1667:
1040:
1662:{\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}}}
2753:
4519:
5266:
2392:
6154:
179:
6175:
4531:
3092:
2748:{\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}}}
5255:
6164:
3519:
2036:
959:
27:
3350:
2852:
1847:
545:
3155:
3087:{\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.}
2031:{\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}}}
954:{\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}}}
3345:{\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,}
3787:
4451:
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:
230:. 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.
4130:
4538:
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
3459:
3404:
4740:"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"
2342:
3508:
2397:
1852:
1045:
550:
2172:
2106:
1704:
3664:
310:
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
1004:
3880:
4267:
508:
4177:
4534:
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.
3977:
1839:
3564:
4317:
4197:
4005:
3900:
2229:
1028:
2839:
2288:
2261:
1790:
363:
2371:
1733:
537:
436:
3587:
4337:
4217:
4025:
3932:
3839:
3813:
3143:
3123:
2800:
2780:
2194:
2128:
2062:
456:
403:
383:
4841:
Kirby, J.T. (2003). "Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents". In Lakhan, V.C. (ed.).
4563:
of the model equations. Scientific reviews and intercomparisons of several
Boussinesq-type equations, their numerical approximation and performance are e.g.
3643:. 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
4522:
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
34:
with a
Boussinesq-type model. The waves propagate over an elliptic-shaped underwater shoal on a plane beach. This example combines several effects of
5894:
4744:
4044:
5884:
4943:
4593:"Tous les ingénieurs connaissent les belles expériences de J. Scott Russell et M. Basin sur la production et la propagation des ondes solitaires"
3415:
20:
4597:"All engineers know the beautiful experiments of J. Scott Russell and M. Basin on the generation and propagation of solitary waves"
3363:
5800:
4758:
4351:
which are referred to as
Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as
6167:
5215:
4739:
4772:
2293:
6210:
5447:
3470:
4936:
4720:"Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire"
4472:
5337:
6042:
5469:
5357:
5889:
5160:
4912:
4850:
4831:
4487:
2137:
2071:
5347:
5307:
1675:
6077:
5063:
3097:
From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the
19:
This article is about the
Boussinesq approximation for waves on a free-moving fluid surface. For other uses, see
6157:
5750:
4929:
4457:
4285:
2378:
311:
170:
is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).
117:
4506:
Besides solitary wave solutions, the
Korteweg–de Vries equation also has periodic and exact solutions, called
2814:
4355:
Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them
226:
from the flow equations, while retaining some of the influences of the vertical structure of the flow under
5205:
3782:{\displaystyle c^{2}\,=\;gh\,{\frac {1\,+\,{\frac {1}{6}}\,k^{2}h^{2}}{1\,+\,{\frac {1}{2}}\,k^{2}h^{2}}},}
91:
5265:
4406:
dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
5402:
3146:
967:
241:). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.
4547:. Some of the free Boussinesq models are Celeris, COULWAVE, and FUNWAVE. Most numerical models employ
3848:
5937:
5342:
5302:
4229:
472:
214:
the difference between crest and trough elevation), and the mean water depth is 5 m, while the
6067:
5442:
5432:
5372:
5008:
4978:
4548:
4544:
4523:
4491:
4382:
4142:
4031:
3651:
3628:
3605:
3102:
183:
125:
3945:
1803:
6200:
6104:
6087:
5924:
5417:
5282:
5220:
5210:
5103:
4296:
2803:
149:
129:
35:
3525:
6099:
6037:
5464:
5150:
4277:
315:
4903:(1972). "Equations for water waves and the approximations behind them". In Meyer, R.E. (ed.).
4302:
4182:
3990:
3885:
2214:
1013:
5932:
5914:
5422:
5317:
4952:
4556:
4439:
4428:
1007:
4693:
6205:
6119:
5952:
5655:
5512:
5377:
5088:
4872:
4552:
2824:
2266:
2246:
1763:
1753:
336:
2347:
1709:
513:
412:
162:
While the Boussinesq approximation is applicable to fairly long waves – that is, when the
8:
6114:
5999:
5994:
5720:
5392:
5352:
5068:
4518:
4348:
2232:
1793:
1031:
303:
295:
283:
279:
238:
133:
87:
4876:
3569:
6057:
5770:
5760:
5725:
5625:
5610:
5507:
4888:
4322:
4202:
4010:
3917:
3824:
3798:
3128:
3108:
2809:
Under some additional approximations, but at the same order of accuracy, the above set
2802:. When the right-hand sides of the above equations are set to zero, they reduce to the
2785:
2765:
2240:
2209:
2179:
2113:
2047:
467:
441:
388:
368:
287:
257:
215:
83:
222:
The essential idea in the Boussinesq approximation is the elimination of the vertical
6139:
6129:
6072:
6052:
5735:
5700:
5635:
5615:
5605:
5487:
5175:
5033:
4908:
4892:
4846:
4827:
4814:
4778:
4768:
4735:
4715:
4675:
1757:
291:
234:
79:
4826:. Cambridge Texts in Applied Mathematics. Vol. 19. Cambridge University Press.
6094:
6062:
6032:
5841:
5826:
5695:
5630:
5522:
5437:
5367:
5292:
5073:
5043:
4973:
4968:
4900:
4880:
4860:
4810:
4798:
4719:
4476:
4461:
4396:
3939:
463:
299:
268:
249:
191:
187:
167:
5899:
5795:
5745:
5710:
5670:
5562:
5532:
5382:
5332:
5242:
5200:
5133:
5058:
5018:
4417:
195:
137:
6009:
6004:
5909:
5904:
5740:
5680:
5675:
5407:
5297:
5118:
5053:
5028:
4560:
4424:
3911:
2374:
330:
55:
5254:
4884:
4370:
Some directions, into which the Boussinesq equations have been extended, are:
6194:
6179:
6027:
5947:
5836:
5755:
5730:
5665:
5595:
5502:
5397:
5274:
5195:
5155:
5128:
5038:
4988:
4410:
3644:
3632:
3098:
2131:
2065:
253:
199:
110:
67:
47:
4125:{\displaystyle c^{2}\,=\,gh\,\left(1\,-\,{\frac {1}{3}}\,k^{2}h^{2}\right).}
6134:
6082:
6022:
5973:
5851:
5846:
5821:
5805:
5780:
5497:
5387:
5327:
5113:
5023:
4998:
4507:
4432:
2818:
2197:
1797:
98:). The 1872 paper of Boussinesq introduces the equations now known as the
6124:
5856:
5785:
5650:
5590:
5557:
5547:
5542:
5427:
5362:
5322:
5312:
5287:
5170:
5143:
5123:
5083:
5048:
4281:
4274:
3980:
3903:
3842:
3816:
3624:
3620:
3616:
2377:
and higher order terms are assumed to be negligible. Then, the following
1749:
1735:. This series may subsequently be truncated to a finite number of terms.
319:
227:
145:
109:
takes into account the vertical structure of the horizontal and vertical
106:
63:
43:
4921:
272:
237:
in 1871, to construct an approximate solution for the solitary wave (or
5942:
5790:
5765:
5660:
5640:
5567:
5552:
5537:
5527:
5492:
5412:
5232:
5227:
5190:
5185:
5180:
5078:
4845:. Elsevier Oceanography Series. Vol. 67. Elsevier. pp. 1–41.
4495:
4400:
4389:
4375:
4367:, since it is used in the analysis in the remainder of his 1872 paper.
4299:
have a relative error in the phase speed less than 4% for wave lengths
406:
223:
203:
163:
141:
114:
71:
39:
6174:
4446:
178:
6014:
5876:
5861:
5775:
5620:
5459:
5454:
5237:
5165:
5093:
5013:
5003:
4960:
4530:
4499:
4480:
4465:
3636:
261:
182:
Periodic waves in the Boussinesq approximation, shown in a vertical
6109:
5831:
5690:
5582:
5572:
5517:
4993:
156:
2758:
This set of equations has been derived for a flat horizontal bed,
26:
5978:
5968:
5138:
5108:
4540:
2201:
95:
3454:{\displaystyle \tau \,=\,{\sqrt {3}}\,t\,{\sqrt {\frac {g}{h}}}}
5685:
5098:
4824:
A modern introduction to the mathematical theory of water waves
4801:(1993). "Wave transformation in the nearshore zone: A review".
2236:
16:
Approximation valid for weakly non-linear and fairly long waves
4510:. These are approximate solutions of the Boussinesq equation.
233:
This elimination of the vertical coordinate was first done by
6047:
5866:
5645:
5600:
3518:
207:
31:
3399:{\displaystyle \psi \,=\,{\frac {1}{2}}\,{\frac {\eta }{h}}}
5479:
4342:
4273:
become physically meaningless, because there are no longer
459:
323:
82:, who first derived them in response to the observation by
4657:
152:
3145:
for non-dimensionalization, this equation reads, after
2337:{\displaystyle u_{b}=\partial \varphi _{b}/\partial x}
4325:
4305:
4232:
4205:
4185:
4145:
4047:
4013:
3993:
3948:
3920:
3888:
3851:
3827:
3801:
3667:
3572:
3528:
3473:
3418:
3366:
3158:
3131:
3111:
2855:
2827:
2788:
2768:
2395:
2350:
2296:
2269:
2249:
2217:
2182:
2140:
2116:
2074:
2050:
1850:
1806:
1766:
1712:
1678:
1043:
1016:
970:
548:
516:
475:
444:
415:
391:
371:
339:
3503:{\displaystyle \xi \,=\,{\sqrt {3}}\,{\frac {x}{h}}}
198:, due to the wave nonlinearity. This case (drawn on
4796:
4572:
4447:
Further approximations for one-way wave propagation
4135:The relative error in the phase speed for equation
136:, Boussinesq-type equations are frequently used in
4331:
4311:
4261:
4211:
4191:
4171:
4124:
4019:
3999:
3971:
3926:
3894:
3874:
3833:
3807:
3781:
3581:
3558:
3502:
3453:
3398:
3344:
3137:
3117:
3086:
2833:
2794:
2774:
2747:
2365:
2336:
2282:
2255:
2223:
2188:
2166:
2122:
2100:
2056:
2030:
1833:
1784:
1727:
1698:
1661:
1022:
998:
953:
531:
502:
450:
430:
397:
377:
357:
5895:North West Shelf Operational Oceanographic System
4905:Waves on Beaches and Resulting Sediment Transport
6192:
4363:Boussinesq equations is the above-mentioned set
1738:
75:
30:Simulation of periodic waves over an underwater
5885:Deep-ocean Assessment and Reporting of Tsunamis
2235:. Further, in the resulting equations only the
1006:is the velocity potential at the bed. Invoking
244:The steps in the Boussinesq approximation are:
4634:
4632:
3611:
2167:{\displaystyle w=\partial \varphi /\partial z}
2101:{\displaystyle u=\partial \varphi /\partial x}
1706:is zero at the – impermeable – horizontal bed
4937:
4658:"Celeria.org - Celeris Boussinesq Wave Model"
3942:, is less than 4% for a relative wave number
4745:Journal de Mathématiques Pures et Appliquées
1699:{\displaystyle \partial \varphi /\partial z}
4662:Celeria.org - Celeris Boussinesq Wave Model
4629:
4591:This paper (Boussinesq, 1872) starts with:
173:
166:is large compared to the water depth – the
4944:
4930:
4734:
4714:
3682:
132:, which are not frequency-dispersive). In
4951:
4899:
4859:
4756:
4724:Comptes Rendus de l'Académie des Sciences
4568:
4093:
4082:
4078:
4069:
4062:
4058:
3752:
3741:
3737:
3711:
3700:
3696:
3689:
3678:
3512:: the dimensionless horizontal position.
3489:
3481:
3477:
3438:
3434:
3426:
3422:
3385:
3374:
3370:
3335:
3331:
3325:
3290:
3286:
3275:
3271:
3236:
3232:
3197:
3193:
3080:
3076:
3036:
3025:
3014:
3010:
2992:
2947:
2940:
2936:
2901:
2894:
2890:
2721:
2689:
2678:
2667:
2663:
2642:
2638:
2634:
2606:
2595:
2587:
2511:
2500:
2489:
2485:
2469:
2444:
2428:
2420:
2208:Now the Boussinesq approximation for the
2017:
2013:
2009:
2005:
2001:
1967:
1956:
1948:
1920:
1916:
1912:
1908:
1887:
1883:
1875:
1646:
1642:
1638:
1596:
1573:
1562:
1558:
1516:
1493:
1482:
1478:
1467:
1459:
1446:
1442:
1438:
1364:
1341:
1330:
1326:
1281:
1265:
1259:
1248:
1242:
1238:
1234:
1192:
1169:
1158:
1154:
1112:
1089:
1078:
1074:
1063:
1055:
1051:
940:
936:
877:
854:
843:
839:
780:
757:
746:
735:
676:
653:
642:
638:
593:
577:
573:
560:
556:
4529:
4517:
4343:Boussinesq-type equations and extensions
4288:(Boussinesq, 1872, equation 25, see set
3517:
2845:set B – Boussinesq (1872), equation (26)
2385:set A – Boussinesq (1872), equation (25)
177:
25:
4821:
4763:. Advanced Series on Ocean Engineering
4292:above) does not have this shortcoming.
3408:: the dimensionless surface elevation,
252:is made of the horizontal and vertical
6193:
5216:one-dimensional Saint-Venant equations
4319:in excess of 13 times the water depth
3566:as a function of relative wave number
2782:is a constant independent of position
2231:, as given above, is applied in these
4925:
4840:
4564:
6163:
4760:Wave propagation over uneven bottoms
4347:There are an overwhelming number of
4199:longer than 7 times the water depth
4007:larger than 4 times the water depth
4907:. Academic Press. pp. 95–122.
4647:Dingemans (1997), p. 473 & 516.
4573:Hamm, Madsen & Peregrine (1993)
4513:
4431:between fluid domains of different
3599:= Boussinesq (1872), equation (26),
3594:= Boussinesq (1872), equation (25),
2344:the horizontal velocity at the bed
78:. The approximation is named after
13:
6043:National Oceanographic Data Center
5470:World Ocean Circulation Experiment
5358:Global Ocean Data Analysis Project
3654:characteristics for the above set
3309:
3295:
3250:
3240:
3216:
3202:
3177:
3163:
3055:
3041:
2961:
2951:
2920:
2906:
2874:
2860:
2722:
2715:
2694:
2654:
2646:
2625:
2610:
2578:
2563:
2537:
2516:
2435:
2431:
2411:
2403:
2328:
2310:
2158:
2147:
2092:
2081:
1939:
1931:
1899:
1891:
1866:
1858:
1690:
1679:
1622:
1601:
1542:
1521:
1410:
1402:
1378:
1368:
1298:
1290:
1218:
1197:
1138:
1117:
901:
887:
804:
790:
700:
686:
610:
602:
14:
6222:
5890:Global Sea Level Observing System
4863:(1967). "Long waves on a beach".
4694:"James T. Kirby, Funwave program"
999:{\displaystyle \varphi _{b}(x,t)}
210:, the wave height is 1.8 m (
105:The Boussinesq approximation for
6173:
6162:
6153:
6152:
5348:Geochemical Ocean Sections Study
5264:
5253:
3875:{\displaystyle k=2\pi /\lambda }
6078:Ocean thermal energy conversion
5801:Vine–Matthews–Morley hypothesis
4767:. World Scientific, Singapore.
4473:Kadomtsev–Petviashvili equation
4262:{\displaystyle k^{2}h^{2}>3}
3604:= full linear wave theory, see
3125:and gravitational acceleration
503:{\displaystyle \varphi (x,z,t)}
4686:
4668:
4650:
4641:
4620:
4611:
4602:
4585:
4488:nonlinear Schrödinger equation
4286:partial differential equations
3553:
3544:
3463:: the dimensionless time, and
2379:partial differential equations
1828:
1816:
1779:
1767:
1587:
1574:
1507:
1494:
1355:
1342:
1278:
1266:
1183:
1170:
1103:
1090:
993:
981:
868:
855:
771:
758:
667:
654:
590:
578:
497:
479:
352:
340:
312:partial differential equations
294:are used, to replace vertical
278:the conservation of mass (see
118:partial differential equations
1:
4708:
4179:, equivalent to wave lengths
4172:{\displaystyle kh<2\pi /7}
3940:linear theory for water waves
2815:partial differential equation
1743:
1739:Original Boussinesq equations
333:over a horizontal bed in the
5338:El Niño–Southern Oscillation
5308:Craik–Leibovich vortex force
5064:Luke's variational principle
4843:Advances in Coastal Modeling
4815:10.1016/0378-3839(93)90044-9
4034:characteristics of equation
3972:{\displaystyle kh<\pi /2}
1834:{\displaystyle z=\eta (x,t)}
1672:since the vertical velocity
7:
6211:Equations of fluid dynamics
4475:for (near uni-directional)
4284:. The original set of two
3641:linear frequency dispersion
3612:Linear frequency dispersion
3522:Linear phase speed squared
2813:can be reduced to a single
190:direction. Notice the flat
10:
6227:
5403:Ocean dynamical thermostat
5251:
4865:Journal of Fluid Mechanics
4458:Korteweg–de Vries equation
4399:around different vertical
3559:{\displaystyle c^{2}/(gh)}
216:gravitational acceleration
18:
6148:
5987:
5961:
5938:Ocean acoustic tomography
5923:
5875:
5814:
5751:Mohorovičić discontinuity
5709:
5581:
5478:
5343:General circulation model
5273:
4979:Benjamin–Feir instability
4959:
4885:10.1017/S0022112067002605
4638:Dingemans (1997), p. 521.
4617:Dingemans (1997), p. 475.
4608:Dingemans (1997), p. 477.
4571:, Part 2, Chapter 5) and
4357:Boussinesq-type equations
3987:is valid for wavelengths
122:Boussinesq-type equations
6068:Ocean surface topography
5443:Thermohaline circulation
5433:Subsurface ocean current
5373:Hydrothermal circulation
5206:Wave–current interaction
4984:Boussinesq approximation
4797:Hamm, L.; Madsen, P.A.;
4757:Dingemans, M.W. (1997).
4578:
4492:complex-valued amplitude
4312:{\displaystyle \lambda }
4192:{\displaystyle \lambda }
4000:{\displaystyle \lambda }
3895:{\displaystyle \lambda }
3627:, a phenomenon known as
3606:dispersion (water waves)
3105:, using the water depth
3103:dimensionless quantities
2224:{\displaystyle \varphi }
1023:{\displaystyle \varphi }
409:. The bed is located at
329:As an example, consider
202:) shows a wave with the
174:Boussinesq approximation
60:Boussinesq approximation
21:Boussinesq approximation
6105:Sea surface temperature
6088:Outline of oceanography
5283:Atmospheric circulation
5221:shallow water equations
5211:Waves and shallow water
5104:Significant wave height
4490:(NLS equation) for the
4297:shallow water equations
2804:shallow water equations
130:shallow water equations
36:waves and shallow water
6100:Sea surface microlayer
5465:Wind generated current
4822:Johnson, R.S. (1997).
4626:Johnson (1997), p. 219
4535:
4527:
4333:
4313:
4263:
4213:
4193:
4173:
4126:
4021:
4001:
3973:
3928:
3896:
3876:
3835:
3809:
3783:
3623:travel with different
3608:
3583:
3560:
3504:
3455:
3400:
3346:
3139:
3119:
3088:
2835:
2796:
2776:
2749:
2367:
2338:
2284:
2257:
2243:terms with respect to
2225:
2190:
2168:
2124:
2102:
2058:
2032:
1835:
1786:
1729:
1700:
1663:
1024:
1000:
955:
533:
504:
452:
432:
399:
379:
359:
219:
51:
5933:Deep scattering layer
5915:World Geodetic System
5423:Princeton Ocean Model
5303:Coriolis–Stokes force
4953:Physical oceanography
4791:See Part 2, Chapter 5
4533:
4521:
4440:variational principle
4359:. Strictly speaking,
4334:
4314:
4264:
4226:For short waves with
4214:
4194:
4174:
4127:
4022:
4002:
3974:
3929:
3897:
3877:
3836:
3810:
3784:
3639:, the terminology is
3584:
3561:
3521:
3505:
3456:
3401:
3347:
3140:
3120:
3089:
2836:
2834:{\displaystyle \eta }
2797:
2777:
2750:
2368:
2339:
2285:
2283:{\displaystyle u_{b}}
2258:
2256:{\displaystyle \eta }
2226:
2191:
2169:
2125:
2103:
2059:
2033:
1836:
1787:
1785:{\displaystyle (x,z)}
1730:
1701:
1664:
1025:
1001:
956:
534:
510:around the bed level
505:
453:
433:
400:
380:
360:
358:{\displaystyle (x,z)}
298:of quantities in the
181:
29:
5953:Underwater acoustics
5513:Perigean spring tide
5378:Langmuir circulation
5089:Rossby-gravity waves
4383:frequency dispersion
4323:
4303:
4230:
4203:
4183:
4143:
4139:is less than 4% for
4045:
4032:frequency dispersion
4011:
3991:
3946:
3918:
3886:
3849:
3825:
3799:
3665:
3652:frequency dispersion
3629:frequency dispersion
3570:
3526:
3471:
3416:
3364:
3156:
3129:
3109:
2853:
2825:
2786:
2766:
2393:
2366:{\displaystyle z=-h}
2348:
2294:
2267:
2247:
2215:
2180:
2138:
2114:
2072:
2048:
1848:
1804:
1764:
1754:incompressible fluid
1728:{\displaystyle z=-h}
1710:
1676:
1041:
1014:
968:
546:
532:{\displaystyle z=-h}
514:
473:
442:
431:{\displaystyle z=-h}
413:
389:
369:
337:
128:(as opposite to the
126:frequency dispersion
124:, which incorporate
100:Boussinesq equations
46:, shoaling and weak
6115:Science On a Sphere
5721:Convergent boundary
5393:Modular Ocean Model
5353:Geostrophic current
5069:Mild-slope equation
4877:1967JFM....27..815P
4803:Coastal Engineering
4559:techniques for the
4349:mathematical models
3938:, as compared with
3914:in the phase speed
2290:are retained (with
2233:boundary conditions
1794:boundary conditions
1032:incompressible flow
385:the horizontal and
304:partial derivatives
296:partial derivatives
284:incompressible flow
280:continuity equation
260:) around a certain
239:wave of translation
206:equal to 39.1
134:coastal engineering
88:wave of translation
5771:Seafloor spreading
5761:Outer trench swell
5726:Divergent boundary
5626:Continental margin
5611:Carbonate platform
5508:Lunitidal interval
4748:. Deuxième Série.
4536:
4528:
4479:in two horizontal
4464:in one horizontal
4438:derivation from a
4329:
4309:
4259:
4209:
4189:
4169:
4122:
4017:
3997:
3983:applications, set
3969:
3924:
3892:
3872:
3831:
3805:
3779:
3658:of equations are:
3631:. For the case of
3609:
3582:{\displaystyle kh}
3579:
3556:
3500:
3451:
3396:
3342:
3135:
3115:
3084:
2831:
2792:
2772:
2745:
2743:
2363:
2334:
2280:
2253:
2221:
2210:velocity potential
2186:
2164:
2120:
2098:
2064:is the horizontal
2054:
2028:
2026:
1831:
1782:
1725:
1696:
1659:
1657:
1020:
1008:Laplace's equation
996:
951:
949:
529:
500:
468:velocity potential
448:
428:
395:
375:
355:
318:of the horizontal
271:is truncated to a
258:velocity potential
220:
113:. This results in
84:John Scott Russell
52:
6188:
6187:
6180:Oceans portal
6140:World Ocean Atlas
6130:Underwater glider
6073:Ocean temperature
5736:Hydrothermal vent
5701:Submarine volcano
5636:Continental shelf
5616:Coastal geography
5606:Bathymetric chart
5488:Amphidromic point
5176:Wave nonlinearity
5034:Infragravity wave
4774:978-981-02-0427-3
4549:finite-difference
4332:{\displaystyle h}
4221:fairly long waves
4212:{\displaystyle h}
4091:
4020:{\displaystyle h}
3927:{\displaystyle c}
3834:{\displaystyle k}
3808:{\displaystyle c}
3774:
3750:
3709:
3516:
3515:
3498:
3487:
3449:
3448:
3432:
3394:
3383:
3323:
3264:
3230:
3191:
3138:{\displaystyle g}
3118:{\displaystyle h}
3069:
3023:
3008:
2990:
2975:
2934:
2888:
2795:{\displaystyle x}
2775:{\displaystyle h}
2736:
2676:
2661:
2632:
2585:
2551:
2498:
2442:
2418:
2189:{\displaystyle g}
2123:{\displaystyle w}
2057:{\displaystyle u}
1965:
1946:
1906:
1873:
1758:irrotational flow
1636:
1571:
1556:
1491:
1417:
1392:
1339:
1305:
1232:
1167:
1152:
1087:
915:
852:
818:
755:
714:
651:
617:
451:{\displaystyle h}
398:{\displaystyle z}
378:{\displaystyle x}
292:irrotational flow
290:condition for an
235:Joseph Boussinesq
218:is 9.81 m/s.
80:Joseph Boussinesq
76:fairly long waves
70:valid for weakly
6218:
6178:
6177:
6166:
6165:
6156:
6155:
6095:Pelagic sediment
6033:Marine pollution
5827:Deep ocean water
5696:Submarine canyon
5631:Continental rise
5523:Rule of twelfths
5438:Sverdrup balance
5368:Humboldt Current
5293:Boundary current
5268:
5257:
5074:Radiation stress
5044:Iribarren number
5019:Equatorial waves
4974:Ballantine scale
4969:Airy wave theory
4946:
4939:
4932:
4923:
4922:
4918:
4896:
4856:
4837:
4818:
4789:
4787:
4786:
4777:. Archived from
4753:
4731:
4702:
4701:
4690:
4684:
4683:
4672:
4666:
4665:
4654:
4648:
4645:
4639:
4636:
4627:
4624:
4618:
4615:
4609:
4606:
4600:
4589:
4514:Numerical models
4477:wave propagation
4462:wave propagation
4397:Taylor expansion
4338:
4336:
4335:
4330:
4318:
4316:
4315:
4310:
4268:
4266:
4265:
4260:
4252:
4251:
4242:
4241:
4218:
4216:
4215:
4210:
4198:
4196:
4195:
4190:
4178:
4176:
4175:
4170:
4165:
4131:
4129:
4128:
4123:
4118:
4114:
4113:
4112:
4103:
4102:
4092:
4084:
4057:
4056:
4026:
4024:
4023:
4018:
4006:
4004:
4003:
3998:
3978:
3976:
3975:
3970:
3965:
3933:
3931:
3930:
3925:
3901:
3899:
3898:
3893:
3881:
3879:
3878:
3873:
3868:
3840:
3838:
3837:
3832:
3814:
3812:
3811:
3806:
3788:
3786:
3785:
3780:
3775:
3773:
3772:
3771:
3762:
3761:
3751:
3743:
3732:
3731:
3730:
3721:
3720:
3710:
3702:
3691:
3677:
3676:
3588:
3586:
3585:
3580:
3565:
3563:
3562:
3557:
3543:
3538:
3537:
3509:
3507:
3506:
3501:
3499:
3491:
3488:
3483:
3460:
3458:
3457:
3452:
3450:
3441:
3440:
3433:
3428:
3405:
3403:
3402:
3397:
3395:
3387:
3384:
3376:
3358:
3357:
3351:
3349:
3348:
3343:
3330:
3326:
3324:
3322:
3321:
3320:
3307:
3303:
3302:
3292:
3285:
3284:
3265:
3263:
3262:
3261:
3248:
3247:
3238:
3231:
3229:
3228:
3227:
3214:
3210:
3209:
3199:
3192:
3190:
3189:
3188:
3175:
3171:
3170:
3160:
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3093:
3091:
3090:
3085:
3075:
3071:
3070:
3068:
3067:
3066:
3053:
3049:
3048:
3038:
3035:
3034:
3024:
3016:
3009:
3004:
3003:
2994:
2991:
2983:
2976:
2974:
2973:
2972:
2959:
2958:
2949:
2935:
2933:
2932:
2931:
2918:
2914:
2913:
2903:
2889:
2887:
2886:
2885:
2872:
2868:
2867:
2857:
2840:
2838:
2837:
2832:
2801:
2799:
2798:
2793:
2781:
2779:
2778:
2773:
2754:
2752:
2751:
2746:
2744:
2737:
2735:
2734:
2733:
2713:
2712:
2711:
2702:
2701:
2691:
2688:
2687:
2677:
2669:
2662:
2660:
2652:
2644:
2633:
2631:
2623:
2622:
2621:
2608:
2605:
2604:
2586:
2584:
2576:
2575:
2574:
2561:
2552:
2550:
2549:
2548:
2535:
2534:
2533:
2524:
2523:
2513:
2510:
2509:
2499:
2491:
2484:
2480:
2479:
2478:
2468:
2464:
2443:
2441:
2430:
2419:
2417:
2409:
2401:
2372:
2370:
2369:
2364:
2343:
2341:
2340:
2335:
2327:
2322:
2321:
2306:
2305:
2289:
2287:
2286:
2281:
2279:
2278:
2262:
2260:
2259:
2254:
2230:
2228:
2227:
2222:
2195:
2193:
2192:
2187:
2173:
2171:
2170:
2165:
2157:
2130:is the vertical
2129:
2127:
2126:
2121:
2107:
2105:
2104:
2099:
2091:
2063:
2061:
2060:
2055:
2037:
2035:
2034:
2029:
2027:
2000:
1996:
1995:
1994:
1982:
1981:
1966:
1958:
1947:
1945:
1937:
1929:
1907:
1905:
1897:
1889:
1874:
1872:
1864:
1856:
1840:
1838:
1837:
1832:
1791:
1789:
1788:
1783:
1734:
1732:
1731:
1726:
1705:
1703:
1702:
1697:
1689:
1668:
1666:
1665:
1660:
1658:
1651:
1647:
1637:
1635:
1634:
1633:
1620:
1619:
1618:
1609:
1608:
1598:
1595:
1594:
1572:
1564:
1557:
1555:
1554:
1553:
1540:
1539:
1538:
1529:
1528:
1518:
1515:
1514:
1492:
1484:
1477:
1476:
1451:
1447:
1437:
1436:
1422:
1418:
1416:
1408:
1400:
1393:
1391:
1390:
1389:
1376:
1375:
1366:
1363:
1362:
1340:
1332:
1325:
1324:
1310:
1306:
1304:
1296:
1288:
1252:
1247:
1243:
1233:
1231:
1230:
1229:
1216:
1215:
1214:
1205:
1204:
1194:
1191:
1190:
1168:
1160:
1153:
1151:
1150:
1149:
1136:
1135:
1134:
1125:
1124:
1114:
1111:
1110:
1088:
1080:
1073:
1072:
1029:
1027:
1026:
1021:
1005:
1003:
1002:
997:
980:
979:
960:
958:
957:
952:
950:
935:
934:
920:
916:
914:
913:
912:
899:
895:
894:
884:
876:
875:
853:
845:
838:
837:
823:
819:
817:
816:
815:
802:
798:
797:
787:
779:
778:
756:
748:
739:
734:
733:
719:
715:
713:
712:
711:
698:
694:
693:
683:
675:
674:
652:
644:
637:
636:
622:
618:
616:
608:
600:
572:
571:
538:
536:
535:
530:
509:
507:
506:
501:
464:Taylor expansion
457:
455:
454:
449:
437:
435:
434:
429:
404:
402:
401:
396:
384:
382:
381:
376:
364:
362:
361:
356:
314:are in terms of
302:with horizontal
300:Taylor expansion
275:number of terms,
269:Taylor expansion
250:Taylor expansion
188:wave propagation
168:Stokes expansion
6226:
6225:
6221:
6220:
6219:
6217:
6216:
6215:
6191:
6190:
6189:
6184:
6172:
6144:
5983:
5957:
5919:
5900:Sea-level curve
5871:
5810:
5796:Transform fault
5746:Mid-ocean ridge
5712:
5705:
5671:Oceanic plateau
5577:
5563:Tidal resonance
5533:Theory of tides
5474:
5383:Longshore drift
5333:Ekman transport
5269:
5263:
5262:
5261:
5260:
5259:
5258:
5249:
5201:Wave turbulence
5134:Trochoidal wave
5059:Longshore drift
4955:
4950:
4915:
4901:Peregrine, D.H.
4861:Peregrine, D.H.
4853:
4834:
4799:Peregrine, D.H.
4784:
4782:
4775:
4711:
4706:
4705:
4692:
4691:
4687:
4676:"ISEC - Models"
4674:
4673:
4669:
4656:
4655:
4651:
4646:
4642:
4637:
4630:
4625:
4621:
4616:
4612:
4607:
4603:
4590:
4586:
4581:
4569:Dingemans (1997
4516:
4449:
4418:surface tension
4345:
4324:
4321:
4320:
4304:
4301:
4300:
4247:
4243:
4237:
4233:
4231:
4228:
4227:
4204:
4201:
4200:
4184:
4181:
4180:
4161:
4144:
4141:
4140:
4108:
4104:
4098:
4094:
4083:
4074:
4070:
4052:
4048:
4046:
4043:
4042:
4012:
4009:
4008:
3992:
3989:
3988:
3961:
3947:
3944:
3943:
3919:
3916:
3915:
3887:
3884:
3883:
3864:
3850:
3847:
3846:
3826:
3823:
3822:
3800:
3797:
3796:
3767:
3763:
3757:
3753:
3742:
3733:
3726:
3722:
3716:
3712:
3701:
3692:
3690:
3672:
3668:
3666:
3663:
3662:
3614:
3600:
3595:
3590:
3571:
3568:
3567:
3539:
3533:
3529:
3527:
3524:
3523:
3490:
3482:
3472:
3469:
3468:
3439:
3427:
3417:
3414:
3413:
3386:
3375:
3365:
3362:
3361:
3316:
3312:
3308:
3298:
3294:
3293:
3291:
3280:
3276:
3270:
3266:
3257:
3253:
3249:
3243:
3239:
3237:
3223:
3219:
3215:
3205:
3201:
3200:
3198:
3184:
3180:
3176:
3166:
3162:
3161:
3159:
3157:
3154:
3153:
3130:
3127:
3126:
3110:
3107:
3106:
3062:
3058:
3054:
3044:
3040:
3039:
3037:
3030:
3026:
3015:
2999:
2995:
2993:
2982:
2981:
2977:
2968:
2964:
2960:
2954:
2950:
2948:
2927:
2923:
2919:
2909:
2905:
2904:
2902:
2881:
2877:
2873:
2863:
2859:
2858:
2856:
2854:
2851:
2850:
2826:
2823:
2822:
2787:
2784:
2783:
2767:
2764:
2763:
2762:the mean depth
2742:
2741:
2729:
2725:
2714:
2707:
2703:
2697:
2693:
2692:
2690:
2683:
2679:
2668:
2653:
2645:
2643:
2624:
2617:
2613:
2609:
2607:
2600:
2596:
2588:
2577:
2570:
2566:
2562:
2560:
2557:
2556:
2544:
2540:
2536:
2529:
2525:
2519:
2515:
2514:
2512:
2505:
2501:
2490:
2474:
2470:
2454:
2450:
2449:
2445:
2434:
2429:
2421:
2410:
2402:
2400:
2396:
2394:
2391:
2390:
2349:
2346:
2345:
2323:
2317:
2313:
2301:
2297:
2295:
2292:
2291:
2274:
2270:
2268:
2265:
2264:
2248:
2245:
2244:
2216:
2213:
2212:
2181:
2178:
2177:
2153:
2139:
2136:
2135:
2115:
2112:
2111:
2087:
2073:
2070:
2069:
2049:
2046:
2045:
2025:
2024:
1990:
1986:
1977:
1973:
1972:
1968:
1957:
1949:
1938:
1930:
1928:
1925:
1924:
1898:
1890:
1888:
1876:
1865:
1857:
1855:
1851:
1849:
1846:
1845:
1805:
1802:
1801:
1765:
1762:
1761:
1746:
1741:
1711:
1708:
1707:
1685:
1677:
1674:
1673:
1656:
1655:
1629:
1625:
1621:
1614:
1610:
1604:
1600:
1599:
1597:
1590:
1586:
1563:
1549:
1545:
1541:
1534:
1530:
1524:
1520:
1519:
1517:
1510:
1506:
1483:
1472:
1468:
1466:
1462:
1460:
1453:
1452:
1423:
1409:
1401:
1399:
1395:
1394:
1385:
1381:
1377:
1371:
1367:
1365:
1358:
1354:
1331:
1311:
1297:
1289:
1287:
1283:
1282:
1264:
1260:
1250:
1249:
1225:
1221:
1217:
1210:
1206:
1200:
1196:
1195:
1193:
1186:
1182:
1159:
1145:
1141:
1137:
1130:
1126:
1120:
1116:
1115:
1113:
1106:
1102:
1079:
1068:
1064:
1062:
1058:
1056:
1044:
1042:
1039:
1038:
1030:, as valid for
1015:
1012:
1011:
975:
971:
969:
966:
965:
948:
947:
921:
908:
904:
900:
890:
886:
885:
883:
879:
878:
871:
867:
844:
824:
811:
807:
803:
793:
789:
788:
786:
782:
781:
774:
770:
747:
737:
736:
720:
707:
703:
699:
689:
685:
684:
682:
678:
677:
670:
666:
643:
623:
609:
601:
599:
595:
594:
567:
563:
561:
549:
547:
544:
543:
515:
512:
511:
474:
471:
470:
466:is made of the
462:water depth. A
443:
440:
439:
414:
411:
410:
390:
387:
386:
370:
367:
366:
338:
335:
334:
176:
138:computer models
90:(also known as
24:
17:
12:
11:
5:
6224:
6214:
6213:
6208:
6203:
6201:Fluid dynamics
6186:
6185:
6183:
6182:
6170:
6160:
6149:
6146:
6145:
6143:
6142:
6137:
6132:
6127:
6122:
6120:Stratification
6117:
6112:
6107:
6102:
6097:
6092:
6091:
6090:
6080:
6075:
6070:
6065:
6060:
6055:
6050:
6045:
6040:
6035:
6030:
6025:
6020:
6012:
6010:Color of water
6007:
6005:Benthic lander
6002:
5997:
5991:
5989:
5985:
5984:
5982:
5981:
5976:
5971:
5965:
5963:
5959:
5958:
5956:
5955:
5950:
5945:
5940:
5935:
5929:
5927:
5921:
5920:
5918:
5917:
5912:
5910:Sea level rise
5907:
5905:Sea level drop
5902:
5897:
5892:
5887:
5881:
5879:
5873:
5872:
5870:
5869:
5864:
5859:
5854:
5849:
5844:
5839:
5834:
5829:
5824:
5818:
5816:
5812:
5811:
5809:
5808:
5803:
5798:
5793:
5788:
5783:
5778:
5773:
5768:
5763:
5758:
5753:
5748:
5743:
5741:Marine geology
5738:
5733:
5728:
5723:
5717:
5715:
5707:
5706:
5704:
5703:
5698:
5693:
5688:
5683:
5681:Passive margin
5678:
5676:Oceanic trench
5673:
5668:
5663:
5658:
5653:
5648:
5643:
5638:
5633:
5628:
5623:
5618:
5613:
5608:
5603:
5598:
5593:
5587:
5585:
5579:
5578:
5576:
5575:
5570:
5565:
5560:
5555:
5550:
5545:
5540:
5535:
5530:
5525:
5520:
5515:
5510:
5505:
5500:
5495:
5490:
5484:
5482:
5476:
5475:
5473:
5472:
5467:
5462:
5457:
5452:
5451:
5450:
5440:
5435:
5430:
5425:
5420:
5415:
5410:
5408:Ocean dynamics
5405:
5400:
5395:
5390:
5385:
5380:
5375:
5370:
5365:
5360:
5355:
5350:
5345:
5340:
5335:
5330:
5325:
5320:
5315:
5310:
5305:
5300:
5298:Coriolis force
5295:
5290:
5285:
5279:
5277:
5271:
5270:
5252:
5250:
5248:
5247:
5246:
5245:
5235:
5230:
5225:
5224:
5223:
5218:
5208:
5203:
5198:
5193:
5188:
5183:
5178:
5173:
5168:
5163:
5158:
5153:
5148:
5147:
5146:
5136:
5131:
5126:
5121:
5119:Stokes problem
5116:
5111:
5106:
5101:
5096:
5091:
5086:
5081:
5076:
5071:
5066:
5061:
5056:
5054:Kinematic wave
5051:
5046:
5041:
5036:
5031:
5026:
5021:
5016:
5011:
5006:
5001:
4996:
4991:
4986:
4981:
4976:
4971:
4965:
4963:
4957:
4956:
4949:
4948:
4941:
4934:
4926:
4920:
4919:
4913:
4897:
4871:(4): 815–827.
4857:
4851:
4838:
4832:
4819:
4794:
4773:
4754:
4736:Boussinesq, J.
4732:
4716:Boussinesq, J.
4710:
4707:
4704:
4703:
4685:
4680:isec.nacse.org
4667:
4649:
4640:
4628:
4619:
4610:
4601:
4583:
4582:
4580:
4577:
4561:discretization
4557:finite element
4515:
4512:
4504:
4503:
4498:waves (slowly
4484:
4469:
4448:
4445:
4444:
4443:
4436:
4425:internal waves
4421:
4414:
4407:
4404:
4393:
4386:
4379:
4344:
4341:
4328:
4308:
4258:
4255:
4250:
4246:
4240:
4236:
4208:
4188:
4168:
4164:
4160:
4157:
4154:
4151:
4148:
4133:
4132:
4121:
4117:
4111:
4107:
4101:
4097:
4090:
4087:
4081:
4077:
4073:
4068:
4065:
4061:
4055:
4051:
4016:
3996:
3968:
3964:
3960:
3957:
3954:
3951:
3923:
3912:relative error
3908:
3907:
3891:
3871:
3867:
3863:
3860:
3857:
3854:
3830:
3820:
3804:
3790:
3789:
3778:
3770:
3766:
3760:
3756:
3749:
3746:
3740:
3736:
3729:
3725:
3719:
3715:
3708:
3705:
3699:
3695:
3688:
3685:
3681:
3675:
3671:
3613:
3610:
3578:
3575:
3555:
3552:
3549:
3546:
3542:
3536:
3532:
3514:
3513:
3510:
3497:
3494:
3486:
3480:
3476:
3465:
3464:
3461:
3447:
3444:
3437:
3431:
3425:
3421:
3410:
3409:
3406:
3393:
3390:
3382:
3379:
3373:
3369:
3353:
3352:
3341:
3338:
3334:
3329:
3319:
3315:
3311:
3306:
3301:
3297:
3289:
3283:
3279:
3274:
3269:
3260:
3256:
3252:
3246:
3242:
3235:
3226:
3222:
3218:
3213:
3208:
3204:
3196:
3187:
3183:
3179:
3174:
3169:
3165:
3134:
3114:
3095:
3094:
3083:
3079:
3074:
3065:
3061:
3057:
3052:
3047:
3043:
3033:
3029:
3022:
3019:
3013:
3007:
3002:
2998:
2989:
2986:
2980:
2971:
2967:
2963:
2957:
2953:
2946:
2943:
2939:
2930:
2926:
2922:
2917:
2912:
2908:
2900:
2897:
2893:
2884:
2880:
2876:
2871:
2866:
2862:
2847:
2846:
2830:
2791:
2771:
2756:
2755:
2740:
2732:
2728:
2724:
2720:
2717:
2710:
2706:
2700:
2696:
2686:
2682:
2675:
2672:
2666:
2659:
2656:
2651:
2648:
2641:
2637:
2630:
2627:
2620:
2616:
2612:
2603:
2599:
2594:
2591:
2589:
2583:
2580:
2573:
2569:
2565:
2559:
2558:
2555:
2547:
2543:
2539:
2532:
2528:
2522:
2518:
2508:
2504:
2497:
2494:
2488:
2483:
2477:
2473:
2467:
2463:
2460:
2457:
2453:
2448:
2440:
2437:
2433:
2427:
2424:
2422:
2416:
2413:
2408:
2405:
2399:
2398:
2387:
2386:
2381:are obtained:
2362:
2359:
2356:
2353:
2333:
2330:
2326:
2320:
2316:
2312:
2309:
2304:
2300:
2277:
2273:
2252:
2220:
2206:
2205:
2185:
2175:
2163:
2160:
2156:
2152:
2149:
2146:
2143:
2119:
2109:
2097:
2094:
2090:
2086:
2083:
2080:
2077:
2053:
2039:
2038:
2023:
2020:
2016:
2012:
2008:
2004:
1999:
1993:
1989:
1985:
1980:
1976:
1971:
1964:
1961:
1955:
1952:
1950:
1944:
1941:
1936:
1933:
1927:
1926:
1923:
1919:
1915:
1911:
1904:
1901:
1896:
1893:
1886:
1882:
1879:
1877:
1871:
1868:
1863:
1860:
1854:
1853:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1781:
1778:
1775:
1772:
1769:
1745:
1742:
1740:
1737:
1724:
1721:
1718:
1715:
1695:
1692:
1688:
1684:
1681:
1670:
1669:
1654:
1650:
1645:
1641:
1632:
1628:
1624:
1617:
1613:
1607:
1603:
1593:
1589:
1585:
1582:
1579:
1576:
1570:
1567:
1561:
1552:
1548:
1544:
1537:
1533:
1527:
1523:
1513:
1509:
1505:
1502:
1499:
1496:
1490:
1487:
1481:
1475:
1471:
1465:
1461:
1458:
1455:
1454:
1450:
1445:
1441:
1435:
1432:
1429:
1426:
1421:
1415:
1412:
1407:
1404:
1398:
1388:
1384:
1380:
1374:
1370:
1361:
1357:
1353:
1350:
1347:
1344:
1338:
1335:
1329:
1323:
1320:
1317:
1314:
1309:
1303:
1300:
1295:
1292:
1286:
1280:
1277:
1274:
1271:
1268:
1263:
1258:
1255:
1253:
1251:
1246:
1241:
1237:
1228:
1224:
1220:
1213:
1209:
1203:
1199:
1189:
1185:
1181:
1178:
1175:
1172:
1166:
1163:
1157:
1148:
1144:
1140:
1133:
1129:
1123:
1119:
1109:
1105:
1101:
1098:
1095:
1092:
1086:
1083:
1077:
1071:
1067:
1061:
1057:
1054:
1050:
1047:
1046:
1019:
995:
992:
989:
986:
983:
978:
974:
962:
961:
946:
943:
939:
933:
930:
927:
924:
919:
911:
907:
903:
898:
893:
889:
882:
874:
870:
866:
863:
860:
857:
851:
848:
842:
836:
833:
830:
827:
822:
814:
810:
806:
801:
796:
792:
785:
777:
773:
769:
766:
763:
760:
754:
751:
745:
742:
740:
738:
732:
729:
726:
723:
718:
710:
706:
702:
697:
692:
688:
681:
673:
669:
665:
662:
659:
656:
650:
647:
641:
635:
632:
629:
626:
621:
615:
612:
607:
604:
598:
592:
589:
586:
583:
580:
576:
570:
566:
562:
559:
555:
552:
551:
528:
525:
522:
519:
499:
496:
493:
490:
487:
484:
481:
478:
447:
427:
424:
421:
418:
394:
374:
354:
351:
348:
345:
342:
331:potential flow
308:
307:
276:
265:
175:
172:
56:fluid dynamics
15:
9:
6:
4:
3:
2:
6223:
6212:
6209:
6207:
6204:
6202:
6199:
6198:
6196:
6181:
6176:
6171:
6169:
6161:
6159:
6151:
6150:
6147:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6116:
6113:
6111:
6108:
6106:
6103:
6101:
6098:
6096:
6093:
6089:
6086:
6085:
6084:
6081:
6079:
6076:
6074:
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6031:
6029:
6028:Marine energy
6026:
6024:
6021:
6019:
6018:
6013:
6011:
6008:
6006:
6003:
6001:
5998:
5996:
5995:Acidification
5993:
5992:
5990:
5986:
5980:
5977:
5975:
5972:
5970:
5967:
5966:
5964:
5960:
5954:
5951:
5949:
5948:SOFAR channel
5946:
5944:
5941:
5939:
5936:
5934:
5931:
5930:
5928:
5926:
5922:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5882:
5880:
5878:
5874:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5845:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5819:
5817:
5813:
5807:
5804:
5802:
5799:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5772:
5769:
5767:
5764:
5762:
5759:
5757:
5756:Oceanic crust
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5731:Fracture zone
5729:
5727:
5724:
5722:
5719:
5718:
5716:
5714:
5708:
5702:
5699:
5697:
5694:
5692:
5689:
5687:
5684:
5682:
5679:
5677:
5674:
5672:
5669:
5667:
5666:Oceanic basin
5664:
5662:
5659:
5657:
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5612:
5609:
5607:
5604:
5602:
5599:
5597:
5596:Abyssal plain
5594:
5592:
5589:
5588:
5586:
5584:
5580:
5574:
5571:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5503:Internal tide
5501:
5499:
5496:
5494:
5491:
5489:
5486:
5485:
5483:
5481:
5477:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5453:
5449:
5446:
5445:
5444:
5441:
5439:
5436:
5434:
5431:
5429:
5426:
5424:
5421:
5419:
5416:
5414:
5411:
5409:
5406:
5404:
5401:
5399:
5398:Ocean current
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5334:
5331:
5329:
5326:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5291:
5289:
5286:
5284:
5281:
5280:
5278:
5276:
5272:
5267:
5256:
5244:
5241:
5240:
5239:
5236:
5234:
5231:
5229:
5226:
5222:
5219:
5217:
5214:
5213:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5196:Wave shoaling
5194:
5192:
5189:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5156:Ursell number
5154:
5152:
5149:
5145:
5142:
5141:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5050:
5047:
5045:
5042:
5040:
5039:Internal wave
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4995:
4992:
4990:
4989:Breaking wave
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4966:
4964:
4962:
4958:
4954:
4947:
4942:
4940:
4935:
4933:
4928:
4927:
4924:
4916:
4914:0-12-493250-9
4910:
4906:
4902:
4898:
4894:
4890:
4886:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4854:
4852:0-444-51149-0
4848:
4844:
4839:
4835:
4833:0-521-59832-X
4829:
4825:
4820:
4816:
4812:
4809:(1–3): 5–39.
4808:
4804:
4800:
4795:
4792:
4781:on 2012-02-08
4780:
4776:
4770:
4766:
4762:
4761:
4755:
4751:
4747:
4746:
4741:
4737:
4733:
4729:
4725:
4721:
4717:
4713:
4712:
4699:
4698:www1.udel.edu
4695:
4689:
4681:
4677:
4671:
4663:
4659:
4653:
4644:
4635:
4633:
4623:
4614:
4605:
4598:
4594:
4588:
4584:
4576:
4574:
4570:
4566:
4562:
4558:
4554:
4553:finite-volume
4550:
4546:
4542:
4532:
4525:
4520:
4511:
4509:
4508:cnoidal waves
4501:
4497:
4493:
4489:
4485:
4482:
4478:
4474:
4470:
4467:
4463:
4459:
4455:
4454:
4453:
4441:
4437:
4434:
4430:
4426:
4423:extension to
4422:
4419:
4416:inclusion of
4415:
4412:
4411:wave breaking
4409:inclusion of
4408:
4405:
4402:
4398:
4394:
4391:
4387:
4384:
4380:
4377:
4373:
4372:
4371:
4368:
4366:
4362:
4358:
4354:
4350:
4340:
4326:
4306:
4298:
4293:
4291:
4287:
4283:
4279:
4276:
4272:
4256:
4253:
4248:
4244:
4238:
4234:
4224:
4222:
4206:
4186:
4166:
4162:
4158:
4155:
4152:
4149:
4146:
4138:
4119:
4115:
4109:
4105:
4099:
4095:
4088:
4085:
4079:
4075:
4071:
4066:
4063:
4059:
4053:
4049:
4041:
4040:
4039:
4037:
4033:
4028:
4014:
3994:
3986:
3982:
3966:
3962:
3958:
3955:
3952:
3949:
3941:
3937:
3921:
3913:
3905:
3889:
3869:
3865:
3861:
3858:
3855:
3852:
3844:
3828:
3821:
3818:
3802:
3795:
3794:
3793:
3776:
3768:
3764:
3758:
3754:
3747:
3744:
3738:
3734:
3727:
3723:
3717:
3713:
3706:
3703:
3697:
3693:
3686:
3683:
3679:
3673:
3669:
3661:
3660:
3659:
3657:
3653:
3648:
3646:
3645:approximation
3642:
3638:
3634:
3633:infinitesimal
3630:
3626:
3622:
3619:of different
3618:
3607:
3603:
3598:
3593:
3576:
3573:
3550:
3547:
3540:
3534:
3530:
3520:
3511:
3495:
3492:
3484:
3478:
3474:
3467:
3466:
3462:
3445:
3442:
3435:
3429:
3423:
3419:
3412:
3411:
3407:
3391:
3388:
3380:
3377:
3371:
3367:
3360:
3359:
3356:
3339:
3336:
3332:
3327:
3317:
3313:
3304:
3299:
3287:
3281:
3277:
3272:
3267:
3258:
3254:
3244:
3233:
3224:
3220:
3211:
3206:
3194:
3185:
3181:
3172:
3167:
3152:
3151:
3150:
3148:
3147:normalization
3132:
3112:
3104:
3100:
3099:Ursell number
3081:
3077:
3072:
3063:
3059:
3050:
3045:
3031:
3027:
3020:
3017:
3011:
3005:
3000:
2996:
2987:
2984:
2978:
2969:
2965:
2955:
2944:
2941:
2937:
2928:
2924:
2915:
2910:
2898:
2895:
2891:
2882:
2878:
2869:
2864:
2849:
2848:
2844:
2843:
2842:
2828:
2820:
2816:
2812:
2807:
2805:
2789:
2769:
2761:
2738:
2730:
2726:
2718:
2708:
2704:
2698:
2684:
2680:
2673:
2670:
2664:
2657:
2649:
2639:
2635:
2628:
2618:
2614:
2601:
2597:
2592:
2590:
2581:
2571:
2567:
2553:
2545:
2541:
2530:
2526:
2520:
2506:
2502:
2495:
2492:
2486:
2481:
2475:
2471:
2465:
2461:
2458:
2455:
2451:
2446:
2438:
2425:
2423:
2414:
2406:
2389:
2388:
2384:
2383:
2382:
2380:
2376:
2360:
2357:
2354:
2351:
2331:
2324:
2318:
2314:
2307:
2302:
2298:
2275:
2271:
2250:
2242:
2238:
2234:
2218:
2211:
2203:
2199:
2183:
2176:
2161:
2154:
2150:
2144:
2141:
2133:
2132:flow velocity
2117:
2110:
2095:
2088:
2084:
2078:
2075:
2067:
2066:flow velocity
2051:
2044:
2043:
2042:
2021:
2018:
2014:
2010:
2006:
2002:
1997:
1991:
1987:
1983:
1978:
1974:
1969:
1962:
1959:
1953:
1951:
1942:
1934:
1921:
1917:
1913:
1909:
1902:
1894:
1884:
1880:
1878:
1869:
1861:
1844:
1843:
1842:
1825:
1822:
1819:
1813:
1810:
1807:
1799:
1795:
1776:
1773:
1770:
1759:
1755:
1751:
1736:
1722:
1719:
1716:
1713:
1693:
1686:
1682:
1652:
1648:
1643:
1639:
1630:
1626:
1615:
1611:
1605:
1591:
1583:
1580:
1577:
1568:
1565:
1559:
1550:
1546:
1535:
1531:
1525:
1511:
1503:
1500:
1497:
1488:
1485:
1479:
1473:
1469:
1463:
1456:
1448:
1443:
1439:
1433:
1430:
1427:
1424:
1419:
1413:
1405:
1396:
1386:
1382:
1372:
1359:
1351:
1348:
1345:
1336:
1333:
1327:
1321:
1318:
1315:
1312:
1307:
1301:
1293:
1284:
1275:
1272:
1269:
1261:
1256:
1254:
1244:
1239:
1235:
1226:
1222:
1211:
1207:
1201:
1187:
1179:
1176:
1173:
1164:
1161:
1155:
1146:
1142:
1131:
1127:
1121:
1107:
1099:
1096:
1093:
1084:
1081:
1075:
1069:
1065:
1059:
1052:
1048:
1037:
1036:
1035:
1033:
1017:
1009:
990:
987:
984:
976:
972:
944:
941:
937:
931:
928:
925:
922:
917:
909:
905:
896:
891:
880:
872:
864:
861:
858:
849:
846:
840:
834:
831:
828:
825:
820:
812:
808:
799:
794:
783:
775:
767:
764:
761:
752:
749:
743:
741:
730:
727:
724:
721:
716:
708:
704:
695:
690:
679:
671:
663:
660:
657:
648:
645:
639:
633:
630:
627:
624:
619:
613:
605:
596:
587:
584:
581:
574:
568:
564:
557:
553:
542:
541:
540:
526:
523:
520:
517:
494:
491:
488:
485:
482:
476:
469:
465:
461:
445:
425:
422:
419:
416:
408:
405:the vertical
392:
372:
349:
346:
343:
332:
327:
325:
321:
317:
313:
305:
301:
297:
293:
289:
286:and the zero-
285:
281:
277:
274:
270:
266:
263:
259:
255:
254:flow velocity
251:
247:
246:
245:
242:
240:
236:
231:
229:
225:
217:
213:
209:
205:
201:
197:
193:
189:
185:
184:cross section
180:
171:
169:
165:
160:
158:
154:
151:
147:
143:
139:
135:
131:
127:
123:
119:
116:
112:
111:flow velocity
108:
103:
101:
97:
93:
92:solitary wave
89:
85:
81:
77:
73:
69:
68:approximation
65:
61:
57:
49:
48:non-linearity
45:
41:
37:
33:
28:
22:
6135:Water column
6083:Oceanography
6058:Observations
6053:Explorations
6023:Marginal sea
6016:
5974:OSTM/Jason-2
5806:Volcanic arc
5781:Slab suction
5498:Head of tide
5388:Loop Current
5328:Ekman spiral
5114:Stokes drift
5024:Gravity wave
4999:Cnoidal wave
4983:
4904:
4868:
4864:
4842:
4823:
4806:
4802:
4790:
4783:. Retrieved
4779:the original
4764:
4759:
4749:
4743:
4727:
4723:
4697:
4688:
4679:
4670:
4661:
4652:
4643:
4622:
4613:
4604:
4596:
4592:
4587:
4565:Kirby (2003)
4537:
4505:
4450:
4433:mass density
4369:
4364:
4360:
4356:
4352:
4346:
4294:
4289:
4270:
4225:
4220:
4136:
4134:
4035:
4029:
3984:
3935:
3909:
3791:
3655:
3649:
3640:
3625:phase speeds
3621:wave lengths
3615:
3601:
3596:
3591:
3354:
3096:
2819:free surface
2810:
2808:
2759:
2757:
2207:
2198:acceleration
2040:
1798:free surface
1747:
1671:
963:
365:plane, with
328:
309:
243:
232:
221:
211:
161:
121:
104:
99:
59:
53:
38:, including
6206:Water waves
6125:Thermocline
5842:Mesopelagic
5815:Ocean zones
5786:Slab window
5651:Hydrography
5591:Abyssal fan
5558:Tidal range
5548:Tidal power
5543:Tidal force
5428:Rip current
5363:Gulf Stream
5323:Ekman layer
5313:Downwelling
5288:Baroclinity
5275:Circulation
5171:Wave height
5161:Wave action
5144:megatsunami
5124:Stokes wave
5084:Rossby wave
5049:Kelvin wave
5029:Green's law
4282:phase speed
4275:real-valued
4030:The linear
3981:engineering
3904:wave length
3843:wave number
3817:phase speed
3650:The linear
3617:Water waves
2134:component:
2068:component:
1792:plane, the
1750:water waves
320:coordinates
228:water waves
146:water waves
107:water waves
64:water waves
44:diffraction
6195:Categories
6063:Reanalysis
5962:Satellites
5943:Sofar bomb
5791:Subduction
5766:Ridge push
5661:Ocean bank
5641:Contourite
5568:Tide gauge
5553:Tidal race
5538:Tidal bore
5528:Slack tide
5493:Earth tide
5413:Ocean gyre
5233:Wind setup
5228:Wind fetch
5191:Wave setup
5186:Wave radar
5181:Wave power
5079:Rogue wave
5009:Dispersion
4785:2008-01-21
4730:: 755–759.
4709:References
4496:narrowband
4481:dimensions
4401:elevations
4390:non-linear
4376:bathymetry
2821:elevation
1800:elevation
1744:Derivation
407:coordinate
224:coordinate
204:wavelength
194:and sharp
164:wavelength
142:simulation
115:non-linear
72:non-linear
40:refraction
5925:Acoustics
5877:Sea level
5776:Slab pull
5713:tectonics
5621:Cold seep
5583:Landforms
5460:Whirlpool
5455:Upwelling
5238:Wind wave
5166:Wave base
5094:Sea state
5014:Edge wave
5004:Cross sea
4893:119385147
4752:: 55–108.
4500:modulated
4466:dimension
4429:interface
4395:making a
4392:behavior,
4388:improved
4381:improved
4307:λ
4278:solutions
4269:equation
4219:, called
4187:λ
4159:π
4080:−
3995:λ
3979:. So, in
3959:π
3890:λ
3870:λ
3862:π
3637:amplitude
3475:ξ
3420:τ
3389:η
3368:ψ
3314:ξ
3310:∂
3305:ψ
3296:∂
3278:ψ
3255:ξ
3251:∂
3241:∂
3234:−
3221:ξ
3217:∂
3212:ψ
3203:∂
3195:−
3182:τ
3178:∂
3173:ψ
3164:∂
3056:∂
3051:η
3042:∂
2997:η
2962:∂
2952:∂
2938:−
2921:∂
2916:η
2907:∂
2892:−
2875:∂
2870:η
2861:∂
2829:η
2723:∂
2716:∂
2695:∂
2655:∂
2650:η
2647:∂
2626:∂
2611:∂
2579:∂
2564:∂
2538:∂
2517:∂
2462:η
2436:∂
2432:∂
2412:∂
2407:η
2404:∂
2358:−
2329:∂
2315:φ
2311:∂
2251:η
2241:quadratic
2219:φ
2159:∂
2151:φ
2148:∂
2093:∂
2085:φ
2082:∂
2011:η
1940:∂
1935:φ
1932:∂
1910:−
1900:∂
1895:η
1892:∂
1867:∂
1862:η
1859:∂
1814:η
1720:−
1691:∂
1683:φ
1680:∂
1644:⋯
1623:∂
1612:φ
1602:∂
1543:∂
1532:φ
1522:∂
1480:−
1470:φ
1444:⋯
1431:−
1411:∂
1406:φ
1403:∂
1379:∂
1369:∂
1328:−
1319:−
1299:∂
1294:φ
1291:∂
1240:⋯
1219:∂
1208:φ
1198:∂
1139:∂
1128:φ
1118:∂
1076:−
1066:φ
1049:φ
1034:, gives:
1018:φ
973:φ
942:⋯
929:−
902:∂
897:φ
888:∂
832:−
805:∂
800:φ
791:∂
728:−
701:∂
696:φ
687:∂
631:−
611:∂
606:φ
603:∂
565:φ
554:φ
524:−
477:φ
423:−
316:functions
282:) for an
262:elevation
120:, called
6158:Category
6110:Seawater
5837:Littoral
5832:Deep sea
5691:Seamount
5573:Tideline
5518:Rip tide
5448:shutdown
5418:Overflow
5151:Undertow
4994:Clapotis
4738:(1872).
4718:(1871).
4374:varying
3934:for set
2817:for the
438:, where
157:harbours
140:for the
6168:Commons
6038:Mooring
5988:Related
5979:Jason-3
5969:Jason-1
5852:Pelagic
5847:Oceanic
5822:Benthic
5139:Tsunami
5109:Soliton
4873:Bibcode
4541:MIKE 21
4502:waves).
4280:of the
3882:, with
2373:). The
2202:gravity
2196:is the
2041:where:
1796:at the
1760:in the
458:is the
192:troughs
186:in the
150:shallow
96:soliton
86:of the
5857:Photic
5686:Seabed
5099:Seiche
4911:
4891:
4849:
4830:
4771:
4427:on an
3792:with:
3355:with:
2237:linear
1752:on an
964:where
273:finite
196:crests
66:is an
58:, the
6048:Ocean
6017:Alvin
5867:Swash
5711:Plate
5656:Knoll
5646:Guyot
5601:Atoll
5480:Tides
5243:model
5129:Swell
4961:Waves
4889:S2CID
4579:Notes
4038:are:
3635:wave
3101:. In
2375:cubic
1841:are:
322:(and
267:this
200:scale
32:shoal
6015:DSV
6000:Argo
5862:Surf
5318:Eddy
4909:ISBN
4847:ISBN
4828:ISBN
4769:ISBN
4543:and
4486:the
4471:the
4460:for
4456:the
4295:The
4254:>
4153:<
3956:<
3910:The
3902:the
3841:the
3815:the
2760:i.e.
2263:and
2239:and
1756:and
1748:For
1010:for
460:mean
324:time
288:curl
256:(or
212:i.e.
155:and
153:seas
74:and
62:for
4881:doi
4811:doi
4555:or
4545:SMS
4524:SMS
4494:of
4361:the
4353:the
4223:.
2200:by
326:).
148:in
144:of
94:or
54:In
6197::
4887:.
4879:.
4869:27
4867:.
4807:21
4805:.
4765:13
4750:17
4742:.
4728:72
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