tere is a flat depression zone followed by a train of cnoidal-like waves. Te oter widely used teoretical model for studying te penomenon of upstream-running solitary waves is te forced Korteweg de Vries (fkdv) equation, wic is simpler tan te g-b model wit te furter assumption of waves propagating in one direction only. Because of its simplicity, tis model as been used to study various types of forcing by Akylas (1984), Cole (1985), Grimsaw & Smyt (1986) among oters and in a more general form by Wu (1987). Te teoretical models admit as eternal forcing disturbances te surface pressure and topograpy, wic are assumed to be positive or negative functions. Lee, Yates & Wu (1989) made an etensive quantitative comparison between numerical results of te fkdv and g-b models and eperiments on transcritical upstream soliton radiation. Teir comparison sowed tat te agreement between te teoretical

3 342 D.-H. Zang and A. T. Cwang models, wen te viscous effect was crudely accounted for, and te eperiments was generally very good. Tey found tat for te fkdv model, te surface pressure and bottom topograpy were entirely equivalent, wereas for te g-b model te surface pressure acted as a stronger disturbance tan te bottom bump of te same distribution, tereby producing larger waves in a sorter period. Te eperimental study of Ertekin, Webster & Weausen (1984) of a sip in a cannel indicated tat te blockage coefficient (defined as te ratio of te mid-sipsectional area to te cross-sectional area of te water cannel) was te dominant forcing factor in governing soliton generation, but not te sole one as te ull displacement also played an important role. Tis was sown in figures 9 and 11 of Ertekin et al. (1984): te blockage coefficients for plots (a) and (b) in tese figures were te same, but te displacements were different due to different draugts. Terefore, te difference between te plots solely indicated te effect of displacement on te wave generation. Grimsaw & Smyt (1986) conducted asymptotic studies on forced flows in a stratified fluid. Tey sowed tat te forcing function of te fkdv model was caracterized by two parameters: forcing amplitude G (in a single-layer flow, wen scaled by water dept, G stands for te blockage coefficient) and a positive parameter ξ tat measures te lengtscale of te forcing. Tey found, based on te mass and energy conservation argument, approimate formulas for te dept of te downstream depression and te amplitude of te upstream solitons as functions of te forcing velocity, te lengt and amplitude of te forcing. Te dependence of te wave mecanical energy and drag on te area and speed of te forcing was eplicitly calculated by Sen (1996). Smyt (1987) modelled te upstream flow by a modulated cnoidal wavetrain, not by a series of equal KdV solitary waves as in Grimsaw & Smyt (1986), and obtained improved solutions for te upstream flow. He found similar approimate formulas for te amplitude and period of soliton generation as Grimsaw & Smyt (1986). Teng & Wu (1997) studied te effect of forcing lengt on te wave amplitude and period of generation based on te Boussinesq and KdV models. Tey sowed tat wen te lengt is muc greater tan te water dept, it as little effect on te wave amplitude and period. If te lengt is of te same order as te water dept, te effect of disturbance lengt becomes important. Sen (1992) also indicated tat te lengt of disturbance migt ave an important effect on solitary wave generation. Wile te surface pressure and te bottom bump sow some difference in te strengt of disturbance in different teoretical models, te features of solitary waves so generated are very similar. However, te features of wave systems generated by positive or negative forcing functions are quite different. Grimsaw & Smyt (1986), Wu (1987), and Camassa & Wu (1991) studied solitary waves generated by a negative forcing function and gave descriptions of teir features. A numerical metod wic solves te Navier Stokes equations (or Euler equations) by direct integration wit consistent free-surface boundary conditions for time-dependent viscous (or inviscid) free-surface flows was developed by Zang & Cwang (1996, 1999). Te metod was validated by te eperimental data of Lee et al. (1989). Te computational results from te direct numerical integration of te Navier Stokes equations were compared wit te eperimental data and te numerical results of te fkdv and g-b models of Lee et al. (1989) at te transcritical range. Since viscous effects are accurately modelled in te Navier Stokes equations, te agreement of te numerical results by Zang & Cwang (1996) wit te eperimental data was ecellent and muc better tan te numerical results of te fkdv and g-b models. Te metod was employed to study te general caracteristics of soli-

4 Generation of solitary waves by forward and backward steps 343 tary waves forced by underwater moving objects including viscous effects. Te pase velocity and te period of solitary wave generation also sowed good agreement wit teoretical predictions (Zang & Cwang 1999). In te present paper, te basic mecanism underlying te features of solitary waves generated by bottom topograpies is eplored by te numerical model of Zang & Cwang (1999). Te geometry of te topograpies (analogous to te forcing function in teoretical models) is simply part of te boundaries of te calculation domain, wic can be easily and accurately simulated in te model. Te investigation as tree aims: (a) to describe te dependence of te upstream-advancing solitons on te eigt and lengt of te topograpy, (b) to eplore weter te waves generated by a bump or a ollow forcing are a superposition of te waves due to two step forcings, and (c) to eplain, wit te new findings in (b), wy te wave systems due to a positive forcing are different from tose due to a negative forcing. Descriptions of te computational metod are provided briefly in 2. Section 3 describes and discusses te results on waves generated by various bottom topograpies including positive, negative, forward-step and backward-step forcings. Quantitative comparisons between numerical solutions for te amplitude of te lead upstream wave for various forcing eigts and te approimate solutions of Smyt (1987) and Grimsaw & Smyt (1986) are made in 3.1. Section 3.2 presents te new finding tat te waves generated by a bump or a ollow forcing are a superposition of te waves due to forward- and backward-step forcings formed by cutting te bump or ollow in alf at te middle and joining te cut ends to a orizontal portion of te bottom (figure 1). Te dependence of te wave amplitude on te forcing lengt is also discussed. In 3.3, te difference between wave systems due to positive and negative forcings is eplained. Finally, conclusions are given in Outline of te computational metod Te model of Zang & Cwang (1999) wit te Euler formulation is employed in te present study. Te following provides a summary of te metod. For more detailed information refer to Zang & Cwang (1996, 1999). Te unsteady Euler and continuity equations are solved wit nonlinear kinematic and dynamic free-surface boundary conditions in a body/free-surface fitted grid. Te grid is allowed to move wit te free surface, providing a more accurate treatment of te free-surface boundary conditions. Te Cartesian coordinates (, y) of te pysical domain are transformed to te numerically generated, boundary-fitted, non-ortogonal curvilinear coordinates (ξ,η) in te computational domain, wile te velocity components (u, v) in te Cartesian coordinate system of te pysical domain remain uncanged. Te transformed equations, in a conservation form in te boundary-fitted coordinate system, are solved using a regular grid. Te second-order central difference is used to discretize te spatial differentials wile te convection terms are discretized by te quick sceme. Velocity components are updated eplicitly by te time-splitting fractional-step metod, wile pressure is updated by solving a Poisson equation for te pressure increment between two adjacent time steps. Te grid is updated at eac time step to conform to bot te body and te free surface. 3. Results and discussion A two-dimensional symmetric and slowly varying bottom topograpy, as a forcing for wave generation, can be looked upon as a combination of fore and aft parts. For

5 344 D.-H. Zang and A. T. Cwang (a) (d) L b m (b) (e) L/2 (c) ( f ) Figure 1. A sketc of bottom topograpies: (a) positive; (d) negative; (b, f) forward step; (c, e) backward step. a positive topograpy (a bump), te fore part is a forward step and te aft part is a backward step. On te oter and, for a negative topograpy (a ollow), te fore part is a backward step and te aft part is a forward step. A sketc of different bottom topograpies is sown in figure 1. Bottom topograpies used for te present numerical computations are eiter b() =± b [ ( )] m 2π 1 + cos 2 L for L/2 L/2, (1) or b() =b m sec 2 for <<. (2) Te undisturbed water dept is normalized to 1, b m is te maimum eigt of te disturbance scaled by water dept, wic is te blockage coefficient in te twodimensional case, and L is te disturbance lengt scaled by water dept. Only te eact linear resonant case is considered in te present study. Tus, te Froude number defined as F = U / g is equal to 1 for all calculations, were U is te upstream uniform inflow velocity and g te gravitational constant. Te calculation domain is of one unit lengt in te vertical direction and 18 unit lengts in te orizontal direction. Forty-one grid points are used in te vertical direction and 397 grid points in te orizontal direction. Time is scaled by /U and te time step is set to be T = Positive forcing Typical numerical results for te topograpy given by equation (2) wit b m =.1 are sown in figure 2(a). If te eigt of te topograpy is assumed to be zero wen it is less tan.4, ten te lengt of te topograpy L is approimately equal to 6.8. Tis is equivalent to setting ξ =.3 (te alf-widt of te forcing ξ 1 3.3) in te study of Grimsaw & Smyt (1986). Te general features of te upstream periodically radiating solitary waves forced by a steadily moving two-dimensional disturbance sustained at resonance are clearly sown. Te teoretical and numerical study of Grimsaw & Smyt (1986) sowed tat te forcing function of te fkdv model is caracterized by two parameters: amplitude G and a positive parameter ξ (ξ measures te lengt scale of te forcing, i.e. ξ 1 is

6 1.5 (a) Generation of solitary waves by forward and backward steps 345 L = T (b) L = T 5 Figure 2. Wave profiles generated by (a) a sec 2 -saped bump wit eigt b m =.1 and lengt L =6.8 and (b) a cosine-saped bump wit eigt b m =.1 and lengt L = 16. its alf-widt). Tese two parameters determine te area of te forcing distribution integrated over te base line, wic represents te intensity of te forcing. Wen scaled by water dept, te amplitude of te forcing function G stands for te blockage coefficient. For a fied value of G, te wider te forcing, te larger te area of te forcing distribution and te stronger te forcing intensity will be. Wen te detuning parameter = (at eact linear resonance) Grimsaw & Smyt (1986) found from teir mass and energy conservation argument tat for a broad obstacle (ξ ), te amplitude of te upstream-advancing solitary waves was determined solely by te blockage coefficient, i.e. a 1.15 G. (3) For a narrow obstacle (ξ ), te amplitude was determined by te dimensionless area of te forcing distribution, i.e. a (G Kξ 1 ) 2/3, (4) were K is te integral of te forcing function from minus infinity to plus infinity. Tey compared te solitary-wave amplitude a obtained from te approimate epression (3) to teir numerical results obtained for forcing function sec 2 (ξx) wit =, ξ =.3, and G =.5,, 1.5 and respectively. Te agreement was good. For ξ 1, Grimsaw & Smyt (1986) found tat te numerical solutions of te fkdv equation wit eiter sec 2 (ξx) or ep ( ξx) 2 forcing functions were quantitatively in agreement. Sen (1991, 1996) approimated a sort bump by Pδ() in te fkdv equation, were P is te dimensionless area of te bump and δ() iste Dirac delta function. He sowed tat, for te locally forced case, te surface wave was independent of te sape of te bottom obstruction, but instead depended on its area. Smyt (1987) modelled te upstream flow by a modulated cnoidal wavetrain and obtained an improved solution for te upstream flow. For a broad forcing (small ξ) wit =, e found tat te amplitude of solitons at te ead of te epansion fan was given as a 1.25 G. (5) To compare te present numerical results for te upstream flow wit tose given by approimate epressions (3) and (5), calculations wit te topograpy given by equation (2) and four forcing amplitudes, i.e. b m =.5,.1,.15 and.2, are carried out. Te amplitudes of te leading upstream waves are presented in table 1 togeter

7 346 D.-H. Zang and A. T. Cwang b m a, numerical results a 1.15 b m a 1.25 b m Table 1. A comparison between numerical results for te solitary-wave amplitude a and tose by te approimate epressions for a broad forcing. Te numerical results were obtained for te forcing function given by equation (2) wit F =. wit te values obtained from approimate epressions (3) and (5). Te agreement is good. Te present numerical results are between te two approimate epressions Forward- and backward-step forcings Sen (1992) indicated tat te lengt of te disturbance migt ave an important effect on te solitary wave generation. Te numerical results of Teng & Wu (1997) based on te Boussinesq and KdV models sowed tat wen te lengt is sufficiently large (i.e. muc greater tan te water dept), it as little effect on te wave amplitude and period, and te dominant forcing factor is indeed te blockage coefficient. However, if te lengt is relatively sort (i.e. same order as te water dept), te blockage coefficient is no longer te sole dominant forcing factor as te effect of te disturbance lengt also becomes important: te sorter te disturbance lengt, te weaker te forcing. In te present study, flows over positive cosine-saped topograpies given by equation (1) wit eigt b m =.1 and lengt L = 2, 4, 8, 12, and 16 respectively are calculated. It is found tat for L less tan 12, te amplitude of te upstreamadvancing solitary waves increases, but te period of wave generation decreases, as L increases. Wen L is equal to or greater tan 12, te waves generated by te topograpy are almost te same: tey are independent of te disturbance lengt. Here, only te results for L = 16 are presented in figure 2(b). Te wave systems sown in figure 2 are generated by two different positive topograpies wit te same forcing amplitude but different forcing lengt. Altoug te number of upstream-advancing solitary waves, teir amplitude and period of generation are different, te general features of te wole wave system are te same: it consists of upstream-advancing solitary waves and a depressed water region of nearly uniform dept, followed by a train of cnoidal-like waves downstream. Te upstream and downstream radiating waves do not interact eac oter, indicating tat te fore and aft parts of te topograpy contribute separately to upstream and downstream radiating waves. Consider a positive topograpy (figure 1a) tat cut at its maimum eigt at te middle and ten joined to a orizontal bottom to form a forward step (figure 1b) and a backward step (figure 1c). In figure 3, waves generated by tese cosine-saped forward- and backward-step forcings wit b m =.1 and L/2 = are presented. It is interesting to see tat te forward-step forcing only generates solitary waves wit practically no depressed water region and trailing wavetrain. Te ecess mass for eac soliton is 4(a/3) 1/2, were a is te solitary wave amplitude. In te present case, a.5, and 5.5 solitons were generated witin T = 2. Te total ecess mass of te 5.5 solitons is 5.5 4(.5/3) 1/2 = 9. Te area swept by te disturbance is b m FT =.1 2 = 2, and so te amount of mass tat flows out downstream

8 1.5 (a) Generation of solitary waves by forward and backward steps 347 Forward-step forcing T (b) Backward-step forcing T 5 Figure 3. Wave profiles generated by (a) forward-step- and (b) backward-step-topograpies wit eigt b m =.1 and te lengt of te cosine portion L/2 =. is ten 11. If it is assumed tat te water surface for > is flat and te water dept is, te outflow velocity at te downstream boundary must be larger tan te inflow velocity F =. By mass conservation, te ecess velocity δu at te downstream boundary satisfies δut( b m ) = 11. Tis gives δu =.6. Te total outflow velocity at te downstream boundary obtained from te numerical calculation is 5. Tis value seems to be rational from te roug analysis of mass conservation. In contrast, te backward-step forcing only generates te depressed water region and te trailing wavetrain witout te upstream-advancing solitary waves. Since te lengts of te forward- or backward-step forcing disturbance are semi-infinite, te combined wave system is similar to tat generated by a long forcing as sown in figure 2(b). Tis indicates tat te wave systems generated by a localized long forcing disturbance can be tougt of as a combination of wave systems generated by a forward- and a backward-step forcing disturbance. Tis finding is important in understanding te role of different parts of te forcing in wave generation, and it will elp to eplain wy te caracteristics of wave systems generated by a negative forcing are different from tose by a positive forcing. Tis will be discussed in te following section Negative forcing Teoretical studies ave sown tat features of wave systems generated by a negative forcing function are quite different from tose of a positive forcing. Te following detailed and vivid description of te features of negative compared to positive forcing was given by Wu (1987): (a) te generation period of solitary waves is sligtly longer; (b) te instantaneous drag fluctuation is skewed in pase (suggesting tat it as two or more armonics); (c) te depressed water region beind te forcing is not as uniform in dept, owing to a succession of very small waves traversing backward across te region, reflected from te forcing; and (d) te local wave continues to be ecited to a relatively quite large amplitude witin te region of te (negative) forcing before it breaks away, and ten settles to a smaller eigt, as if overcoming a tresold. Te above description summarizes te overall features of wave systems generated by a negative forcing from teoretical models. In te present study, flow over a negative cosine-saped topograpy given by equation (1) wit lengt L = 2 is calculated and te wave system so generated is sown in figure 4 togeter wit te wave-resistance coefficient of te topograpy. It is seen tat te features of te wave systems are different from tose of a positive forcing and are in accord wit te description by Wu (1987). A negative forcing (figure 1d) can be looked upon as a concave backwardstep forcing (figure 1e) followed by a concave forward-step forcing (figure 1f). In contrast to a positive forcing, te forward-step forms te aft part of te forcing

9 348 D.-H. Zang and A. T. Cwang 1.5 (a) L = T 1 (b).4.2 C DW T Figure 4. (a) Wave profiles and (b) wave resistance coefficient C DW due to a negative cosine-saped topograpy wit eigt b m =.1 and lengt L =2. disturbance wile te backward-step forms te fore part. As discussed in te previous section, a forward-step forcing contributes to te generation of upstream-advancing solitary waves. In te case of a negative forcing, waves start to be generated just beind te aft part, first propagating downstream a small distance and ten upstream across te negative forcing region. In te meantime, waves are generated at te fore part, radiating downstream and merging wit te upstream-advancing waves. Te wave continues to grow to relatively large amplitude witin te region of te negative forcing as te energy acquired by local fluid at te rate of work done by moving disturbance accumulates. Wen te wave reaces a certain tresold amplitude, te increase in pase velocity wit increasing amplitude will be sufficient to make te wave radiate out and finally settle to an upstream-advancing solitary wave of smaller eigt. However, in te case of a positive forcing, waves generated at te fore and aft parts radiate upstream and downstream respectively witout mutual interaction. Te difference in te process of wave generation and propagation between positive and negative forcings makes te generation period of solitary waves sligtly longer in te case of a negative forcing. Wave resistance of te forcing disturbance varies periodically wit time, and te variation is related to te relative positions of te wave and te forcing. Wen a wave starts to be generated beind te aft part of te forcing, te corresponding wave resistance is small and positive. As te wave moves upstream, te wave resistance increases slowly at first, and ten rapidly reaces a maimum wen te igest elevation of te free surface is at te trailing edge of te forcing disturbance. Tis makes te resistance curve skew. As te wave moves furter upstream away from te trailing edge, te wave resistance decreases rapidly and reaces zero wen te wave is at te middle of te forcing region. It continues to drop to a negative maimum wen te wave is at te leading edge of te forcing, and ten starts to increase again as te wave breaks away from te forcing. Tis is followed by anoter solitary wave going troug te same cycle. Altoug te instantaneous wave drag fluctuation is skewed, te periodic nature of te solitary wave generation is clearly locked to te evolution of te wave resistance coefficient. For positive forcings, wen L is not sufficiently large, an increase or decrease of L would strengten or weaken te intensity of forcing, resulting in a cange of te wave amplitude and period of wave generation, but no cange in te features of wave generation. Tis does not seem to be true for a negative forcing. Figure 5 sows te wave system generated by a negative cosine-saped distribution wit lengt L = 4 togeter wit te wave-resistance coefficient of te forcing. It is seen from figure 5 tat te features of te wave system are somewat different from tose

10 (a) 1.5 Generation of solitary waves by forward and backward steps 349 L = T 1 (b).8.4 C DW T 3 4 Figure 5. As figure 4 but for lengt L =4. wit lengt L = 2 sown in figure 4. Wen L becomes larger, te nonlinear effect as a significant influence on te incipient emission time. Te first wave generated at te aft part of te forcing disturbance dies out. Te second wave probably gains mass and energy from te dying first wave as it propagates upstream. After tis incipient emission, te regular sequential emission will ten be ensured wit every second incubating wave in te vicinity of te aft part of te forcing emitting a solitary wave. In tis case, te oscillation of te wave-resistance coefficient wit time does not coincide wit te period of solitary wave emission. Te emission occurs at every second peak of te wave-resistance curve. Tese salient features of te process were first eplored by Camassa & Wu (1991) based on numerical solutions of te fkdv model. Adopting te fkdv model and using two of its steady forced solitary wave solutions as primary flows, te stability of tese two transcritical steady motions was investigated. It was found tat if te rest state was cosen as te initial condition, te period of oscillation in te wave-resistance coefficient coincides wit te period of birt of upstream-running solitary waves, wereas te incipient emission of an upstream-advancing solitary wave was delayed wit a non-zero initial condition due to te nonlinear effect. Te smaller te initial departure from te stationary response, te longer te local fluctuation would last before te first solitary wave became mature and was emitted. 4. Conclusions Te features of water waves generated by different bottom topograpies moving steadily wit te critical velocity troug a layer of sallow water are studied numerically wit te Euler equations. It is found tat a forward-step forcing only generates upstream-running solitary waves and a backward-step forcing only generates a downstream-radiating depressed water region followed by a cnoidal wavetrain. Tus, a positive forcing disturbance a combination of a forward-step forcing as te fore part and a backward-forcing as te aft part would generate wave systems at its fore and aft parts respectively radiating out in opposite directions (upstream and downstream) separately witout interaction. In contrast, a negative forcing disturbance a combination of a concave backward-step forcing as te fore part and a concave forward-step forcing as te aft part would generate upstream-running waves from its aft part (forward-step forcing) and downstream-radiating waves from its fore part (backward-step forcing). Tese two wave systems would interact in te negative forcing region causing te resultant waves to be quite different from tose due to a positive forcing.

11 35 D.-H. Zang and A. T. Cwang Tis researc was sponsored by te Hong Kong Researc Grants Council under Grant Numbers HKU 568/96E, HKU 766/97E and NSFC/HKU 8. REFERENCES Akylas, T. R On te ecitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mec. 141, Camassa, R. & Wu, T. Y Stability of forced steady solitary waves. Pil. Trans. R. Soc. Lond. A 337, Cole, S. L Transient waves produced by flow past a bump. Wave Motion 7, Ertekin, R. C., Webster, W. C. & Weausen, J. V Sip-generated solitons. Proc. 15t Symp. Naval Hydrodyn., Hamburg, pp Ertekin, R. C., Webster, W. C. & Weausen, J. V Waves caused by a moving disturbance in a sallow cannel of finite widt. J. Fluid Mec. 169, Grimsaw, R. H. J. & Smyt, N. F Resonant flow of a stratified fluid over topograpy. J. Fluid Mec. 169, Huang, D. B., Sibul, O. J., Webster, W. C., Weausen, J. V., Wu, D. M. & Wu, T. Y Sips moving in te transcritical range. Proc. Conf. on Beavior of Sips in Restricted Waters, Vol. 2, pp Bulgarian Sip Hydrodynamic Center. Lee, S. J., Yates, G. T. & Wu, T. Y Eperiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mec. 199, Mei, C. C Radiation of solitons by slender bodies advancing in a sallow cannel. J. Fluid Mec. 162, Sen,S.S.P.1991 Locally forced critical surface waves in cannels of arbitrary cross section. Z. Angew. Mat. Pys. 42, Sen,S.S.P.1992 Forced solitary waves and ydraulic falls in two-layer flows. J. Fluid Mec. 234, Sen,S.S.P.1996 Energy distribution for waves in transcritical flows over a bump. Wave Motion 23, Smyt, N. F Modulation teory solution for resonant flow over topograpy. Proc.R.Soc. Lond. A49, Teng, M. H. & Wu, T. Y Effect of disturbance lengt on resonantly forced nonlinear sallow water waves. Intl J. Offsore and Polar Engng 7, Wu, D. M. & Wu, T. Y Tree-dimensional nonlinear long waves due to moving surface pressure. Proc. 14t Symp. on Naval Hydrodyn., pp National Academy Press. Wu,T.Y.1981 Long waves in ocean and coastal waters. J. Engng Mec. Div. ASCE 17, Wu,T.Y.1987 Generation of upstream advancing solitons by moving disturbances. J. Fluid Mec. 184, Zang, D. H. & Cwang, A. T Numerical study of nonlinear sallow water waves produced by a submerged moving disturbance in viscous flow. Pys Fluids 8, Zang, D. H. & Cwang, A. T On solitary waves forced by underwater moving objects. J. Fluid Mec. 389,