1678:
1051:
1673:{\displaystyle {\begin{aligned}\varphi \,=\,&\left\{\,\varphi _{b}\,-\,{\frac {1}{2}}\,(z+h)^{2}\,{\frac {\partial ^{2}\varphi _{b}}{\partial x^{2}}}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,{\frac {\partial ^{4}\varphi _{b}}{\partial x^{4}}}\,+\,\cdots \,\right\}\,\\&+\,\left\{\,(z+h)\,\left_{z=-h}\,-\,{\frac {1}{6}}\,(z+h)^{3}\,{\frac {\partial ^{2}}{\partial x^{2}}}\left_{z=-h}\,+\,\cdots \,\right\}\\=\,&\left\{\,\varphi _{b}\,-\,{\frac {1}{2}}\,(z+h)^{2}\,{\frac {\partial ^{2}\varphi _{b}}{\partial x^{2}}}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,{\frac {\partial ^{4}\varphi _{b}}{\partial x^{4}}}\,+\,\cdots \,\right\},\end{aligned}}}
2764:
4530:
2403:
5277:
3103:
2759:{\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}\,&+\,{\frac {\partial }{\partial x}}\,\left\,=\,{\frac {1}{6}}\,h^{3}\,{\frac {\partial ^{3}u_{b}}{\partial x^{3}}},\\{\frac {\partial u_{b}}{\partial t}}\,&+\,u_{b}\,{\frac {\partial u_{b}}{\partial x}}\,+\,g\,{\frac {\partial \eta }{\partial x}}\,=\,{\frac {1}{2}}\,h^{2}\,{\frac {\partial ^{3}u_{b}}{\partial t\,\partial x^{2}}}.\end{aligned}}}
6165:
190:
6186:
4542:
2047:
970:
3361:
5266:
3530:
6175:
2863:
38:
1858:
556:
3166:
3098:{\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}\,-\,gh\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}\,-\,gh\,{\frac {\partial ^{2}}{\partial x^{2}}}\left({\frac {3}{2}}\,{\frac {\eta ^{2}}{h}}\,+\,{\frac {1}{3}}\,h^{2}\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}\right)\,=\,0.}
2042:{\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}\,&+\,u\,{\frac {\partial \eta }{\partial x}}\,-\,w\,=\,0\\{\frac {\partial \varphi }{\partial t}}\,&+\,{\frac {1}{2}}\,\left(u^{2}+w^{2}\right)\,+\,g\,\eta \,=\,0,\end{aligned}}}
965:{\displaystyle {\begin{aligned}\varphi \,=\,&\varphi _{b}\,+\,(z+h)\,\left_{z=-h}\,+\,{\frac {1}{2}}\,(z+h)^{2}\,\left_{z=-h}\,\\&+\,{\frac {1}{6}}\,(z+h)^{3}\,\left_{z=-h}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,\left_{z=-h}\,+\,\cdots ,\end{aligned}}}
3356:{\displaystyle {\frac {\partial ^{2}\psi }{\partial \tau ^{2}}}\,-\,{\frac {\partial ^{2}\psi }{\partial \xi ^{2}}}\,-\,{\frac {\partial ^{2}}{\partial \xi ^{2}}}\left(\,3\,\psi ^{2}\,+\,{\frac {\partial ^{2}\psi }{\partial \xi ^{2}}}\,\right)\,=\,0,}
3798:
4462:
While the
Boussinesq equations allow for waves traveling simultaneously in opposing directions, it is often advantageous to only consider waves traveling in one direction. Under small additional assumptions, the Boussinesq equations reduce to:
241:. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation.
4141:
4549:
For the simulation of wave motion near coasts and harbours, numerical models – both commercial and academic – employing
Boussinesq-type equations exist. Some commercial examples are the Boussinesq-type wave modules in
3470:
3415:
4751:"Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond"
2353:
3519:
2408:
1863:
1056:
561:
2183:
2117:
1715:
3675:
321:
Thereafter, the
Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the resulting
1015:
3891:
4278:
519:
4188:
4755:
4545:
Faster than real-time simulation with the
Boussinesq module of Celeris, showing wave breaking and refraction near the beach. The model provides an interactive environment.
3988:
1850:
5905:
3575:
4328:
4208:
4016:
3911:
2240:
1039:
5895:
4954:
2850:
2299:
2272:
1801:
374:
2382:
1744:
548:
447:
3598:
4348:
4228:
4036:
3943:
3850:
3824:
3154:
3134:
2811:
2791:
2205:
2139:
2073:
467:
414:
394:
4852:
Kirby, J.T. (2003). "Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents". In Lakhan, V.C. (ed.).
4574:
of the model equations. Scientific reviews and intercomparisons of several
Boussinesq-type equations, their numerical approximation and performance are e.g.
3654:. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid
4533:
A simulation with a
Boussinesq-type wave model of nearshore waves travelling towards a harbour entrance. The simulation is with the BOUSS-2D module of
45:
with a
Boussinesq-type model. The waves propagate over an elliptic-shaped underwater shoal on a plane beach. This example combines several effects of
31:
5811:
4055:
4604:"Tous les ingénieurs connaissent les belles expériences de J. Scott Russell et M. Basin sur la production et la propagation des ondes solitaires"
5226:
4783:
6221:
5458:
4483:
5348:
4947:
6053:
5480:
5368:
3426:
5900:
5171:
4923:
4861:
4842:
5318:
4608:"All engineers know the beautiful experiments of J. Scott Russell and M. Basin on the generation and propagation of solitary waves"
4498:
3374:
5358:
4769:
4362:
which are referred to as
Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as
6178:
6088:
5074:
5761:
4750:
4468:
6168:
4940:
4296:
2389:
322:
128:
2304:
3481:
2825:
5216:
4731:"Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire"
5276:
102:
2148:
2082:
5413:
3157:
5313:
1686:
5948:
5353:
3108:
From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the
30:
This article is about the
Boussinesq approximation for waves on a free-moving fluid surface. For other uses, see
6078:
5453:
5443:
5383:
5019:
4989:
4559:
4555:
4534:
4502:
4393:
4042:
3662:
3639:
3616:
3113:
194:
181:
is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).
136:
4517:
Besides solitary wave solutions, the
Korteweg–de Vries equation also has periodic and exact solutions, called
6211:
6115:
6098:
5935:
5428:
5293:
5231:
5221:
5114:
4529:
4366:
Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them
4307:
2814:
237:
from the flow equations, while retaining some of the influences of the vertical structure of the flow under
160:
140:
46:
3793:{\displaystyle c^{2}\,=\;gh\,{\frac {1\,+\,{\frac {1}{6}}\,k^{2}h^{2}}{1\,+\,{\frac {1}{2}}\,k^{2}h^{2}}},}
6110:
6048:
5475:
5161:
4288:
326:
5943:
5925:
5433:
5328:
4963:
4567:
4450:
4439:
4417:
dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
1018:
978:
252:). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.
4558:. Some of the free Boussinesq models are Celeris, COULWAVE, and FUNWAVE. Most numerical models employ
3859:
6216:
6130:
5963:
5666:
5523:
5388:
5099:
4883:
4563:
4240:
1764:
483:
225:
the difference between crest and trough elevation), and the mean water depth is 5 m, while the
6125:
6010:
6005:
5731:
5403:
5363:
5079:
4359:
4153:
2243:
1804:
1042:
314:
306:
294:
290:
249:
144:
98:
3956:
1814:
6068:
5781:
5771:
5736:
5636:
5621:
5518:
4899:
2251:
2220:
478:
298:
268:
94:
3536:
226:
6150:
6140:
6083:
6063:
5746:
5711:
5646:
5626:
5616:
5498:
5186:
5044:
4919:
4914:(1972). "Equations for water waves and the approximations behind them". In Meyer, R.E. (ed.).
4857:
4838:
4779:
4746:
4726:
4313:
4193:
4001:
3896:
2225:
1768:
1024:
302:
245:
90:
6105:
6073:
6043:
5852:
5837:
5706:
5641:
5533:
5448:
5378:
5303:
5084:
5054:
4984:
4979:
4911:
4891:
4871:
4821:
4809:
4704:
4487:
4472:
4407:
3950:
474:
310:
279:
260:
202:
198:
178:
2835:
2277:
2257:
1774:
347:
5910:
5806:
5756:
5721:
5681:
5573:
5543:
5393:
5343:
5253:
5211:
5144:
5069:
5029:
4428:
2358:
1720:
524:
423:
206:
173:
While the Boussinesq approximation is applicable to fairly long waves – that is, when the
148:
5265:
4887:
3580:
17:
6020:
6015:
5920:
5915:
5751:
5691:
5686:
5418:
5308:
5129:
5064:
5039:
4571:
4435:
4333:
4213:
4021:
3928:
3922:
3835:
3809:
3139:
3119:
2820:
Under some additional approximations, but at the same order of accuracy, the above set
2813:. When the right-hand sides of the above equations are set to zero, they reduce to the
2796:
2776:
2385:
2190:
2124:
2058:
452:
399:
379:
341:
66:
233:
The essential idea in the Boussinesq approximation is the elimination of the vertical
6205:
6190:
6038:
5958:
5847:
5766:
5741:
5676:
5606:
5513:
5408:
5285:
5206:
5166:
5139:
5049:
4999:
4903:
4825:
4789:
4686:
4421:
3655:
3643:
3109:
2142:
2076:
264:
210:
121:
78:
58:
4837:. Cambridge Texts in Applied Mathematics. Vol. 19. Cambridge University Press.
6185:
6145:
6093:
6033:
5984:
5862:
5857:
5832:
5816:
5791:
5508:
5398:
5338:
5124:
5034:
5009:
4730:
4518:
4443:
2829:
2208:
1808:
189:
4541:
6135:
5867:
5796:
5661:
5601:
5568:
5558:
5553:
5438:
5373:
5333:
5323:
5298:
5181:
5154:
5134:
5094:
5059:
4292:
4285:
3991:
3914:
3853:
3827:
3635:
3631:
3627:
1760:
330:
238:
156:
117:
74:
54:
5953:
5801:
5776:
5671:
5651:
5578:
5563:
5548:
5538:
5503:
5423:
5243:
5238:
5201:
5196:
5191:
5089:
4895:
4506:
4411:
4400:
4386:
4381:
Some directions, into which the Boussinesq equations have been extended, are:
417:
234:
214:
174:
152:
125:
82:
50:
37:
6025:
5887:
5872:
5786:
5631:
5470:
5465:
5248:
5176:
5104:
5024:
5014:
4971:
4510:
4491:
4476:
4136:{\displaystyle c^{2}\,=\,gh\,\left(1\,-\,{\frac {1}{3}}\,k^{2}h^{2}\right).}
3647:
272:
109:). The 1872 paper of Boussinesq introduces the equations now known as the
6120:
5842:
5701:
5593:
5583:
5528:
5004:
2388:
and higher order terms are assumed to be negligible. Then, the following
1746:. This series may subsequently be truncated to a finite number of terms.
167:
120:
takes into account the vertical structure of the horizontal and vertical
4932:
283:
248:
in 1871, to construct an approximate solution for the solitary wave (or
5989:
5979:
5149:
5119:
4856:. Elsevier Oceanography Series. Vol. 67. Elsevier. pp. 1–41.
4551:
4378:, since it is used in the analysis in the remainder of his 1872 paper.
4310:
have a relative error in the phase speed less than 4% for wave lengths
3529:
2212:
106:
5696:
5109:
2247:
193:
Periodic waves in the Boussinesq approximation, shown in a vertical
6058:
5877:
5656:
5611:
4540:
4528:
3528:
2769:
This set of equations has been derived for a flat horizontal bed,
218:
188:
42:
36:
5490:
3465:{\displaystyle \tau \,=\,{\sqrt {3}}\,t\,{\sqrt {\frac {g}{h}}}}
470:
334:
4936:
4835:
A modern introduction to the mathematical theory of water waves
4812:(1993). "Wave transformation in the nearshore zone: A review".
27:
Approximation valid for weakly non-linear and fairly long waves
4521:. These are approximate solutions of the Boussinesq equation.
244:
This elimination of the vertical coordinate was first done by
163:
3410:{\displaystyle \psi \,=\,{\frac {1}{2}}\,{\frac {\eta }{h}}}
4284:
become physically meaningless, because there are no longer
93:, who first derived them in response to the observation by
4668:
3156:
for non-dimensionalization, this equation reads, after
2348:{\displaystyle u_{b}=\partial \varphi _{b}/\partial x}
4336:
4316:
4243:
4216:
4196:
4156:
4058:
4024:
4004:
3959:
3931:
3899:
3862:
3838:
3812:
3678:
3583:
3539:
3484:
3429:
3377:
3169:
3142:
3122:
2866:
2838:
2799:
2779:
2406:
2361:
2307:
2280:
2260:
2228:
2193:
2151:
2127:
2085:
2061:
1861:
1817:
1777:
1723:
1689:
1054:
1027:
981:
559:
527:
486:
455:
426:
402:
382:
350:
3514:{\displaystyle \xi \,=\,{\sqrt {3}}\,{\frac {x}{h}}}
209:, due to the wave nonlinearity. This case (drawn on
5998:
5972:
5934:
5886:
5825:
5720:
5592:
5489:
5284:
4970:
4583:
4458:
Further approximations for one-way wave propagation
4146:The relative error in the phase speed for equation
147:, Boussinesq-type equations are frequently used in
4342:
4322:
4272:
4222:
4202:
4182:
4135:
4030:
4010:
3982:
3937:
3905:
3885:
3844:
3818:
3792:
3592:
3569:
3513:
3464:
3409:
3355:
3148:
3128:
3097:
2844:
2805:
2785:
2758:
2376:
2347:
2293:
2266:
2234:
2199:
2177:
2133:
2111:
2067:
2041:
1844:
1795:
1738:
1709:
1672:
1033:
1009:
964:
542:
513:
461:
441:
408:
388:
368:
5906:North West Shelf Operational Oceanographic System
4916:Waves on Beaches and Resulting Sediment Transport
4374:Boussinesq equations is the above-mentioned set
86:
41:Simulation of periodic waves over an underwater
5896:Deep-ocean Assessment and Reporting of Tsunamis
2246:. Further, in the resulting equations only the
1017:is the velocity potential at the bed. Invoking
255:The steps in the Boussinesq approximation are:
4645:
4643:
2178:{\displaystyle w=\partial \varphi /\partial z}
2112:{\displaystyle u=\partial \varphi /\partial x}
1717:is zero at the – impermeable – horizontal bed
4948:
4669:"Celeria.org - Celeris Boussinesq Wave Model"
3953:, is less than 4% for a relative wave number
8:
4756:Journal de Mathématiques Pures et Appliquées
1710:{\displaystyle \partial \varphi /\partial z}
4673:Celeria.org - Celeris Boussinesq Wave Model
4602:This paper (Boussinesq, 1872) starts with:
177:is large compared to the water depth – the
4955:
4941:
4933:
3693:
143:, which are not frequency-dispersive). In
4735:Comptes Rendus de l'Académie des Sciences
4579:
4335:
4315:
4258:
4248:
4242:
4215:
4195:
4172:
4155:
4119:
4109:
4104:
4094:
4093:
4089:
4080:
4073:
4069:
4063:
4057:
4023:
4003:
3972:
3958:
3930:
3898:
3875:
3861:
3837:
3811:
3778:
3768:
3763:
3753:
3752:
3748:
3737:
3727:
3722:
3712:
3711:
3707:
3701:
3700:
3689:
3683:
3677:
3582:
3550:
3544:
3538:
3523:: the dimensionless horizontal position.
3501:
3500:
3493:
3492:
3488:
3483:
3450:
3449:
3445:
3438:
3437:
3433:
3428:
3397:
3396:
3386:
3385:
3381:
3376:
3346:
3342:
3336:
3327:
3309:
3302:
3301:
3297:
3291:
3286:
3282:
3268:
3254:
3248:
3247:
3243:
3234:
3216:
3209:
3208:
3204:
3195:
3177:
3170:
3168:
3141:
3121:
3091:
3087:
3073:
3055:
3048:
3047:
3041:
3036:
3026:
3025:
3021:
3010:
3004:
3003:
2993:
2979:
2965:
2959:
2958:
2951:
2947:
2938:
2920:
2913:
2912:
2905:
2901:
2892:
2874:
2867:
2865:
2837:
2798:
2778:
2740:
2732:
2718:
2708:
2701:
2700:
2694:
2689:
2679:
2678:
2674:
2654:
2653:
2649:
2645:
2628:
2618:
2617:
2611:
2606:
2598:
2581:
2571:
2555:
2540:
2530:
2523:
2522:
2516:
2511:
2501:
2500:
2496:
2485:
2480:
2455:
2440:
2439:
2431:
2411:
2407:
2405:
2360:
2334:
2328:
2312:
2306:
2285:
2279:
2259:
2227:
2219:Now the Boussinesq approximation for the
2192:
2164:
2150:
2126:
2098:
2084:
2060:
2028:
2024:
2020:
2016:
2012:
2001:
1988:
1978:
1968:
1967:
1959:
1939:
1931:
1927:
1923:
1919:
1899:
1898:
1894:
1886:
1866:
1862:
1860:
1816:
1776:
1722:
1696:
1688:
1657:
1653:
1649:
1640:
1625:
1615:
1608:
1607:
1601:
1584:
1574:
1573:
1569:
1560:
1545:
1535:
1528:
1527:
1521:
1504:
1494:
1493:
1489:
1483:
1478:
1470:
1457:
1453:
1449:
1434:
1410:
1396:
1382:
1376:
1375:
1369:
1352:
1342:
1341:
1337:
1322:
1298:
1292:
1276:
1270:
1259:
1253:
1249:
1245:
1236:
1221:
1211:
1204:
1203:
1197:
1180:
1170:
1169:
1165:
1156:
1141:
1131:
1124:
1123:
1117:
1100:
1090:
1089:
1085:
1079:
1074:
1066:
1062:
1055:
1053:
1026:
986:
980:
951:
947:
932:
919:
901:
894:
888:
882:
865:
855:
854:
850:
835:
822:
804:
797:
791:
785:
768:
758:
757:
746:
731:
718:
700:
693:
687:
681:
664:
654:
653:
649:
634:
610:
604:
588:
584:
578:
571:
567:
560:
558:
526:
485:
454:
425:
401:
381:
349:
4354:Boussinesq-type equations and extensions
4299:(Boussinesq, 1872, equation 25, see set
2856:set B – Boussinesq (1872), equation (26)
2396:set A – Boussinesq (1872), equation (25)
4774:. Advanced Series on Ocean Engineering
4595:
4303:above) does not have this shortcoming.
3419:: the dimensionless surface elevation,
263:is made of the horizontal and vertical
5227:one-dimensional Saint-Venant equations
4330:in excess of 13 times the water depth
3577:as a function of relative wave number
2793:is a constant independent of position
2242:, as given above, is applied in these
4575:
7:
6174:
4771:Wave propagation over uneven bottoms
4358:There are an overwhelming number of
4210:longer than 7 times the water depth
4018:larger than 4 times the water depth
4918:. Academic Press. pp. 95–122.
4658:Dingemans (1997), p. 473 & 516.
4584:Hamm, Madsen & Peregrine (1993)
4442:between fluid domains of different
3610:= Boussinesq (1872), equation (26),
3605:= Boussinesq (1872), equation (25),
2355:the horizontal velocity at the bed
89:. The approximation is named after
6054:National Oceanographic Data Center
5481:World Ocean Circulation Experiment
5369:Global Ocean Data Analysis Project
3665:characteristics for the above set
3320:
3306:
3261:
3251:
3227:
3213:
3188:
3174:
3066:
3052:
2972:
2962:
2931:
2917:
2885:
2871:
2733:
2726:
2705:
2665:
2657:
2636:
2621:
2589:
2574:
2548:
2527:
2446:
2442:
2422:
2414:
2339:
2321:
2169:
2158:
2103:
2092:
1950:
1942:
1910:
1902:
1877:
1869:
1701:
1690:
1633:
1612:
1553:
1532:
1421:
1413:
1389:
1379:
1309:
1301:
1229:
1208:
1149:
1128:
912:
898:
815:
801:
711:
697:
621:
613:
25:
5901:Global Sea Level Observing System
4874:(1967). "Long waves on a beach".
4705:"James T. Kirby, Funwave program"
1010:{\displaystyle \varphi _{b}(x,t)}
221:, the wave height is 1.8 m (
116:The Boussinesq approximation for
6184:
6173:
6164:
6163:
5359:Geochemical Ocean Sections Study
5275:
5264:
3886:{\displaystyle k=2\pi /\lambda }
6089:Ocean thermal energy conversion
5812:Vine–Matthews–Morley hypothesis
4778:. World Scientific, Singapore.
4484:Kadomtsev–Petviashvili equation
4273:{\displaystyle k^{2}h^{2}>3}
3615:= full linear wave theory, see
3136:and gravitational acceleration
514:{\displaystyle \varphi (x,z,t)}
4499:nonlinear Schrödinger equation
4297:partial differential equations
3564:
3555:
3474:: the dimensionless time, and
2390:partial differential equations
1839:
1827:
1790:
1778:
1598:
1585:
1518:
1505:
1366:
1353:
1289:
1277:
1194:
1181:
1114:
1101:
1004:
992:
879:
866:
782:
769:
678:
665:
601:
589:
508:
490:
363:
351:
323:partial differential equations
305:are used, to replace vertical
289:the conservation of mass (see
129:partial differential equations
1:
4190:, equivalent to wave lengths
4183:{\displaystyle kh<2\pi /7}
3951:linear theory for water waves
2826:partial differential equation
1750:Original Boussinesq equations
344:over a horizontal bed in the
5349:El Niño–Southern Oscillation
5319:Craik–Leibovich vortex force
5075:Luke's variational principle
4854:Advances in Coastal Modeling
4826:10.1016/0378-3839(93)90044-9
4045:characteristics of equation
3983:{\displaystyle kh<\pi /2}
1845:{\displaystyle z=\eta (x,t)}
1683:since the vertical velocity
6222:Equations of fluid dynamics
4486:for (near uni-directional)
4295:. The original set of two
3652:linear frequency dispersion
3623:Linear frequency dispersion
3533:Linear phase speed squared
2824:can be reduced to a single
201:direction. Notice the flat
6238:
5414:Ocean dynamical thermostat
5262:
4876:Journal of Fluid Mechanics
4469:Korteweg–de Vries equation
4410:around different vertical
3570:{\displaystyle c^{2}/(gh)}
227:gravitational acceleration
29:
6159:
5949:Ocean acoustic tomography
5762:Mohorovičić discontinuity
5354:General circulation model
4990:Benjamin–Feir instability
4896:10.1017/S0022112067002605
4649:Dingemans (1997), p. 521.
4628:Dingemans (1997), p. 475.
4619:Dingemans (1997), p. 477.
4582:, Part 2, Chapter 5) and
4368:Boussinesq-type equations
3998:is valid for wavelengths
133:Boussinesq-type equations
6079:Ocean surface topography
5454:Thermohaline circulation
5444:Subsurface ocean current
5384:Hydrothermal circulation
5217:Wave–current interaction
4995:Boussinesq approximation
4808:Hamm, L.; Madsen, P.A.;
4768:Dingemans, M.W. (1997).
4503:complex-valued amplitude
4323:{\displaystyle \lambda }
4203:{\displaystyle \lambda }
4011:{\displaystyle \lambda }
3906:{\displaystyle \lambda }
3638:, a phenomenon known as
3617:dispersion (water waves)
3116:, using the water depth
3114:dimensionless quantities
2235:{\displaystyle \varphi }
1034:{\displaystyle \varphi }
420:. The bed is located at
340:As an example, consider
213:) shows a wave with the
185:Boussinesq approximation
71:Boussinesq approximation
32:Boussinesq approximation
18:Boussinesq type equation
6116:Sea surface temperature
6099:Outline of oceanography
5294:Atmospheric circulation
5232:shallow water equations
5222:Waves and shallow water
5115:Significant wave height
4501:(NLS equation) for the
4308:shallow water equations
2815:shallow water equations
141:shallow water equations
47:waves and shallow water
6111:Sea surface microlayer
5476:Wind generated current
4833:Johnson, R.S. (1997).
4637:Johnson (1997), p. 219
4546:
4538:
4344:
4324:
4274:
4224:
4204:
4184:
4137:
4032:
4012:
3984:
3939:
3907:
3887:
3846:
3820:
3794:
3634:travel with different
3619:
3594:
3571:
3515:
3466:
3411:
3357:
3150:
3130:
3099:
2846:
2807:
2787:
2760:
2378:
2349:
2295:
2268:
2254:terms with respect to
2236:
2201:
2179:
2135:
2113:
2069:
2043:
1846:
1797:
1740:
1711:
1674:
1035:
1011:
966:
544:
515:
463:
443:
410:
390:
370:
230:
62:
5944:Deep scattering layer
5926:World Geodetic System
5434:Princeton Ocean Model
5314:Coriolis–Stokes force
4964:Physical oceanography
4802:See Part 2, Chapter 5
4544:
4532:
4451:variational principle
4370:. Strictly speaking,
4345:
4325:
4275:
4237:For short waves with
4225:
4205:
4185:
4138:
4033:
4013:
3985:
3940:
3908:
3888:
3847:
3821:
3795:
3650:, the terminology is
3595:
3572:
3532:
3516:
3467:
3412:
3358:
3151:
3131:
3100:
2847:
2845:{\displaystyle \eta }
2808:
2788:
2761:
2379:
2350:
2296:
2294:{\displaystyle u_{b}}
2269:
2267:{\displaystyle \eta }
2237:
2202:
2180:
2136:
2114:
2070:
2044:
1847:
1798:
1796:{\displaystyle (x,z)}
1741:
1712:
1675:
1036:
1012:
967:
545:
521:around the bed level
516:
464:
444:
411:
391:
371:
369:{\displaystyle (x,z)}
309:of quantities in the
192:
40:
5964:Underwater acoustics
5524:Perigean spring tide
5389:Langmuir circulation
5100:Rossby-gravity waves
4394:frequency dispersion
4334:
4314:
4241:
4214:
4194:
4154:
4150:is less than 4% for
4056:
4043:frequency dispersion
4022:
4002:
3957:
3929:
3897:
3860:
3836:
3810:
3676:
3663:frequency dispersion
3640:frequency dispersion
3581:
3537:
3482:
3427:
3375:
3167:
3140:
3120:
2864:
2836:
2797:
2777:
2404:
2377:{\displaystyle z=-h}
2359:
2305:
2278:
2258:
2226:
2191:
2149:
2125:
2083:
2059:
1859:
1815:
1775:
1765:incompressible fluid
1739:{\displaystyle z=-h}
1721:
1687:
1052:
1025:
979:
557:
543:{\displaystyle z=-h}
525:
484:
453:
442:{\displaystyle z=-h}
424:
400:
380:
348:
139:(as opposite to the
137:frequency dispersion
135:, which incorporate
111:Boussinesq equations
57:, shoaling and weak
6126:Science On a Sphere
5732:Convergent boundary
5404:Modular Ocean Model
5364:Geostrophic current
5080:Mild-slope equation
4888:1967JFM....27..815P
4814:Coastal Engineering
4570:techniques for the
4360:mathematical models
3949:, as compared with
3925:in the phase speed
2301:are retained (with
2244:boundary conditions
1805:boundary conditions
1043:incompressible flow
396:the horizontal and
315:partial derivatives
307:partial derivatives
295:incompressible flow
291:continuity equation
271:) around a certain
250:wave of translation
217:equal to 39.1
145:coastal engineering
99:wave of translation
5782:Seafloor spreading
5772:Outer trench swell
5737:Divergent boundary
5637:Continental margin
5622:Carbonate platform
5519:Lunitidal interval
4759:. Deuxième Série.
4547:
4539:
4490:in two horizontal
4475:in one horizontal
4449:derivation from a
4340:
4320:
4270:
4220:
4200:
4180:
4133:
4028:
4008:
3994:applications, set
3980:
3935:
3903:
3883:
3842:
3816:
3790:
3669:of equations are:
3642:. For the case of
3620:
3593:{\displaystyle kh}
3590:
3567:
3511:
3462:
3407:
3353:
3146:
3126:
3095:
2842:
2803:
2783:
2756:
2754:
2374:
2345:
2291:
2264:
2232:
2221:velocity potential
2197:
2175:
2131:
2109:
2075:is the horizontal
2065:
2039:
2037:
1842:
1793:
1736:
1707:
1670:
1668:
1031:
1019:Laplace's equation
1007:
962:
960:
540:
511:
479:velocity potential
459:
439:
406:
386:
366:
329:of the horizontal
282:is truncated to a
269:velocity potential
231:
124:. This results in
95:John Scott Russell
63:
6199:
6198:
6191:Oceans portal
6151:World Ocean Atlas
6141:Underwater glider
6084:Ocean temperature
5747:Hydrothermal vent
5712:Submarine volcano
5647:Continental shelf
5627:Coastal geography
5617:Bathymetric chart
5499:Amphidromic point
5187:Wave nonlinearity
5045:Infragravity wave
4785:978-981-02-0427-3
4560:finite-difference
4343:{\displaystyle h}
4232:fairly long waves
4223:{\displaystyle h}
4102:
4031:{\displaystyle h}
3938:{\displaystyle c}
3845:{\displaystyle k}
3819:{\displaystyle c}
3785:
3761:
3720:
3527:
3526:
3509:
3498:
3460:
3459:
3443:
3405:
3394:
3334:
3275:
3241:
3202:
3149:{\displaystyle g}
3129:{\displaystyle h}
3080:
3034:
3019:
3001:
2986:
2945:
2899:
2806:{\displaystyle x}
2786:{\displaystyle h}
2747:
2687:
2672:
2643:
2596:
2562:
2509:
2453:
2429:
2200:{\displaystyle g}
2134:{\displaystyle w}
2068:{\displaystyle u}
1976:
1957:
1917:
1884:
1769:irrotational flow
1647:
1582:
1567:
1502:
1428:
1403:
1350:
1316:
1243:
1178:
1163:
1098:
926:
863:
829:
766:
725:
662:
628:
462:{\displaystyle h}
409:{\displaystyle z}
389:{\displaystyle x}
303:irrotational flow
301:condition for an
246:Joseph Boussinesq
229:is 9.81 m/s.
91:Joseph Boussinesq
87:fairly long waves
81:valid for weakly
16:(Redirected from
6229:
6189:
6188:
6177:
6176:
6167:
6166:
6106:Pelagic sediment
6044:Marine pollution
5838:Deep ocean water
5707:Submarine canyon
5642:Continental rise
5534:Rule of twelfths
5449:Sverdrup balance
5379:Humboldt Current
5304:Boundary current
5279:
5268:
5085:Radiation stress
5055:Iribarren number
5030:Equatorial waves
4985:Ballantine scale
4980:Airy wave theory
4957:
4950:
4943:
4934:
4929:
4907:
4867:
4848:
4829:
4800:
4798:
4797:
4788:. Archived from
4764:
4742:
4713:
4712:
4701:
4695:
4694:
4683:
4677:
4676:
4665:
4659:
4656:
4650:
4647:
4638:
4635:
4629:
4626:
4620:
4617:
4611:
4600:
4525:Numerical models
4488:wave propagation
4473:wave propagation
4408:Taylor expansion
4349:
4347:
4346:
4341:
4329:
4327:
4326:
4321:
4279:
4277:
4276:
4271:
4263:
4262:
4253:
4252:
4229:
4227:
4226:
4221:
4209:
4207:
4206:
4201:
4189:
4187:
4186:
4181:
4176:
4142:
4140:
4139:
4134:
4129:
4125:
4124:
4123:
4114:
4113:
4103:
4095:
4068:
4067:
4037:
4035:
4034:
4029:
4017:
4015:
4014:
4009:
3989:
3987:
3986:
3981:
3976:
3944:
3942:
3941:
3936:
3912:
3910:
3909:
3904:
3892:
3890:
3889:
3884:
3879:
3851:
3849:
3848:
3843:
3825:
3823:
3822:
3817:
3799:
3797:
3796:
3791:
3786:
3784:
3783:
3782:
3773:
3772:
3762:
3754:
3743:
3742:
3741:
3732:
3731:
3721:
3713:
3702:
3688:
3687:
3599:
3597:
3596:
3591:
3576:
3574:
3573:
3568:
3554:
3549:
3548:
3520:
3518:
3517:
3512:
3510:
3502:
3499:
3494:
3471:
3469:
3468:
3463:
3461:
3452:
3451:
3444:
3439:
3416:
3414:
3413:
3408:
3406:
3398:
3395:
3387:
3369:
3368:
3362:
3360:
3359:
3354:
3341:
3337:
3335:
3333:
3332:
3331:
3318:
3314:
3313:
3303:
3296:
3295:
3276:
3274:
3273:
3272:
3259:
3258:
3249:
3242:
3240:
3239:
3238:
3225:
3221:
3220:
3210:
3203:
3201:
3200:
3199:
3186:
3182:
3181:
3171:
3155:
3153:
3152:
3147:
3135:
3133:
3132:
3127:
3104:
3102:
3101:
3096:
3086:
3082:
3081:
3079:
3078:
3077:
3064:
3060:
3059:
3049:
3046:
3045:
3035:
3027:
3020:
3015:
3014:
3005:
3002:
2994:
2987:
2985:
2984:
2983:
2970:
2969:
2960:
2946:
2944:
2943:
2942:
2929:
2925:
2924:
2914:
2900:
2898:
2897:
2896:
2883:
2879:
2878:
2868:
2851:
2849:
2848:
2843:
2812:
2810:
2809:
2804:
2792:
2790:
2789:
2784:
2765:
2763:
2762:
2757:
2755:
2748:
2746:
2745:
2744:
2724:
2723:
2722:
2713:
2712:
2702:
2699:
2698:
2688:
2680:
2673:
2671:
2663:
2655:
2644:
2642:
2634:
2633:
2632:
2619:
2616:
2615:
2597:
2595:
2587:
2586:
2585:
2572:
2563:
2561:
2560:
2559:
2546:
2545:
2544:
2535:
2534:
2524:
2521:
2520:
2510:
2502:
2495:
2491:
2490:
2489:
2479:
2475:
2454:
2452:
2441:
2430:
2428:
2420:
2412:
2383:
2381:
2380:
2375:
2354:
2352:
2351:
2346:
2338:
2333:
2332:
2317:
2316:
2300:
2298:
2297:
2292:
2290:
2289:
2273:
2271:
2270:
2265:
2241:
2239:
2238:
2233:
2206:
2204:
2203:
2198:
2184:
2182:
2181:
2176:
2168:
2141:is the vertical
2140:
2138:
2137:
2132:
2118:
2116:
2115:
2110:
2102:
2074:
2072:
2071:
2066:
2048:
2046:
2045:
2040:
2038:
2011:
2007:
2006:
2005:
1993:
1992:
1977:
1969:
1958:
1956:
1948:
1940:
1918:
1916:
1908:
1900:
1885:
1883:
1875:
1867:
1851:
1849:
1848:
1843:
1802:
1800:
1799:
1794:
1745:
1743:
1742:
1737:
1716:
1714:
1713:
1708:
1700:
1679:
1677:
1676:
1671:
1669:
1662:
1658:
1648:
1646:
1645:
1644:
1631:
1630:
1629:
1620:
1619:
1609:
1606:
1605:
1583:
1575:
1568:
1566:
1565:
1564:
1551:
1550:
1549:
1540:
1539:
1529:
1526:
1525:
1503:
1495:
1488:
1487:
1462:
1458:
1448:
1447:
1433:
1429:
1427:
1419:
1411:
1404:
1402:
1401:
1400:
1387:
1386:
1377:
1374:
1373:
1351:
1343:
1336:
1335:
1321:
1317:
1315:
1307:
1299:
1263:
1258:
1254:
1244:
1242:
1241:
1240:
1227:
1226:
1225:
1216:
1215:
1205:
1202:
1201:
1179:
1171:
1164:
1162:
1161:
1160:
1147:
1146:
1145:
1136:
1135:
1125:
1122:
1121:
1099:
1091:
1084:
1083:
1040:
1038:
1037:
1032:
1016:
1014:
1013:
1008:
991:
990:
971:
969:
968:
963:
961:
946:
945:
931:
927:
925:
924:
923:
910:
906:
905:
895:
887:
886:
864:
856:
849:
848:
834:
830:
828:
827:
826:
813:
809:
808:
798:
790:
789:
767:
759:
750:
745:
744:
730:
726:
724:
723:
722:
709:
705:
704:
694:
686:
685:
663:
655:
648:
647:
633:
629:
627:
619:
611:
583:
582:
549:
547:
546:
541:
520:
518:
517:
512:
475:Taylor expansion
468:
466:
465:
460:
448:
446:
445:
440:
415:
413:
412:
407:
395:
393:
392:
387:
375:
373:
372:
367:
325:are in terms of
313:with horizontal
311:Taylor expansion
286:number of terms,
280:Taylor expansion
261:Taylor expansion
199:wave propagation
179:Stokes expansion
21:
6237:
6236:
6232:
6231:
6230:
6228:
6227:
6226:
6202:
6201:
6200:
6195:
6183:
6155:
5994:
5968:
5930:
5911:Sea-level curve
5882:
5821:
5807:Transform fault
5757:Mid-ocean ridge
5723:
5716:
5682:Oceanic plateau
5588:
5574:Tidal resonance
5544:Theory of tides
5485:
5394:Longshore drift
5344:Ekman transport
5280:
5274:
5273:
5272:
5271:
5270:
5269:
5260:
5212:Wave turbulence
5145:Trochoidal wave
5070:Longshore drift
4966:
4961:
4926:
4912:Peregrine, D.H.
4910:
4872:Peregrine, D.H.
4870:
4864:
4851:
4845:
4832:
4810:Peregrine, D.H.
4807:
4795:
4793:
4786:
4767:
4745:
4725:
4722:
4717:
4716:
4703:
4702:
4698:
4687:"ISEC - Models"
4685:
4684:
4680:
4667:
4666:
4662:
4657:
4653:
4648:
4641:
4636:
4632:
4627:
4623:
4618:
4614:
4601:
4597:
4592:
4580:Dingemans (1997
4527:
4460:
4429:surface tension
4356:
4332:
4331:
4312:
4311:
4254:
4244:
4239:
4238:
4212:
4211:
4192:
4191:
4152:
4151:
4115:
4105:
4085:
4081:
4059:
4054:
4053:
4020:
4019:
4000:
3999:
3955:
3954:
3927:
3926:
3895:
3894:
3858:
3857:
3834:
3833:
3808:
3807:
3774:
3764:
3744:
3733:
3723:
3703:
3679:
3674:
3673:
3625:
3611:
3606:
3601:
3579:
3578:
3540:
3535:
3534:
3480:
3479:
3425:
3424:
3373:
3372:
3323:
3319:
3305:
3304:
3287:
3281:
3277:
3264:
3260:
3250:
3230:
3226:
3212:
3211:
3191:
3187:
3173:
3172:
3165:
3164:
3138:
3137:
3118:
3117:
3069:
3065:
3051:
3050:
3037:
3006:
2992:
2988:
2975:
2971:
2961:
2934:
2930:
2916:
2915:
2888:
2884:
2870:
2869:
2862:
2861:
2834:
2833:
2795:
2794:
2775:
2774:
2773:the mean depth
2753:
2752:
2736:
2725:
2714:
2704:
2703:
2690:
2664:
2656:
2635:
2624:
2620:
2607:
2599:
2588:
2577:
2573:
2568:
2567:
2551:
2547:
2536:
2526:
2525:
2512:
2481:
2465:
2461:
2460:
2456:
2445:
2432:
2421:
2413:
2402:
2401:
2357:
2356:
2324:
2308:
2303:
2302:
2281:
2276:
2275:
2256:
2255:
2224:
2223:
2189:
2188:
2147:
2146:
2123:
2122:
2081:
2080:
2057:
2056:
2036:
2035:
1997:
1984:
1983:
1979:
1960:
1949:
1941:
1936:
1935:
1909:
1901:
1887:
1876:
1868:
1857:
1856:
1813:
1812:
1773:
1772:
1757:
1752:
1719:
1718:
1685:
1684:
1667:
1666:
1636:
1632:
1621:
1611:
1610:
1597:
1556:
1552:
1541:
1531:
1530:
1517:
1479:
1477:
1473:
1471:
1464:
1463:
1420:
1412:
1406:
1405:
1392:
1388:
1378:
1365:
1308:
1300:
1294:
1293:
1275:
1271:
1261:
1260:
1232:
1228:
1217:
1207:
1206:
1193:
1152:
1148:
1137:
1127:
1126:
1113:
1075:
1073:
1069:
1067:
1050:
1049:
1041:, as valid for
1023:
1022:
982:
977:
976:
959:
958:
915:
911:
897:
896:
890:
889:
878:
818:
814:
800:
799:
793:
792:
781:
748:
747:
714:
710:
696:
695:
689:
688:
677:
620:
612:
606:
605:
574:
572:
555:
554:
523:
522:
482:
481:
477:is made of the
473:water depth. A
451:
450:
422:
421:
398:
397:
378:
377:
346:
345:
187:
149:computer models
101:(also known as
35:
28:
23:
22:
15:
12:
11:
5:
6235:
6233:
6225:
6224:
6219:
6214:
6212:Fluid dynamics
6204:
6203:
6197:
6196:
6194:
6193:
6181:
6171:
6160:
6157:
6156:
6154:
6153:
6148:
6143:
6138:
6133:
6131:Stratification
6128:
6123:
6118:
6113:
6108:
6103:
6102:
6101:
6091:
6086:
6081:
6076:
6071:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
6023:
6021:Color of water
6018:
6016:Benthic lander
6013:
6008:
6002:
6000:
5996:
5995:
5993:
5992:
5987:
5982:
5976:
5974:
5970:
5969:
5967:
5966:
5961:
5956:
5951:
5946:
5940:
5938:
5932:
5931:
5929:
5928:
5923:
5921:Sea level rise
5918:
5916:Sea level drop
5913:
5908:
5903:
5898:
5892:
5890:
5884:
5883:
5881:
5880:
5875:
5870:
5865:
5860:
5855:
5850:
5845:
5840:
5835:
5829:
5827:
5823:
5822:
5820:
5819:
5814:
5809:
5804:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5764:
5759:
5754:
5752:Marine geology
5749:
5744:
5739:
5734:
5728:
5726:
5718:
5717:
5715:
5714:
5709:
5704:
5699:
5694:
5692:Passive margin
5689:
5687:Oceanic trench
5684:
5679:
5674:
5669:
5664:
5659:
5654:
5649:
5644:
5639:
5634:
5629:
5624:
5619:
5614:
5609:
5604:
5598:
5596:
5590:
5589:
5587:
5586:
5581:
5576:
5571:
5566:
5561:
5556:
5551:
5546:
5541:
5536:
5531:
5526:
5521:
5516:
5511:
5506:
5501:
5495:
5493:
5487:
5486:
5484:
5483:
5478:
5473:
5468:
5463:
5462:
5461:
5451:
5446:
5441:
5436:
5431:
5426:
5421:
5419:Ocean dynamics
5416:
5411:
5406:
5401:
5396:
5391:
5386:
5381:
5376:
5371:
5366:
5361:
5356:
5351:
5346:
5341:
5336:
5331:
5326:
5321:
5316:
5311:
5309:Coriolis force
5306:
5301:
5296:
5290:
5288:
5282:
5281:
5263:
5261:
5259:
5258:
5257:
5256:
5246:
5241:
5236:
5235:
5234:
5229:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5164:
5159:
5158:
5157:
5147:
5142:
5137:
5132:
5130:Stokes problem
5127:
5122:
5117:
5112:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5065:Kinematic wave
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5027:
5022:
5017:
5012:
5007:
5002:
4997:
4992:
4987:
4982:
4976:
4974:
4968:
4967:
4962:
4960:
4959:
4952:
4945:
4937:
4931:
4930:
4924:
4908:
4882:(4): 815–827.
4868:
4862:
4849:
4843:
4830:
4805:
4784:
4765:
4747:Boussinesq, J.
4743:
4727:Boussinesq, J.
4721:
4718:
4715:
4714:
4696:
4691:isec.nacse.org
4678:
4660:
4651:
4639:
4630:
4621:
4612:
4594:
4593:
4591:
4588:
4572:discretization
4568:finite element
4526:
4523:
4515:
4514:
4509:waves (slowly
4495:
4480:
4459:
4456:
4455:
4454:
4447:
4436:internal waves
4432:
4425:
4418:
4415:
4404:
4397:
4390:
4355:
4352:
4339:
4319:
4269:
4266:
4261:
4257:
4251:
4247:
4219:
4199:
4179:
4175:
4171:
4168:
4165:
4162:
4159:
4144:
4143:
4132:
4128:
4122:
4118:
4112:
4108:
4101:
4098:
4092:
4088:
4084:
4079:
4076:
4072:
4066:
4062:
4027:
4007:
3979:
3975:
3971:
3968:
3965:
3962:
3934:
3923:relative error
3919:
3918:
3902:
3882:
3878:
3874:
3871:
3868:
3865:
3841:
3831:
3815:
3801:
3800:
3789:
3781:
3777:
3771:
3767:
3760:
3757:
3751:
3747:
3740:
3736:
3730:
3726:
3719:
3716:
3710:
3706:
3699:
3696:
3692:
3686:
3682:
3624:
3621:
3589:
3586:
3566:
3563:
3560:
3557:
3553:
3547:
3543:
3525:
3524:
3521:
3508:
3505:
3497:
3491:
3487:
3476:
3475:
3472:
3458:
3455:
3448:
3442:
3436:
3432:
3421:
3420:
3417:
3404:
3401:
3393:
3390:
3384:
3380:
3364:
3363:
3352:
3349:
3345:
3340:
3330:
3326:
3322:
3317:
3312:
3308:
3300:
3294:
3290:
3285:
3280:
3271:
3267:
3263:
3257:
3253:
3246:
3237:
3233:
3229:
3224:
3219:
3215:
3207:
3198:
3194:
3190:
3185:
3180:
3176:
3145:
3125:
3106:
3105:
3094:
3090:
3085:
3076:
3072:
3068:
3063:
3058:
3054:
3044:
3040:
3033:
3030:
3024:
3018:
3013:
3009:
3000:
2997:
2991:
2982:
2978:
2974:
2968:
2964:
2957:
2954:
2950:
2941:
2937:
2933:
2928:
2923:
2919:
2911:
2908:
2904:
2895:
2891:
2887:
2882:
2877:
2873:
2858:
2857:
2841:
2802:
2782:
2767:
2766:
2751:
2743:
2739:
2735:
2731:
2728:
2721:
2717:
2711:
2707:
2697:
2693:
2686:
2683:
2677:
2670:
2667:
2662:
2659:
2652:
2648:
2641:
2638:
2631:
2627:
2623:
2614:
2610:
2605:
2602:
2600:
2594:
2591:
2584:
2580:
2576:
2570:
2569:
2566:
2558:
2554:
2550:
2543:
2539:
2533:
2529:
2519:
2515:
2508:
2505:
2499:
2494:
2488:
2484:
2478:
2474:
2471:
2468:
2464:
2459:
2451:
2448:
2444:
2438:
2435:
2433:
2427:
2424:
2419:
2416:
2410:
2409:
2398:
2397:
2392:are obtained:
2373:
2370:
2367:
2364:
2344:
2341:
2337:
2331:
2327:
2323:
2320:
2315:
2311:
2288:
2284:
2263:
2231:
2217:
2216:
2196:
2186:
2174:
2171:
2167:
2163:
2160:
2157:
2154:
2130:
2120:
2108:
2105:
2101:
2097:
2094:
2091:
2088:
2064:
2050:
2049:
2034:
2031:
2027:
2023:
2019:
2015:
2010:
2004:
2000:
1996:
1991:
1987:
1982:
1975:
1972:
1966:
1963:
1961:
1955:
1952:
1947:
1944:
1938:
1937:
1934:
1930:
1926:
1922:
1915:
1912:
1907:
1904:
1897:
1893:
1890:
1888:
1882:
1879:
1874:
1871:
1865:
1864:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1792:
1789:
1786:
1783:
1780:
1756:
1753:
1751:
1748:
1735:
1732:
1729:
1726:
1706:
1703:
1699:
1695:
1692:
1681:
1680:
1665:
1661:
1656:
1652:
1643:
1639:
1635:
1628:
1624:
1618:
1614:
1604:
1600:
1596:
1593:
1590:
1587:
1581:
1578:
1572:
1563:
1559:
1555:
1548:
1544:
1538:
1534:
1524:
1520:
1516:
1513:
1510:
1507:
1501:
1498:
1492:
1486:
1482:
1476:
1472:
1469:
1466:
1465:
1461:
1456:
1452:
1446:
1443:
1440:
1437:
1432:
1426:
1423:
1418:
1415:
1409:
1399:
1395:
1391:
1385:
1381:
1372:
1368:
1364:
1361:
1358:
1355:
1349:
1346:
1340:
1334:
1331:
1328:
1325:
1320:
1314:
1311:
1306:
1303:
1297:
1291:
1288:
1285:
1282:
1279:
1274:
1269:
1266:
1264:
1262:
1257:
1252:
1248:
1239:
1235:
1231:
1224:
1220:
1214:
1210:
1200:
1196:
1192:
1189:
1186:
1183:
1177:
1174:
1168:
1159:
1155:
1151:
1144:
1140:
1134:
1130:
1120:
1116:
1112:
1109:
1106:
1103:
1097:
1094:
1088:
1082:
1078:
1072:
1068:
1065:
1061:
1058:
1057:
1030:
1006:
1003:
1000:
997:
994:
989:
985:
973:
972:
957:
954:
950:
944:
941:
938:
935:
930:
922:
918:
914:
909:
904:
900:
893:
885:
881:
877:
874:
871:
868:
862:
859:
853:
847:
844:
841:
838:
833:
825:
821:
817:
812:
807:
803:
796:
788:
784:
780:
777:
774:
771:
765:
762:
756:
753:
751:
749:
743:
740:
737:
734:
729:
721:
717:
713:
708:
703:
699:
692:
684:
680:
676:
673:
670:
667:
661:
658:
652:
646:
643:
640:
637:
632:
626:
623:
618:
615:
609:
603:
600:
597:
594:
591:
587:
581:
577:
573:
570:
566:
563:
562:
539:
536:
533:
530:
510:
507:
504:
501:
498:
495:
492:
489:
458:
438:
435:
432:
429:
405:
385:
365:
362:
359:
356:
353:
342:potential flow
319:
318:
287:
276:
186:
183:
67:fluid dynamics
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6234:
6223:
6220:
6218:
6215:
6213:
6210:
6209:
6207:
6192:
6187:
6182:
6180:
6172:
6170:
6162:
6161:
6158:
6152:
6149:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6117:
6114:
6112:
6109:
6107:
6104:
6100:
6097:
6096:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6039:Marine energy
6037:
6035:
6032:
6030:
6029:
6024:
6022:
6019:
6017:
6014:
6012:
6009:
6007:
6006:Acidification
6004:
6003:
6001:
5997:
5991:
5988:
5986:
5983:
5981:
5978:
5977:
5975:
5971:
5965:
5962:
5960:
5959:SOFAR channel
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5941:
5939:
5937:
5933:
5927:
5924:
5922:
5919:
5917:
5914:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5893:
5891:
5889:
5885:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5836:
5834:
5831:
5830:
5828:
5824:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5767:Oceanic crust
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5742:Fracture zone
5740:
5738:
5735:
5733:
5730:
5729:
5727:
5725:
5719:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5677:Oceanic basin
5675:
5673:
5670:
5668:
5665:
5663:
5660:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5607:Abyssal plain
5605:
5603:
5600:
5599:
5597:
5595:
5591:
5585:
5582:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5555:
5552:
5550:
5547:
5545:
5542:
5540:
5537:
5535:
5532:
5530:
5527:
5525:
5522:
5520:
5517:
5515:
5514:Internal tide
5512:
5510:
5507:
5505:
5502:
5500:
5497:
5496:
5494:
5492:
5488:
5482:
5479:
5477:
5474:
5472:
5469:
5467:
5464:
5460:
5457:
5456:
5455:
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5415:
5412:
5410:
5409:Ocean current
5407:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5355:
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5335:
5332:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5295:
5292:
5291:
5289:
5287:
5283:
5278:
5267:
5255:
5252:
5251:
5250:
5247:
5245:
5242:
5240:
5237:
5233:
5230:
5228:
5225:
5224:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5207:Wave shoaling
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5167:Ursell number
5165:
5163:
5160:
5156:
5153:
5152:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5050:Internal wave
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
5000:Breaking wave
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4975:
4973:
4969:
4965:
4958:
4953:
4951:
4946:
4944:
4939:
4938:
4935:
4927:
4925:0-12-493250-9
4921:
4917:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4885:
4881:
4877:
4873:
4869:
4865:
4863:0-444-51149-0
4859:
4855:
4850:
4846:
4844:0-521-59832-X
4840:
4836:
4831:
4827:
4823:
4820:(1–3): 5–39.
4819:
4815:
4811:
4806:
4803:
4792:on 2012-02-08
4791:
4787:
4781:
4777:
4773:
4772:
4766:
4762:
4758:
4757:
4752:
4748:
4744:
4740:
4736:
4732:
4728:
4724:
4723:
4719:
4710:
4709:www1.udel.edu
4706:
4700:
4697:
4692:
4688:
4682:
4679:
4674:
4670:
4664:
4661:
4655:
4652:
4646:
4644:
4640:
4634:
4631:
4625:
4622:
4616:
4613:
4609:
4605:
4599:
4596:
4589:
4587:
4585:
4581:
4577:
4573:
4569:
4565:
4564:finite-volume
4561:
4557:
4553:
4543:
4536:
4531:
4524:
4522:
4520:
4519:cnoidal waves
4512:
4508:
4504:
4500:
4496:
4493:
4489:
4485:
4481:
4478:
4474:
4470:
4466:
4465:
4464:
4457:
4452:
4448:
4445:
4441:
4437:
4434:extension to
4433:
4430:
4427:inclusion of
4426:
4423:
4422:wave breaking
4420:inclusion of
4419:
4416:
4413:
4409:
4405:
4402:
4398:
4395:
4391:
4388:
4384:
4383:
4382:
4379:
4377:
4373:
4369:
4365:
4361:
4353:
4351:
4337:
4317:
4309:
4304:
4302:
4298:
4294:
4290:
4287:
4283:
4267:
4264:
4259:
4255:
4249:
4245:
4235:
4233:
4217:
4197:
4177:
4173:
4169:
4166:
4163:
4160:
4157:
4149:
4130:
4126:
4120:
4116:
4110:
4106:
4099:
4096:
4090:
4086:
4082:
4077:
4074:
4070:
4064:
4060:
4052:
4051:
4050:
4048:
4044:
4039:
4025:
4005:
3997:
3993:
3977:
3973:
3969:
3966:
3963:
3960:
3952:
3948:
3932:
3924:
3916:
3900:
3880:
3876:
3872:
3869:
3866:
3863:
3855:
3839:
3832:
3829:
3813:
3806:
3805:
3804:
3787:
3779:
3775:
3769:
3765:
3758:
3755:
3749:
3745:
3738:
3734:
3728:
3724:
3717:
3714:
3708:
3704:
3697:
3694:
3690:
3684:
3680:
3672:
3671:
3670:
3668:
3664:
3659:
3657:
3656:approximation
3653:
3649:
3645:
3644:infinitesimal
3641:
3637:
3633:
3630:of different
3629:
3622:
3618:
3614:
3609:
3604:
3587:
3584:
3561:
3558:
3551:
3545:
3541:
3531:
3522:
3506:
3503:
3495:
3489:
3485:
3478:
3477:
3473:
3456:
3453:
3446:
3440:
3434:
3430:
3423:
3422:
3418:
3402:
3399:
3391:
3388:
3382:
3378:
3371:
3370:
3367:
3350:
3347:
3343:
3338:
3328:
3324:
3315:
3310:
3298:
3292:
3288:
3283:
3278:
3269:
3265:
3255:
3244:
3235:
3231:
3222:
3217:
3205:
3196:
3192:
3183:
3178:
3163:
3162:
3161:
3159:
3158:normalization
3143:
3123:
3115:
3111:
3110:Ursell number
3092:
3088:
3083:
3074:
3070:
3061:
3056:
3042:
3038:
3031:
3028:
3022:
3016:
3011:
3007:
2998:
2995:
2989:
2980:
2976:
2966:
2955:
2952:
2948:
2939:
2935:
2926:
2921:
2909:
2906:
2902:
2893:
2889:
2880:
2875:
2860:
2859:
2855:
2854:
2853:
2839:
2831:
2827:
2823:
2818:
2816:
2800:
2780:
2772:
2749:
2741:
2737:
2729:
2719:
2715:
2709:
2695:
2691:
2684:
2681:
2675:
2668:
2660:
2650:
2646:
2639:
2629:
2625:
2612:
2608:
2603:
2601:
2592:
2582:
2578:
2564:
2556:
2552:
2541:
2537:
2531:
2517:
2513:
2506:
2503:
2497:
2492:
2486:
2482:
2476:
2472:
2469:
2466:
2462:
2457:
2449:
2436:
2434:
2425:
2417:
2400:
2399:
2395:
2394:
2393:
2391:
2387:
2371:
2368:
2365:
2362:
2342:
2335:
2329:
2325:
2318:
2313:
2309:
2286:
2282:
2261:
2253:
2249:
2245:
2229:
2222:
2214:
2210:
2194:
2187:
2172:
2165:
2161:
2155:
2152:
2144:
2143:flow velocity
2128:
2121:
2106:
2099:
2095:
2089:
2086:
2078:
2077:flow velocity
2062:
2055:
2054:
2053:
2032:
2029:
2025:
2021:
2017:
2013:
2008:
2002:
1998:
1994:
1989:
1985:
1980:
1973:
1970:
1964:
1962:
1953:
1945:
1932:
1928:
1924:
1920:
1913:
1905:
1895:
1891:
1889:
1880:
1872:
1855:
1854:
1853:
1836:
1833:
1830:
1824:
1821:
1818:
1810:
1806:
1787:
1784:
1781:
1770:
1766:
1762:
1754:
1749:
1747:
1733:
1730:
1727:
1724:
1704:
1697:
1693:
1663:
1659:
1654:
1650:
1641:
1637:
1626:
1622:
1616:
1602:
1594:
1591:
1588:
1579:
1576:
1570:
1561:
1557:
1546:
1542:
1536:
1522:
1514:
1511:
1508:
1499:
1496:
1490:
1484:
1480:
1474:
1467:
1459:
1454:
1450:
1444:
1441:
1438:
1435:
1430:
1424:
1416:
1407:
1397:
1393:
1383:
1370:
1362:
1359:
1356:
1347:
1344:
1338:
1332:
1329:
1326:
1323:
1318:
1312:
1304:
1295:
1286:
1283:
1280:
1272:
1267:
1265:
1255:
1250:
1246:
1237:
1233:
1222:
1218:
1212:
1198:
1190:
1187:
1184:
1175:
1172:
1166:
1157:
1153:
1142:
1138:
1132:
1118:
1110:
1107:
1104:
1095:
1092:
1086:
1080:
1076:
1070:
1063:
1059:
1048:
1047:
1046:
1044:
1028:
1020:
1001:
998:
995:
987:
983:
955:
952:
948:
942:
939:
936:
933:
928:
920:
916:
907:
902:
891:
883:
875:
872:
869:
860:
857:
851:
845:
842:
839:
836:
831:
823:
819:
810:
805:
794:
786:
778:
775:
772:
763:
760:
754:
752:
741:
738:
735:
732:
727:
719:
715:
706:
701:
690:
682:
674:
671:
668:
659:
656:
650:
644:
641:
638:
635:
630:
624:
616:
607:
598:
595:
592:
585:
579:
575:
568:
564:
553:
552:
551:
537:
534:
531:
528:
505:
502:
499:
496:
493:
487:
480:
476:
472:
456:
436:
433:
430:
427:
419:
416:the vertical
403:
383:
360:
357:
354:
343:
338:
336:
332:
328:
324:
316:
312:
308:
304:
300:
297:and the zero-
296:
292:
288:
285:
281:
277:
274:
270:
266:
265:flow velocity
262:
258:
257:
256:
253:
251:
247:
242:
240:
236:
228:
224:
220:
216:
212:
208:
204:
200:
196:
195:cross section
191:
184:
182:
180:
176:
171:
169:
165:
162:
158:
154:
150:
146:
142:
138:
134:
130:
127:
123:
122:flow velocity
119:
114:
112:
108:
104:
103:solitary wave
100:
96:
92:
88:
84:
80:
79:approximation
76:
72:
68:
60:
59:non-linearity
56:
52:
48:
44:
39:
33:
19:
6146:Water column
6094:Oceanography
6069:Observations
6064:Explorations
6034:Marginal sea
6027:
5985:OSTM/Jason-2
5817:Volcanic arc
5792:Slab suction
5509:Head of tide
5399:Loop Current
5339:Ekman spiral
5125:Stokes drift
5035:Gravity wave
5010:Cnoidal wave
4994:
4915:
4879:
4875:
4853:
4834:
4817:
4813:
4801:
4794:. Retrieved
4790:the original
4775:
4770:
4760:
4754:
4738:
4734:
4708:
4699:
4690:
4681:
4672:
4663:
4654:
4633:
4624:
4615:
4607:
4603:
4598:
4576:Kirby (2003)
4548:
4516:
4461:
4444:mass density
4380:
4375:
4371:
4367:
4363:
4357:
4305:
4300:
4281:
4236:
4231:
4147:
4145:
4046:
4040:
3995:
3946:
3920:
3802:
3666:
3660:
3651:
3636:phase speeds
3632:wave lengths
3626:
3612:
3607:
3602:
3365:
3107:
2830:free surface
2821:
2819:
2770:
2768:
2218:
2209:acceleration
2051:
1809:free surface
1758:
1682:
974:
376:plane, with
339:
320:
254:
243:
232:
222:
172:
132:
115:
110:
70:
64:
49:, including
6217:Water waves
6136:Thermocline
5853:Mesopelagic
5826:Ocean zones
5797:Slab window
5662:Hydrography
5602:Abyssal fan
5569:Tidal range
5559:Tidal power
5554:Tidal force
5439:Rip current
5374:Gulf Stream
5334:Ekman layer
5324:Downwelling
5299:Baroclinity
5286:Circulation
5182:Wave height
5172:Wave action
5155:megatsunami
5135:Stokes wave
5095:Rossby wave
5060:Kelvin wave
5040:Green's law
4293:phase speed
4286:real-valued
4041:The linear
3992:engineering
3915:wave length
3854:wave number
3828:phase speed
3661:The linear
3628:Water waves
2145:component:
2079:component:
1803:plane, the
1761:water waves
331:coordinates
239:water waves
157:water waves
118:water waves
75:water waves
55:diffraction
6206:Categories
6074:Reanalysis
5973:Satellites
5954:Sofar bomb
5802:Subduction
5777:Ridge push
5672:Ocean bank
5652:Contourite
5579:Tide gauge
5564:Tidal race
5549:Tidal bore
5539:Slack tide
5504:Earth tide
5424:Ocean gyre
5244:Wind setup
5239:Wind fetch
5202:Wave setup
5197:Wave radar
5192:Wave power
5090:Rogue wave
5020:Dispersion
4796:2008-01-21
4741:: 755–759.
4720:References
4507:narrowband
4492:dimensions
4412:elevations
4401:non-linear
4387:bathymetry
2832:elevation
1811:elevation
1755:Derivation
418:coordinate
235:coordinate
215:wavelength
205:and sharp
175:wavelength
153:simulation
126:non-linear
83:non-linear
51:refraction
5936:Acoustics
5888:Sea level
5787:Slab pull
5724:tectonics
5632:Cold seep
5594:Landforms
5471:Whirlpool
5466:Upwelling
5249:Wind wave
5177:Wave base
5105:Sea state
5025:Edge wave
5015:Cross sea
4904:119385147
4763:: 55–108.
4511:modulated
4477:dimension
4440:interface
4406:making a
4403:behavior,
4399:improved
4392:improved
4318:λ
4289:solutions
4280:equation
4230:, called
4198:λ
4170:π
4091:−
4006:λ
3990:. So, in
3970:π
3901:λ
3881:λ
3873:π
3648:amplitude
3486:ξ
3431:τ
3400:η
3379:ψ
3325:ξ
3321:∂
3316:ψ
3307:∂
3289:ψ
3266:ξ
3262:∂
3252:∂
3245:−
3232:ξ
3228:∂
3223:ψ
3214:∂
3206:−
3193:τ
3189:∂
3184:ψ
3175:∂
3067:∂
3062:η
3053:∂
3008:η
2973:∂
2963:∂
2949:−
2932:∂
2927:η
2918:∂
2903:−
2886:∂
2881:η
2872:∂
2840:η
2734:∂
2727:∂
2706:∂
2666:∂
2661:η
2658:∂
2637:∂
2622:∂
2590:∂
2575:∂
2549:∂
2528:∂
2473:η
2447:∂
2443:∂
2423:∂
2418:η
2415:∂
2369:−
2340:∂
2326:φ
2322:∂
2262:η
2252:quadratic
2230:φ
2170:∂
2162:φ
2159:∂
2104:∂
2096:φ
2093:∂
2022:η
1951:∂
1946:φ
1943:∂
1921:−
1911:∂
1906:η
1903:∂
1878:∂
1873:η
1870:∂
1825:η
1731:−
1702:∂
1694:φ
1691:∂
1655:⋯
1634:∂
1623:φ
1613:∂
1554:∂
1543:φ
1533:∂
1491:−
1481:φ
1455:⋯
1442:−
1422:∂
1417:φ
1414:∂
1390:∂
1380:∂
1339:−
1330:−
1310:∂
1305:φ
1302:∂
1251:⋯
1230:∂
1219:φ
1209:∂
1150:∂
1139:φ
1129:∂
1087:−
1077:φ
1060:φ
1045:, gives:
1029:φ
984:φ
953:⋯
940:−
913:∂
908:φ
899:∂
843:−
816:∂
811:φ
802:∂
739:−
712:∂
707:φ
698:∂
642:−
622:∂
617:φ
614:∂
576:φ
565:φ
535:−
488:φ
434:−
327:functions
293:) for an
273:elevation
131:, called
6169:Category
6121:Seawater
5848:Littoral
5843:Deep sea
5702:Seamount
5584:Tideline
5529:Rip tide
5459:shutdown
5429:Overflow
5162:Undertow
5005:Clapotis
4749:(1872).
4729:(1871).
4385:varying
3945:for set
2828:for the
449:, where
168:harbours
151:for the
6179:Commons
6049:Mooring
5999:Related
5990:Jason-3
5980:Jason-1
5863:Pelagic
5858:Oceanic
5833:Benthic
5150:Tsunami
5120:Soliton
4884:Bibcode
4552:MIKE 21
4513:waves).
4291:of the
3893:, with
2384:). The
2213:gravity
2207:is the
2052:where:
1807:at the
1771:in the
469:is the
203:troughs
197:in the
161:shallow
107:soliton
97:of the
5868:Photic
5697:Seabed
5110:Seiche
4922:
4902:
4860:
4841:
4782:
4438:on an
3803:with:
3366:with:
2248:linear
1763:on an
975:where
284:finite
207:crests
77:is an
69:, the
6059:Ocean
6028:Alvin
5878:Swash
5722:Plate
5667:Knoll
5657:Guyot
5612:Atoll
5491:Tides
5254:model
5140:Swell
4972:Waves
4900:S2CID
4590:Notes
4049:are:
3646:wave
3112:. In
2386:cubic
1852:are:
333:(and
278:this
211:scale
43:shoal
6026:DSV
6011:Argo
5873:Surf
5329:Eddy
4920:ISBN
4858:ISBN
4839:ISBN
4780:ISBN
4554:and
4497:the
4482:the
4471:for
4467:the
4306:The
4265:>
4164:<
3967:<
3921:The
3913:the
3852:the
3826:the
2771:i.e.
2274:and
2250:and
1767:and
1759:For
1021:for
471:mean
335:time
299:curl
267:(or
223:i.e.
166:and
164:seas
85:and
73:for
4892:doi
4822:doi
4566:or
4556:SMS
4535:SMS
4505:of
4372:the
4364:the
4234:.
2211:by
337:).
159:in
155:of
105:or
65:In
6208::
4898:.
4890:.
4880:27
4878:.
4818:21
4816:.
4776:13
4761:17
4753:.
4739:72
4737:.
4733:.
4707:.
4689:.
4671:.
4642:^
4610:).
4586:.
4578:,
4562:,
4350:.
4038:.
3917:).
3658:.
3160::
3093:0.
2852::
2817:.
1580:24
1176:24
861:24
550::
259:a
170:.
113:.
53:,
4956:e
4949:t
4942:v
4928:.
4906:.
4894::
4886::
4866:.
4847:.
4828:.
4824::
4804:.
4799:.
4711:.
4693:.
4675:.
4606:(
4537:.
4494:,
4479:,
4453:.
4446:,
4431:,
4424:,
4414:,
4396:,
4389:,
4376:B
4338:h
4301:A
4282:B
4268:3
4260:2
4256:h
4250:2
4246:k
4218:h
4178:7
4174:/
4167:2
4161:h
4158:k
4148:B
4131:.
4127:)
4121:2
4117:h
4111:2
4107:k
4100:3
4097:1
4087:1
4083:(
4078:h
4075:g
4071:=
4065:2
4061:c
4047:B
4026:h
3996:A
3978:2
3974:/
3964:h
3961:k
3947:A
3933:c
3877:/
3870:2
3867:=
3864:k
3856:(
3840:k
3830:,
3814:c
3788:,
3780:2
3776:h
3770:2
3766:k
3759:2
3756:1
3750:+
3746:1
3739:2
3735:h
3729:2
3725:k
3718:6
3715:1
3709:+
3705:1
3698:h
3695:g
3691:=
3685:2
3681:c
3667:A
3613:C
3608:B
3603:A
3600:.
3588:h
3585:k
3565:)
3562:h
3559:g
3556:(
3552:/
3546:2
3542:c
3507:h
3504:x
3496:3
3490:=
3457:h
3454:g
3447:t
3441:3
3435:=
3403:h
3392:2
3389:1
3383:=
3351:,
3348:0
3344:=
3339:)
3329:2
3311:2
3299:+
3293:2
3284:3
3279:(
3270:2
3256:2
3236:2
3218:2
3197:2
3179:2
3144:g
3124:h
3089:=
3084:)
3075:2
3071:x
3057:2
3043:2
3039:h
3032:3
3029:1
3023:+
3017:h
3012:2
2999:2
2996:3
2990:(
2981:2
2977:x
2967:2
2956:h
2953:g
2940:2
2936:x
2922:2
2910:h
2907:g
2894:2
2890:t
2876:2
2822:A
2801:x
2781:h
2750:.
2742:2
2738:x
2730:t
2720:b
2716:u
2710:3
2696:2
2692:h
2685:2
2682:1
2676:=
2669:x
2651:g
2647:+
2640:x
2630:b
2626:u
2613:b
2609:u
2604:+
2593:t
2583:b
2579:u
2565:,
2557:3
2553:x
2542:b
2538:u
2532:3
2518:3
2514:h
2507:6
2504:1
2498:=
2493:]
2487:b
2483:u
2477:)
2470:+
2467:h
2463:(
2458:[
2450:x
2437:+
2426:t
2372:h
2366:=
2363:z
2343:x
2336:/
2330:b
2319:=
2314:b
2310:u
2287:b
2283:u
2215:.
2195:g
2185:,
2173:z
2166:/
2156:=
2153:w
2129:w
2119:,
2107:x
2100:/
2090:=
2087:u
2063:u
2033:,
2030:0
2026:=
2018:g
2014:+
2009:)
2003:2
1999:w
1995:+
1990:2
1986:u
1981:(
1974:2
1971:1
1965:+
1954:t
1933:0
1929:=
1925:w
1914:x
1896:u
1892:+
1881:t
1840:)
1837:t
1834:,
1831:x
1828:(
1822:=
1819:z
1791:)
1788:z
1785:,
1782:x
1779:(
1734:h
1728:=
1725:z
1705:z
1698:/
1664:,
1660:}
1651:+
1642:4
1638:x
1627:b
1617:4
1603:4
1599:)
1595:h
1592:+
1589:z
1586:(
1577:1
1571:+
1562:2
1558:x
1547:b
1537:2
1523:2
1519:)
1515:h
1512:+
1509:z
1506:(
1500:2
1497:1
1485:b
1475:{
1468:=
1460:}
1451:+
1445:h
1439:=
1436:z
1431:]
1425:z
1408:[
1398:2
1394:x
1384:2
1371:3
1367:)
1363:h
1360:+
1357:z
1354:(
1348:6
1345:1
1333:h
1327:=
1324:z
1319:]
1313:z
1296:[
1290:)
1287:h
1284:+
1281:z
1278:(
1273:{
1268:+
1256:}
1247:+
1238:4
1234:x
1223:b
1213:4
1199:4
1195:)
1191:h
1188:+
1185:z
1182:(
1173:1
1167:+
1158:2
1154:x
1143:b
1133:2
1119:2
1115:)
1111:h
1108:+
1105:z
1102:(
1096:2
1093:1
1081:b
1071:{
1064:=
1005:)
1002:t
999:,
996:x
993:(
988:b
956:,
949:+
943:h
937:=
934:z
929:]
921:4
917:z
903:4
892:[
884:4
880:)
876:h
873:+
870:z
867:(
858:1
852:+
846:h
840:=
837:z
832:]
824:3
820:z
806:3
795:[
787:3
783:)
779:h
776:+
773:z
770:(
764:6
761:1
755:+
742:h
736:=
733:z
728:]
720:2
716:z
702:2
691:[
683:2
679:)
675:h
672:+
669:z
666:(
660:2
657:1
651:+
645:h
639:=
636:z
631:]
625:z
608:[
602:)
599:h
596:+
593:z
590:(
586:+
580:b
569:=
538:h
532:=
529:z
509:)
506:t
503:,
500:z
497:,
494:x
491:(
457:h
437:h
431:=
428:z
404:z
384:x
364:)
361:z
358:,
355:x
352:(
317:.
275:,
219:m
61:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
