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Jul 22, 2009 - the regular perturbation of the free surface and the bottom about their ... analysis leads to a reflection coefficient, Kr, defined as the reflected wave height divided ..... The writers would like to thank Seung-nam Seo, who carried.
WATER WAVES OVER RIPPLES By Robert A. Dairymple, 1 M . ASCE a n d James T. Kirby, 2 A. M . ASCE ABSTRACT: The interaction of small amplitude water waves with a patch of bottom ripples is studied using the Boundary Integral Equation Method (BIEM). Normal and oblique wave incidence is examined for reflection coefficients and the laboratory data of Davies and Heathershaw are compared to the numerical results, with the conclusion that the analytical model of Davies and Heathershaw overestimates the reflection coefficient at resonance. Comparison of BIEM results for oblique incidence is made with an extension of the recent work of Mei showing good agreement, even for large amplitude ripples. INTRODUCTION

In a series of papers, Davies a n d Heathershaw (2-6) studied the propagation of normally incident water waves over ripples that m a y extend over infinite distances or are confined to patches of length 2L on an otherwise constant depth seabed. The mathematical analysis is based on the regular perturbation of the free surface a n d the bottom about their mean positions using the same small parameter. For a ripple patch, their analysis leads to a reflection coefficient, Kr, defined as the reflected wave height divided by the incident height, given as

2bk sin 2kL Kr

~

(-i)'"l 7 (1)

sinh 2kh + 2kh,

#

"

and the transmission coefficient, KT , is unity for an integer n u m b e r of ripples of sinusoidal shape and amplitude b. Here, k and (, are t h e wave numbers of the surface wave a n d the bottom ripples, respectively, h is the mean water depth (assumed constant), and m is the n u m b e r of ripples in the patch. See Fig. 1 for a definition sketch. It is interesting to note that the first parenthesis represents the reflection from a rectangular obstacle of length 2L (13), corresponding to the horizontal extent to which the bottom is perturbed, a n d of height b. This portion of the reflection coefficient accounts for all the zeros a n d the brackets are due to the ripples. The maximum reflection occurs for 2k/t equal to unity. This is the Bragg scattering condition corresponding to the case in which the ripples have half the wave length of the surface waves, a n d thus a reflecting wave is reinforced exactly by the reflected portions of the previous waves. 'Prof., Dept. Civ. Engrg., College of Engrg. and College of Marine Studies, Univ. of Delaware, Newark, DE 19716. 2 Asst. Prof., Coastal and Oceanographic Engrg. Dept., Univ. of Florida, Gainesville, FL 32611. Note.—Discussion open until August 1, 1986. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 17, 1985. This paper is part of the Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 112, No. 2, March, 1986. ©ASCE, ISSN 0733-950X/ 86/0002-0309/$01.00. Paper No. 20474. 309

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27T/k

Zir/1

^%^%^^%^K^^. I 2L J FIG. 1 .—Definition Sketch of Waves over Ripple Patch, m = 4

(See Ref. 1 for an application of Bragg's law to obliquely incident water waves.) This is denoted the resonant condition. There are, however, several shortcomings of the Davies and Heathershaw result. 1. As the number of ripples in the patch increases to infinity at the Bragg condition, Kr does as well. 2. The transmission coefficient is always unity, even as the ripple bed becomes very long. At the order of the analysis, K, + K ! ? 4 1. 3. Experiments by Heathershaw (6) show that the wave height clearly decreases across a ripple patch for normal wave incidence, and hence, KT must be less than unity. To rectify the second and third problem, Davies (2) has developed an approximate technique that fits the laboratory data better but does not satisfy the first-order governing equations. Recently Mei (14), using a multiple scales approach and including directly at the first order the reflected wave, developed a more accurate first-order theory, valid near resonance, which eliminates all three problems. Comparison of his normal incident results to the experiments of Heathershaw show excellent agreement. His results will be extended here for obliquely incident waves over a ripple patch. This paper utilizes the Boundary Integral Equation Method (BIEM) to study both the effect of the finite amplitude of the ripples and oblique wave incidence, thus extending the work of Davies and Heathershaw. With the BIEM approach, the problem is linear in all the boundary conditions but, by placing the bottom boundary elements directly on the ripples themselves, the bottom is represented exactly in the model. In the analytical models, the perturbation of the bottom condition about the mean location of the bottom introduces higher order terms, which are discussed later. THEORETICAL CONSIDERATIONS

BIEM.—The BIEM has been used to solve a variety of problems in theoretical hydrodynamics and elasticity theory (e.g., 7, 11). For water 310

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waves the BIEM has been used by Yeung (18), Liu and coworkers (10,12,17), and Raichlan a n d Lee (15). In this study the constant element model used by Kirby and Dalrymple (8) w a s improved by the inclusion of analytical lateral b o u n d a r y elements (18). More sophisticated models exist (12); however as long as a sufficient n u m b e r of elements are used, the accuracy is the same. The basis of the BIEM lies in the representation of the velocity potential at any location in a fluid in terms of the values of and its normal derivative on the boundaries (9), b y using Green's theorem: 1 4>(*o/Zo) = - — 4IT Js

86 G(r)fdn

dG — 6 ds 8n

(2)

in which G(r) is a simple Green's function a n d r is the distance from the arbitrary interior point (x0,z0) to the boundary, S. The first part of the integral represents a distribution of sources along t h e boundaries, while the second part is a distribution of doublets. In order to obtain 6(x0 ,z 0 ), the boundary values of c(> a n d d/3n m u s t be k n o w n . For a properly posed boundary value problem, either 6 or 86/dn is prescribed on all the boundaries and therefore only t h e u n k n o w n 6 or dS/dn is to be determined. As the cited references provide detailed explanations of the BIEM technique, only a brief discussion is provided here. For a linear water wave, Eq. 2 is written for a surface point, (x,, z s ), 1 2TT JS

36 G(r)fdn

dG — 6 ds dn

(3)

For normal incidence, G(r) = 1/r, a n d for oblique incidence G{r) = -K0(Kr), w h e r e \ = k sin 6, the w a v e n u m b e r component in t h e y direction, a n d K0( ) is the modified Bessel function of the second kind. For the u p w a v e a n d d o w n w a v e lateral boundaries, w h e r e t h e bottom is assumed constant, the velocity potential for the linear wave can be represented analytically using an eigenfunction expansion with u n known coefficients (18), with the exception of the incident wave. The integration in Eq. 3 is then carried out numerically for these boundaries using Gaussian quadrature (including a special form for logarithmic functions). The advantage of this hybrid m e t h o d is that fewer elements are needed a n d the reflection a n d transmission coefficients are obtained directly. For the results presented here, the boundaries were represented by 122 elements (60 at the surface a n d 60 o n the bottom). By placing the lateral boundaries sufficiently far from the ripple patch, only the progressive waves in the eigenfunction expansion were necessary. Oblique Incidence.—By consideration of Bragg's scattering it w a s anticipated (1) that the wave number component normal to the ripple patch was instrumental in determining scattering a n d therefore the resonant (or maximum) reflection could be predicted by (2k cos 6) , =1

=

(4)

311

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in which 0 is the angle of incidence measured from the normal direction. For given wave numbers k and t, the maximum reflection occurs at angles greater than zero. Mitra and Greenberg (15) show this for a doubly infinite rippled bed. However Mei's (14) result for a semi-infinite ripple patch does not follow the Bragg condition. In Appendix I an extension of Mei's result to a finite patch of ripples with oblique wave incidence predicts that the maximum reflection occurs for normal incidence for a given k and i. One of the goals of this study was to determine which of the two results is correct. From Appendix I for waves incident to angle 9 to the x axis, the reflection coefficient for a ripple patch is sin 2kL Kr =

(5)

1/2

2

— ) sin 2fcL + "0/

-

L\"6.

2

- i cos 2kL

for 0,/ills > 1. Here ukb

/cos 26

(6)

ni = smhlkh \cos 2 8 *

2

= l ^

9i

cos2 6,

(7)

in which Cg is the wave group velocity and ft = cr — w

(8)

in which

k+i

3 Q.

e


= 2*i 0 a„e",f", a„ = «64-n, / = 0.1995 nr 1 ), Showing Approximate Locations of Higher BottomInduced Harmonics, for (2fc/€) = 0.48

8. For both models a zero reflection coefficient occurs for 45°, however for the first case the BIEM predicts that the maximum reflection occurs at 17.5°, approximately half the value predicted by oblique Bragg scattering (Eq. 4), which is clearly not applicable. For the second case (XI < lio) both models predict the maximum scattering at normal incidence.

0.40

0.30

0.20

BIEM Eqn ( 9 ) m--2

0,10

(f).o,s.

\\

t> = 0,05 in

1V 10

20

30

40

20

50

FIG. 7.—Reflection Coefficient for Obliquely Incident Waves over Ripple Patch; BIEM versus Eq. 5. Dots Show Result of Eq. 1

30

40

50

FIG. 8.—Reflection Coefficient for Obliquely Incident Waves over Ripple Patch; BIEM versus Eq. 9. Dot Shows Result of Eq. 1

315

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(At zero degrees, normal incidence, the Davies solution, Eq. 1, is also shown.) CONCLUSIONS

For water waves propagating over a ripple patch the linear solution of Davies and Heathershaw overpredicts the reflection coefficient at the Bragg condition (2k/€ equal to one), particularly as m increases. The multiple scales approach of Mei is more accurate and allows for oblique incidence, as given in Appendix II. For a ripple patch and for values of b/h u p to 0.4, the reflection coefficients given by Eqs. 5 a n d 9 are accurate, provided fl/fto < 1/ a s s h o w n by comparison to data for normal incidence (14) and by comparison to the numerical results of this study. Classical Bragg scattering, as given by Eq. 4, does not hold for oblique wave incidence on a ripple patch. The BIEM is a useful m e t h o d for determining wave properties a n d provides either a useful check to analytical solutions or actual data. ACKNOWLEDGMENT

This study was partially funded by the Office of Naval Research, Coastal Sciences. The writers would like to thank Seung-nam Seo, w h o carried out some of the calculations in this paper. APPENDIX I.—OBLIQUE WAVE REFLECTION FROM RIPPLE PATCH

Following Mei (13) closely, w e require small amplitude waves over a ripple patch to satisfy the following equations dA dA BA — + C„s cos 6 — + C„ sin 9 — = -i£la cos 26B 8 dt dx dy dB dt

dB dB C„s cos 0 — + C,s sin 6 — = -i'ft„ cos 26A dx dy

(10)

(11)

in which A and B are the amplitudes of the forward and backward scattered waves of frequency co where, for example, the forward scattered wave is given by w cosh k(h + z) .,,,, . , ,. . . ,. +(x,y,z) = — - • d/lei(p*coSe*fa+*Sii.ey-o.o 2co cosh kh and

co2 = gk tanh kh :

(ft) (13)

(okb A =

(14) v ; 2sinh2fc/j which is an important parameter measuring the a m o u n t of interaction between the waves a n d ripples, b is the ripple amplitude, a n d h is the mean water depth. Combining these equations we have 316

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[id

d

2 — + C„s sin 6 — - Cl + ill cos 2 20 g cos 0 —. dt by) dx2

B = °



If we assume that co and k are chosen to be resonant with the ripples, then 2k = t (the ripple wave number) and 0,2 = g

\i)tanh

(16)

(jjr

Waves of different frequencies (nonresonant) will be represented by frequencies a = co + fl, such that fl measures the amount of detuning and it is assumed to be small. In front of the ripple patch, which extends from x = 0 to Lp, the incident wave amplitude is A _

A „i(Kcos8x+Ksin(ty-nt)

H 7\

in which K = 0,/Cg, the small correction to the wave number due to the small change in o\ The reflected wave is B = B n g'(~ K c o s 9 : , ^ + K s i n e J'~ n , )

(]»]

Downwave of the patch, x > Lp A = A

giCKcosOi+Ksiney-fU)")

B =0



i

Over the patch A = ^(x)e'' (Ksiney_nt) B = B(x)e

(20)

Substituting Eq. 20 into Eq. 15 yields

£ +p )(i;" in which

2

k =\ —

fi^ For 0 < x < Lp, we assume A(x) = Ax sin kx + A2 cos kx B(x) = Bx sin kx + B2 cos kx 317

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Using Eqs. 10 and 11, a n d the b o u n d a r y conditions, at the u p w a v e a n d downwave ends of the ripple patch, w e find 2

B0

K: = —

sin khv

:

1/2

A0

(24)

2

sin kL„ +

cos kL,

or with Lp = 2L, Eq. 5 results. Case 2.—H < ill For il < ill, A(x) and B(x) are assumed to vary as hyperbolic sines and cosines as in Mei's analysis. A n analysis similar to the preceding one yields,

s n h ; IkL 2

il\ —

ill/

, 2

sinh 2A:L +

1-

fil [ill

1/2

(25)

2

cosh 2kL

APPENDIX II.—REFERENCES

1. Dalrymple, R. A., and Fowler, J. F., "Bragg Scattering by Pile-Supported Structures," Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, Vol. 108, No. WW3, 1982, pp. 426-429. 2. Davies, A. D., "The Reflection of Wave Energy by Undulations on the Seabed," Dynamics of Atmos. and Oceans, Vol. 6, 1982, pp. 207-232. 3. Davies, A. D., "On the Interaction between Surface Waves and Undulations on the Seabed," /. Marine Research, Vol. 40, No. 2, 1982, pp. 331-368. 4. Davies, A. D., and Heathershaw, A. D., "Surface Wave Propagation Over Sinusoidaily Varying Topography: Theory and Observation," Part. I, II, Report No. 159, Institute of Oceanographic Sciences, Taunton, 1983. 5. Davies, A. D., and Heathershaw, A. D., "Surface Wave Propagation Over Sinusoidaily Varying Topography," /. Fluid Mech., Vol. 144, 1984, pp. 419443. 6. Heathershaw, A. D., "Seabed-Wave Resonance and Sand Bar Growth," Nature, Vol. 296, No. 5855, 1982, pp. 343-345. 7. Jaswon, M. A., and Ponter, A. R., "An Integral Equation Solution of the Torsion Problem," Proc. Royal Society, Vol. 273, Series A, 1963, pp. 237-246. 8. Kirby, J. T., and Dalrymple, R. A., "Propagation of Obliquely Incident Water Waves Over a Trench," /. Fluid Mech., Vol. 133, 1983, pp. 47-63. 9. Lamb, Sir H , Hydrodynamics, Dover Press, New York, 1945. 10. Lennon, G. P., Liu, P. L.-F., and Ligget, J. A., "Boundary Integral Solutions of Water Wave Problems," Journal of the Hydraulics Division, ASCE, Vol. 108, No. HY8, 1982, pp. 921-931. 11. Ligget, J. A., "Location of the Free Surface in Porous Media," Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY4, 1977, pp. 353-365. 12. Liu, P. L.-F., and Abbaspour, M., "An Integral Equation Method for the Diffraction of Oblique Waves by an Infinite Cylinder," Int. J. Num. Methods Engrg., Vol. 18, No. 10, 1982, pp. 1497-1504. 13. Mei, C. C , "Weak Reflection of Water Waves by Bottom Obstacles," Journal of the Engineering Mechanics Division, ASCE, Vol. 95, No. EMI, 1969, pp. 183194. 14. Mei, C. C , "Resonant Reflection of Surface Water Waves by Periodic Sandbars," /. Fluid Mech., Vol. 152, 1985, pp. 315-336. 15. Mitra, A., and Greenberg, M., "Slow Interactions of Gravity Waves and a Corrugated Sea Bed," /. Applied Mech., Vol. 51, 1984, pp. 251-255. 318

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16. Raichlan, F., and Lee, J. J., "An Inclined-Plate Wave Generator," Proc. 16th Int. Conf. Coastal Engrg., ASCE, Hamburg, 1978, pp. 388-399. 17. Salmon, }. R., Liu, P. L.-F., and Ligget, J. A., "Integral Equation Method for Linear Water Waves," Journal of the Hydraulics Division, ASCE, Vol. 106, No. HY12, 1980, pp. 1995-2010. 18. Yeung, R. W., "A Hybrid Integral Equation Method for Time Harmonic Free Surface Flow," Proc. 1st Int. Conf. on Num. Ship Hydrod., 1975, pp. 581-607.

319

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