4759:
4774:
51:
519:
74:
3950:
59:
3659:
1281:
3715:
1067:) waves (e.g. 2 mm for the waterâair interface), which are proper capillary waves, do the opposite: an individual wave appears at the front of the group, grows when moving towards the group center and finally disappears at the back of the group. Phase velocity is two thirds of group velocity in this limit.
2131:
1075:
Between these two limits is a point at which the dispersion caused by gravity cancels out the dispersion due to the capillary effect. At a certain wavelength, the group velocity equals the phase velocity, and there is no dispersion. At precisely this same wavelength, the phase velocity of
3414:
4508:
795:
2439:
3945:{\displaystyle {\begin{aligned}V_{\text{g}}&={\frac {1}{4}}(\rho -\rho ')ga^{2}\lambda ,\\V_{\text{st}}&={\frac {1}{4}}\sigma k^{2}a^{2}\lambda ,\\T&={\frac {1}{4}}(\rho +\rho '){\frac {\omega ^{2}}{|k|}}a^{2}\lambda .\end{aligned}}}
1442:(waves are not high enough for gravitation to change appreciably). For surface tension, the deviations from planarity (as measured by derivatives of the surface) are supposed to be small. For common waves both approximations are good enough.
1136:
3372:
3307:
1468:) can be solved with the proper boundary conditions. On one hand, the velocity must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more involved result is obtained, see
3097:
648:
272:
3654:{\displaystyle {\begin{aligned}\Phi (x,y,z,t)&=+{\frac {1}{|k|}}{\text{e}}^{+|k|z}\,\omega a\,\sin \,\theta ,\\\Phi '(x,y,z,t)&=-{\frac {1}{|k|}}{\text{e}}^{-|k|z}\,\omega a\,\sin \,\theta .\end{aligned}}}
1527:
Consider two fluid domains, separated by an interface with surface tension. The mean interface position is horizontal. It separates the upper from the lower fluid, both having a different constant mass density,
1973:
3180:
508:
662:
When capillary waves are also affected substantially by gravity, they are called gravityâcapillary waves. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:
4664:
970:
2640:
595:
4218:
3720:
3419:
4380:
1369:
that are easily seen by everyone and which are usually used as an example of waves in elementary courses are the worst possible example ; they have all the complications that waves can have.
669:
441:
2826:
3246:. The kinematic boundary condition at the interface, relating the potentials to the interface motion, is that the vertical velocity components must match the motion of the surface:
908:
186:
in waves. Distinction can be made between pure capillary waves â fully dominated by the effects of surface tension â and gravityâcapillary waves which are also affected by gravity.
2742:
2171:
1737:
4276:
3244:
2785:
1638:
3699:
1025:
1453:
framework. Incompressibility is again involved (which is satisfied if the speed of the waves is much less than the speed of sound in the media), together with the flow being
4554:
4337:
2690:
1965:
1685:
1918:
1276:{\displaystyle \lambda _{m}=2\pi {\sqrt {\frac {\sigma }{(\rho -\rho ')g}}}\quad {\text{and}}\quad c_{m}={\sqrt {\frac {2{\sqrt {(\rho -\rho ')g\sigma }}}{\rho +\rho '}}}.}
551:
4372:
1610:
1319:
1101:
2874:
1571:
867:
372:
2937:
1887:
1420:
319:
295:
4065:
1807:
4035:
3988:
2849:
1861:
1773:
1546:
1346:
1128:
842:
343:
4699:
3398:
2911:
2160:
1039:: following a single wave's crest in a group one can see the wave appearing at the back of the group, growing and finally disappearing at the front of the group.
4296:
4008:
3312:
3203:
1834:
1514:
1490:
1440:
1400:
1065:
818:
392:
3252:
4758:
1076:
gravityâcapillary waves as a function of wavelength (or wave number) has a minimum. Waves with wavelengths much smaller than this critical wavelength
4773:
2945:
600:
5069:
4831:
5201:
1351:
If one drops a small stone or droplet into liquid, the waves then propagate outside an expanding circle of fluid at rest; this circle is a
200:
2126:{\displaystyle V_{\mathrm {g} }=\iint dx\,dy\;\int _{0}^{\eta }dz\;(\rho -\rho ')gz={\frac {1}{2}}(\rho -\rho ')g\iint dx\,dy\;\eta ^{2},}
1472:.) On the other, its vertical component must match the motion of the surface. This contribution ends up being responsible for the extra
3116:
1382:. The first two are potential energies, and responsible for the two terms inside the parenthesis, as is clear from the appearance of
5110:
5091:
4947:
449:
154:
waves. Light breezes which stir up such small ripples are also sometimes referred to as cat's paws. On the open ocean, much larger
1103:
are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed
5130:
4851:
2645:
Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both
4562:
4503:{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}\,g+{\frac {\sigma }{\rho +\rho '}}\,k^{2}\right),}
2461:
556:
4073:
1422:. For gravity, an assumption is made of the density of the fluids being constant (i.e., incompressibility), and likewise
790:{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}g+{\frac {\sigma }{\rho +\rho '}}k^{2}\right),}
5009:
913:
405:
5105:. Advanced Series on Ocean Engineering. Vol. 13. World Scientific, Singapore. pp. 2 Parts, 967 pages.
4719:
2434:{\displaystyle V_{\mathrm {st} }=\sigma \iint dx\,dy\;\left\approx {\frac {1}{2}}\sigma \iint dx\,dy\;\left,}
5191:
3405:
43:
4734:
2790:
63:
4230:
3211:
2448:'s) representation, and the second applies for small values of the derivatives (surfaces not too rough).
4671:
2754:
1522:
Dispersion relation for gravityâcapillary waves on an interface between two semiâinfinite fluid domains
876:
3667:
993:
522:
Dispersion of gravityâcapillary waves on the surface of deep water (zero mass density of upper layer,
5196:
5142:
5079:
5021:
3956:
2695:
1690:
98:
4519:
4301:
2648:
2165:
An increase in area of the surface causes a proportional increase of energy due to surface tension:
1923:
1643:
4224:
2880:
1615:
175:
39:
4342:
1592:
1297:
1079:
5045:
4744:
4729:
2876:
must vanish well away from the surface (in the "deep water" case, which is the one we consider).
1469:
4724:
5176:
658: • Dash-dotted lines: dispersion relation valid for deep-water capillary waves.
5158:
5106:
5087:
5065:
5037:
4943:
4935:
4827:
1892:
525:
298:
2916:
1866:
1405:
443:
For the boundary between fluid and vacuum (free surface), the dispersion relation reduces to
304:
280:
5150:
5029:
4980:
4714:
4040:
2854:
2745:
1782:
1776:
1746:
1551:
1465:
1352:
985:
847:
821:
352:
136:
4013:
3961:
2834:
1839:
1751:
1531:
1324:
1106:
827:
328:
4846:
3367:{\displaystyle {\frac {\partial \Phi '}{\partial z}}={\frac {\partial \eta }{\partial t}}}
1810:
1375:
1364:
322:
106:
50:
4678:
3377:
3302:{\displaystyle {\frac {\partial \Phi }{\partial z}}={\frac {\partial \eta }{\partial t}}}
2890:
2139:
1863:
due to gravity is the simplest: integrating the potential energy density due to gravity,
5146:
5025:
4739:
4281:
3993:
3664:
Then the contributions to the wave energy, horizontally integrated over one wavelength
3188:
2452:
1819:
1814:
1740:
1586:
1578:
1516:, and high ones (except around the one value at which the two dispersions cancel out.)
1499:
1475:
1458:
1446:
1425:
1385:
1050:
1036:
1032:
803:
377:
102:
94:
73:
5185:
5049:
4964:
4765:
3206:
2445:
1450:
1379:
973:
163:
17:
518:
4374:
is just the expression in the square brackets, so that the dispersion relation is:
1582:
1454:
1028:
870:
144:
32:
656: • Dashed lines: dispersion relation for deep-water gravity waves.
1371:" The derivation of the general dispersion relation is therefore quite involved.
652: • Blue lines (A): phase velocity, Red lines (B): group velocity.
5057:
654: • Drawn lines: dispersion relation for gravityâcapillary waves.
125:
5154:
5033:
4669:
As usual for linear wave motions, the potential and kinetic energy are equal (
3103:
399:
395:
179:
121:
58:
5041:
4984:
3092:{\displaystyle T\approx {\frac {1}{2}}\iint dx\,dy\;\left_{{\text{at }}z=0}.}
124:
of capillary waves on water is typically less than a few centimeters, with a
643:{\displaystyle \scriptstyle {\frac {1}{\lambda }}{\sqrt {\sigma /(\rho g)}}}
183:
159:
155:
5162:
2883:, and assuming the deviations of the surface elevation to be small (so the
1574:
1573:
for the lower and upper domain respectively. The fluid is assumed to be
150:
When generated by light wind in open water, a nautical name for them is
346:
140:
78:
1461:. These are typically also good approximations for common situations.
1589:, and the velocity in the lower and upper layer can be obtained from
267:{\displaystyle \omega ^{2}={\frac {\sigma }{\rho +\rho '}}\,|k|^{3},}
113:
67:
5012:(1963). "The generation of capillary waves by steep gravity waves".
4968:
2831:
These equations can be solved with the proper boundary conditions:
166:) may result from coalescence of smaller wind-caused ripple-waves.
1291:
517:
72:
57:
49:
5122:
Statistical thermodynamics of surfaces, interfaces, and membranes
3175:{\displaystyle \eta =a\,\cos \,(kx-\omega t)=a\,\cos \,\theta ,}
3102:
To find the dispersion relation, it is sufficient to consider a
1374:
There are three contributions to the energy, due to gravity, to
135:
which are influenced by both the effects of surface tension and
90:
4516:
As a result, the average wave energy per unit horizontal area,
1287:
503:{\displaystyle \omega ^{2}={\frac {\sigma }{\rho }}\,|k|^{3}.}
4278:, so that variation with respect to the only free parameter,
3404:
To tackle the problem of finding the potentials, one may try
4826:. Cambridge University Press. Section 3.1 and Exercise 3.3.
27:
Wave on the surface of a fluid, dominated by surface tension
1920:) from a reference height to the position of the surface,
1449:
of the fluids. It is the most complicated and calls for a
5129:
Tufillaro, N. B.; Ramshankar, R.; Gollub, J. P. (1989).
4659:{\displaystyle {\bar {E}}={\frac {1}{2}}\,\left\,a^{2}.}
131:
A longer wavelength on a fluid interface will result in
31:"Rippled waves" redirects here. Not to be confused with
4865:
See e.g. Safran (1994) for a more detailed description.
2887:âintegrations may be approximated by integrating up to
1518:
604:
560:
4681:
4565:
4522:
4383:
4345:
4304:
4284:
4233:
4076:
4043:
4016:
3996:
3964:
3955:
The dispersion relation can now be obtained from the
3718:
3670:
3417:
3380:
3315:
3255:
3214:
3191:
3119:
2948:
2919:
2893:
2857:
2837:
2793:
2757:
2698:
2651:
2464:
2174:
2142:
1976:
1926:
1895:
1869:
1842:
1822:
1785:
1754:
1693:
1646:
1618:
1595:
1554:
1534:
1502:
1478:
1428:
1408:
1388:
1327:
1300:
1139:
1109:
1082:
1053:
996:
916:
879:
850:
830:
806:
672:
603:
559:
528:
452:
408:
380:
355:
331:
307:
283:
203:
2635:{\displaystyle T={\frac {1}{2}}\iint dx\,dy\;\left.}
1745:
Three contributions to the energy are involved: the
590:{\displaystyle \scriptstyle {\sqrt{g\sigma /\rho }}}
4779:
Light breeze ripples in the surface water of a lake
4213:{\displaystyle L={\frac {1}{4}}\lefta^{2}\lambda .}
1464:The resulting equation for the potential (which is
4693:
4658:
4548:
4502:
4366:
4331:
4290:
4270:
4212:
4059:
4029:
4002:
3982:
3944:
3693:
3653:
3392:
3366:
3301:
3238:
3197:
3174:
3091:
2931:
2905:
2868:
2843:
2820:
2779:
2736:
2684:
2634:
2433:
2154:
2125:
1959:
1912:
1881:
1855:
1828:
1801:
1767:
1731:
1679:
1632:
1604:
1565:
1540:
1508:
1484:
1434:
1414:
1394:
1340:
1313:
1275:
1122:
1095:
1059:
1019:
964:
902:
861:
836:
812:
789:
642:
589:
545:
502:
435:
386:
366:
337:
313:
289:
266:
4223:For sinusoidal waves and linear wave theory, the
1355:which corresponds to the minimal group velocity.
5131:"Order-disorder transition in capillary ripples"
1496:regimes to be dispersive, both at low values of
1321:is found to be 1.7 cm (0.67 in), and
1027:), only the first term is relevant and one has
194:The dispersion relation for capillary waves is
81:impacts on the interface between water and air.
4973:Proceedings of the London Mathematical Society
4824:Fluid Mechanics, a short course for physicists
2444:where the first equality is the area in this (
4010:the sum of the potential energies by gravity
965:{\displaystyle (\rho -\rho ')/(\rho +\rho ')}
597:as a function of inverse relative wavelength
8:
5086:(2nd ed.). Cambridge University Press.
5064:(6th ed.). Cambridge University Press.
4810:Dingemans (1997), Section 2.1.1, p. 45.
436:{\displaystyle \lambda ={\frac {2\pi }{k}}.}
2136:assuming the mean interface position is at
5103:Water wave propagation over uneven bottoms
4930:
4928:
3106:wave on the interface, propagating in the
3020:
3016:
2981:
2584:
2527:
2497:
2344:
2212:
2109:
2030:
2008:
4900:
4898:
4888:
4886:
4884:
4882:
4880:
4812:Phillips (1977), Section 3.2, p. 37.
4680:
4647:
4642:
4631:
4617:
4591:
4581:
4567:
4566:
4564:
4538:
4521:
4486:
4481:
4458:
4451:
4415:
4405:
4397:
4388:
4382:
4344:
4303:
4283:
4262:
4232:
4198:
4183:
4139:
4131:
4124:
4118:
4083:
4075:
4048:
4042:
4021:
4015:
3995:
3963:
3926:
3914:
3906:
3899:
3893:
3863:
3837:
3827:
3810:
3797:
3777:
3740:
3727:
3719:
3717:
3705:âdirection, and over a unit width in the
3683:
3669:
3640:
3636:
3629:
3619:
3611:
3607:
3602:
3593:
3585:
3579:
3523:
3519:
3512:
3502:
3494:
3490:
3485:
3476:
3468:
3462:
3418:
3416:
3379:
3344:
3316:
3314:
3279:
3256:
3254:
3213:
3190:
3165:
3161:
3133:
3129:
3118:
3070:
3069:
3039:
3038:
3029:
2996:
2995:
2991:
2974:
2955:
2947:
2939:), the kinetic energy can be written as:
2918:
2892:
2856:
2836:
2798:
2792:
2762:
2756:
2697:
2650:
2618:
2600:
2593:
2569:
2564:
2551:
2538:
2531:
2515:
2507:
2490:
2471:
2463:
2417:
2393:
2379:
2355:
2337:
2315:
2293:
2269:
2255:
2231:
2218:
2205:
2180:
2179:
2173:
2141:
2114:
2102:
2060:
2018:
2013:
2001:
1982:
1981:
1975:
1925:
1894:
1868:
1847:
1841:
1821:
1790:
1784:
1759:
1753:
1692:
1645:
1617:
1594:
1553:
1533:
1501:
1477:
1427:
1407:
1387:
1332:
1326:
1305:
1299:
1220:
1213:
1204:
1194:
1159:
1144:
1138:
1114:
1108:
1087:
1081:
1052:
1009:
995:
937:
915:
878:
849:
829:
805:
773:
746:
704:
694:
686:
677:
671:
620:
615:
605:
602:
579:
570:
561:
558:
527:
491:
486:
477:
476:
466:
457:
451:
415:
407:
379:
354:
330:
306:
282:
255:
250:
241:
240:
217:
208:
202:
3408:, when both fields can be expressed as:
4996:, MacMillan, 2nd revised edition, 1894.
4856:Addison-Wesley. Volume I, Chapter 51-4.
4789:
4754:
553:). Phase and group velocity divided by
4849:, R.B. Leighton, and M. Sands (1963).
4797:
4795:
4793:
1492:outside the parenthesis, which causes
374:the density of the lighter fluid and
7:
2821:{\displaystyle \nabla ^{2}\Phi '=0.}
1445:The third contribution involves the
4271:{\displaystyle L=D(\omega ,k)a^{2}}
3239:{\displaystyle \theta =kx-\omega t}
2451:The last contribution involves the
1348:is 0.23 m/s (0.75 ft/s).
178:describes the relationship between
128:in excess of 0.2â0.3 meter/second.
5177:Capillary waves entry at sklogwiki
3535:
3422:
3355:
3347:
3332:
3323:
3319:
3290:
3282:
3267:
3262:
3259:
3055:
3046:
3042:
3031:
3007:
3002:
2999:
2992:
2805:
2795:
2780:{\displaystyle \nabla ^{2}\Phi =0}
2768:
2759:
2606:
2601:
2573:
2543:
2539:
2511:
2404:
2396:
2366:
2358:
2280:
2272:
2242:
2234:
2184:
2181:
1983:
1619:
1596:
1031:. In this limit, the waves have a
54:Capillary waves (ripples) in water
25:
4298:, gives the dispersion relation
903:{\displaystyle (\rho >\rho ')}
4801:Lamb (1994), §267, page 458â460.
4772:
4757:
3694:{\displaystyle \lambda =2\pi /k}
1581:, and the flow is assumed to be
1020:{\displaystyle k=2\pi /\lambda }
147:have a still longer wavelength.
105:are dominated by the effects of
38:For ripples in electricity, see
5084:The dynamics of the upper ocean
4852:The Feynman Lectures on Physics
2737:{\displaystyle \phi '(x,y,z,t)}
1732:{\displaystyle \phi '(x,y,z,t)}
1199:
1193:
116:, and are often referred to as
4614:
4597:
4572:
4549:{\displaystyle (T+V)/\lambda }
4535:
4523:
4406:
4398:
4361:
4349:
4332:{\displaystyle D(\omega ,k)=0}
4320:
4308:
4255:
4243:
4167:
4150:
4140:
4132:
4115:
4098:
3915:
3907:
3890:
3873:
3767:
3750:
3620:
3612:
3594:
3586:
3566:
3542:
3503:
3495:
3477:
3469:
3449:
3425:
3152:
3134:
2731:
2707:
2685:{\displaystyle \phi (x,y,z,t)}
2679:
2655:
2087:
2070:
2048:
2031:
1960:{\displaystyle z=\eta (x,y,t)}
1954:
1936:
1726:
1702:
1680:{\displaystyle \phi (x,y,z,t)}
1674:
1650:
1239:
1222:
1183:
1166:
959:
942:
934:
917:
897:
880:
695:
687:
634:
625:
487:
478:
251:
242:
112:Capillary waves are common in
1:
1633:{\displaystyle \nabla \phi '}
990:For large wavelengths (small
4965:Lord Rayleigh (J. W. Strutt)
4764:Ripples on water created by
4367:{\displaystyle D(\omega ,k)}
1605:{\displaystyle \nabla \phi }
1314:{\displaystyle \lambda _{m}}
1096:{\displaystyle \lambda _{m}}
77:Capillary waves produced by
5202:Oceanographical terminology
4874:Lamb (1994), §174 and §230.
820:is the acceleration due to
5218:
5155:10.1103/PhysRevLett.62.422
5014:Journal of Fluid Mechanics
4989:Reprinted as Appendix in:
4940:Linear and nonlinear waves
983:
29:
5101:Dingemans, M. W. (1997).
5034:10.1017/S0022112063000641
4225:phaseâaveraged Lagrangian
972:in the first term is the
4720:Dispersion (water waves)
1913:{\displaystyle \rho 'gz}
546:{\displaystyle \rho '=0}
5135:Physical Review Letters
5120:Safran, Samuel (1994).
3406:separation of variables
2932:{\displaystyle z=\eta }
1882:{\displaystyle \rho gz}
1779:, the potential energy
1415:{\displaystyle \sigma }
514:Gravityâcapillary waves
349:of the heavier fluid,
314:{\displaystyle \sigma }
290:{\displaystyle \omega }
190:Capillary waves, proper
133:gravityâcapillary waves
44:Ripple (disambiguation)
5010:Longuet-Higgins, M. S.
4985:10.1112/plms/s1-9.1.21
4969:"On progressive waves"
4942:. Wiley-Interscience.
4822:Falkovich, G. (2011).
4735:Thermal capillary wave
4695:
4660:
4550:
4504:
4368:
4333:
4292:
4272:
4227:is always of the form
4214:
4061:
4060:{\displaystyle V_{st}}
4031:
4004:
3984:
3946:
3695:
3655:
3394:
3368:
3303:
3240:
3199:
3176:
3093:
2933:
2907:
2870:
2869:{\displaystyle \phi '}
2845:
2822:
2781:
2738:
2686:
2636:
2435:
2156:
2127:
1961:
1914:
1883:
1857:
1836:of the flow. The part
1830:
1803:
1802:{\displaystyle V_{st}}
1769:
1733:
1681:
1634:
1606:
1567:
1566:{\displaystyle \rho '}
1542:
1510:
1486:
1436:
1416:
1396:
1342:
1315:
1277:
1124:
1097:
1071:Phase velocity minimum
1061:
1021:
966:
904:
863:
862:{\displaystyle \rho '}
838:
814:
791:
659:
644:
591:
547:
504:
437:
388:
368:
367:{\displaystyle \rho '}
339:
315:
291:
268:
139:, as well as by fluid
82:
70:
62:Ripples on Lifjord in
55:
42:. For other uses, see
4696:
4661:
4551:
4505:
4369:
4334:
4293:
4273:
4215:
4062:
4032:
4030:{\displaystyle V_{g}}
4005:
3985:
3983:{\displaystyle L=T-V}
3947:
3696:
3656:
3395:
3369:
3304:
3241:
3200:
3177:
3094:
2934:
2908:
2871:
2846:
2844:{\displaystyle \phi }
2823:
2782:
2739:
2687:
2637:
2436:
2157:
2128:
1962:
1915:
1884:
1858:
1856:{\displaystyle V_{g}}
1831:
1804:
1770:
1768:{\displaystyle V_{g}}
1734:
1682:
1640:, respectively. Here
1635:
1607:
1585:. Then the flows are
1568:
1543:
1541:{\displaystyle \rho }
1511:
1487:
1437:
1417:
1397:
1343:
1341:{\displaystyle c_{m}}
1316:
1278:
1125:
1123:{\displaystyle c_{m}}
1098:
1062:
1043:Capillary wave regime
1022:
984:Further information:
967:
905:
864:
839:
837:{\displaystyle \rho }
815:
792:
645:
592:
548:
521:
505:
438:
389:
369:
340:
338:{\displaystyle \rho }
316:
292:
269:
76:
61:
53:
18:Cat's paw (wave)
4679:
4563:
4520:
4381:
4343:
4302:
4282:
4231:
4074:
4041:
4037:and surface tension
4014:
3994:
3962:
3716:
3709:âdirection, become:
3668:
3415:
3378:
3313:
3253:
3212:
3189:
3117:
2946:
2917:
2891:
2855:
2835:
2791:
2755:
2696:
2649:
2462:
2172:
2140:
1974:
1924:
1893:
1867:
1840:
1820:
1783:
1752:
1691:
1644:
1616:
1593:
1552:
1532:
1500:
1476:
1426:
1406:
1386:
1325:
1298:
1137:
1107:
1080:
1051:
994:
914:
877:
848:
828:
804:
670:
601:
557:
526:
450:
406:
378:
353:
329:
305:
281:
201:
93:traveling along the
5147:1989PhRvL..62..422T
5026:1963JFM....16..138L
4694:{\displaystyle T=V}
4513:the same as above.
3393:{\displaystyle z=0}
2906:{\displaystyle z=0}
2577:
2520:
2155:{\displaystyle z=0}
2023:
1741:velocity potentials
1470:Ocean surface waves
1457:â the flow is then
980:Gravity wave regime
176:dispersion relation
170:Dispersion relation
156:ocean surface waves
64:Ăksnes Municipality
40:Ripple (electrical)
4922:Lamb (1994), §230.
4892:Lamb (1994), §266.
4745:Wave-formed ripple
4730:Ocean surface wave
4691:
4656:
4546:
4500:
4364:
4329:
4288:
4268:
4210:
4057:
4027:
4000:
3980:
3942:
3940:
3691:
3651:
3649:
3390:
3364:
3309: and
3299:
3236:
3195:
3172:
3089:
2929:
2903:
2866:
2841:
2818:
2787: and
2777:
2734:
2682:
2632:
2560:
2503:
2431:
2152:
2123:
2009:
1957:
1910:
1879:
1853:
1826:
1799:
1765:
1729:
1677:
1630:
1602:
1563:
1538:
1506:
1482:
1432:
1412:
1392:
1338:
1311:
1273:
1120:
1093:
1057:
1017:
962:
900:
873:of the two fluids
859:
834:
810:
787:
660:
640:
639:
587:
586:
543:
500:
433:
384:
364:
335:
311:
287:
264:
97:of a fluid, whose
83:
71:
56:
5124:. Addison-Wesley.
5071:978-0-521-45868-9
4954:See section 11.7.
4904:Lamb (1994), §61.
4833:978-1-107-00575-4
4706:
4705:
4589:
4575:
4479:
4449:
4291:{\displaystyle a}
4145:
4091:
4003:{\displaystyle V}
3920:
3871:
3818:
3800:
3748:
3730:
3605:
3599:
3488:
3482:
3362:
3339:
3297:
3274:
3198:{\displaystyle a}
3073:
3062:
3014:
2963:
2744:must satisfy the
2479:
2411:
2373:
2323:
2299:
2287:
2249:
2068:
1829:{\displaystyle T}
1509:{\displaystyle k}
1485:{\displaystyle k}
1435:{\displaystyle g}
1395:{\displaystyle g}
1268:
1267:
1248:
1197:
1191:
1190:
1060:{\displaystyle k}
813:{\displaystyle g}
767:
738:
637:
613:
584:
474:
428:
387:{\displaystyle k}
299:angular frequency
238:
16:(Redirected from
5209:
5166:
5125:
5116:
5097:
5075:
5053:
4997:
4988:
4961:
4955:
4953:
4932:
4923:
4920:
4914:
4913:Lamb (1994), §20
4911:
4905:
4902:
4893:
4890:
4875:
4872:
4866:
4863:
4857:
4844:
4838:
4837:
4819:
4813:
4808:
4802:
4799:
4776:
4761:
4715:Capillary action
4700:
4698:
4697:
4692:
4665:
4663:
4662:
4657:
4652:
4651:
4641:
4637:
4636:
4635:
4613:
4590:
4582:
4577:
4576:
4568:
4555:
4553:
4552:
4547:
4542:
4509:
4507:
4506:
4501:
4496:
4492:
4491:
4490:
4480:
4478:
4477:
4459:
4450:
4448:
4447:
4432:
4431:
4416:
4409:
4401:
4393:
4392:
4373:
4371:
4370:
4365:
4338:
4336:
4335:
4330:
4297:
4295:
4294:
4289:
4277:
4275:
4274:
4269:
4267:
4266:
4219:
4217:
4216:
4211:
4203:
4202:
4193:
4189:
4188:
4187:
4166:
4146:
4144:
4143:
4135:
4129:
4128:
4119:
4114:
4092:
4084:
4066:
4064:
4063:
4058:
4056:
4055:
4036:
4034:
4033:
4028:
4026:
4025:
4009:
4007:
4006:
4001:
3989:
3987:
3986:
3981:
3951:
3949:
3948:
3943:
3941:
3931:
3930:
3921:
3919:
3918:
3910:
3904:
3903:
3894:
3889:
3872:
3864:
3842:
3841:
3832:
3831:
3819:
3811:
3802:
3801:
3798:
3782:
3781:
3766:
3749:
3741:
3732:
3731:
3728:
3700:
3698:
3697:
3692:
3687:
3660:
3658:
3657:
3652:
3650:
3628:
3627:
3623:
3615:
3606:
3603:
3600:
3598:
3597:
3589:
3580:
3541:
3511:
3510:
3506:
3498:
3489:
3486:
3483:
3481:
3480:
3472:
3463:
3399:
3397:
3396:
3391:
3373:
3371:
3370:
3365:
3363:
3361:
3353:
3345:
3340:
3338:
3330:
3329:
3317:
3308:
3306:
3305:
3300:
3298:
3296:
3288:
3280:
3275:
3273:
3265:
3257:
3245:
3243:
3242:
3237:
3204:
3202:
3201:
3196:
3181:
3179:
3178:
3173:
3098:
3096:
3095:
3090:
3085:
3084:
3074:
3071:
3068:
3064:
3063:
3061:
3053:
3052:
3040:
3037:
3028:
3015:
3013:
3005:
2997:
2964:
2956:
2938:
2936:
2935:
2930:
2912:
2910:
2909:
2904:
2881:Green's identity
2875:
2873:
2872:
2867:
2865:
2850:
2848:
2847:
2842:
2827:
2825:
2824:
2819:
2811:
2803:
2802:
2786:
2784:
2783:
2778:
2767:
2766:
2746:Laplace equation
2743:
2741:
2740:
2735:
2706:
2691:
2689:
2688:
2683:
2641:
2639:
2638:
2633:
2628:
2624:
2623:
2622:
2617:
2613:
2612:
2604:
2592:
2576:
2568:
2556:
2555:
2550:
2546:
2542:
2519:
2514:
2480:
2472:
2440:
2438:
2437:
2432:
2427:
2423:
2422:
2421:
2416:
2412:
2410:
2402:
2394:
2384:
2383:
2378:
2374:
2372:
2364:
2356:
2324:
2316:
2311:
2307:
2300:
2298:
2297:
2292:
2288:
2286:
2278:
2270:
2260:
2259:
2254:
2250:
2248:
2240:
2232:
2219:
2189:
2188:
2187:
2161:
2159:
2158:
2153:
2132:
2130:
2129:
2124:
2119:
2118:
2086:
2069:
2061:
2047:
2022:
2017:
1988:
1987:
1986:
1966:
1964:
1963:
1958:
1919:
1917:
1916:
1911:
1903:
1888:
1886:
1885:
1880:
1862:
1860:
1859:
1854:
1852:
1851:
1835:
1833:
1832:
1827:
1808:
1806:
1805:
1800:
1798:
1797:
1774:
1772:
1771:
1766:
1764:
1763:
1747:potential energy
1738:
1736:
1735:
1730:
1701:
1686:
1684:
1683:
1678:
1639:
1637:
1636:
1631:
1629:
1611:
1609:
1608:
1603:
1572:
1570:
1569:
1564:
1562:
1547:
1545:
1544:
1539:
1519:
1515:
1513:
1512:
1507:
1491:
1489:
1488:
1483:
1466:Laplace equation
1447:kinetic energies
1441:
1439:
1438:
1433:
1421:
1419:
1418:
1413:
1401:
1399:
1398:
1393:
1347:
1345:
1344:
1339:
1337:
1336:
1320:
1318:
1317:
1312:
1310:
1309:
1282:
1280:
1279:
1274:
1269:
1266:
1265:
1250:
1249:
1238:
1221:
1215:
1214:
1209:
1208:
1198:
1195:
1192:
1189:
1182:
1161:
1160:
1149:
1148:
1129:
1127:
1126:
1121:
1119:
1118:
1102:
1100:
1099:
1094:
1092:
1091:
1066:
1064:
1063:
1058:
1026:
1024:
1023:
1018:
1013:
986:Airy wave theory
971:
969:
968:
963:
958:
941:
933:
909:
907:
906:
901:
896:
868:
866:
865:
860:
858:
843:
841:
840:
835:
819:
817:
816:
811:
796:
794:
793:
788:
783:
779:
778:
777:
768:
766:
765:
747:
739:
737:
736:
721:
720:
705:
698:
690:
682:
681:
649:
647:
646:
641:
638:
624:
616:
614:
606:
596:
594:
593:
588:
585:
583:
578:
574:
562:
552:
550:
549:
544:
536:
509:
507:
506:
501:
496:
495:
490:
481:
475:
467:
462:
461:
442:
440:
439:
434:
429:
424:
416:
393:
391:
390:
385:
373:
371:
370:
365:
363:
344:
342:
341:
336:
320:
318:
317:
312:
296:
294:
293:
288:
273:
271:
270:
265:
260:
259:
254:
245:
239:
237:
236:
218:
213:
212:
47:
36:
21:
5217:
5216:
5212:
5211:
5210:
5208:
5207:
5206:
5182:
5181:
5173:
5128:
5119:
5113:
5100:
5094:
5080:Phillips, O. M.
5078:
5072:
5056:
5008:
5005:
5000:
4991:Theory of Sound
4963:
4962:
4958:
4950:
4934:
4933:
4926:
4921:
4917:
4912:
4908:
4903:
4896:
4891:
4878:
4873:
4869:
4864:
4860:
4845:
4841:
4834:
4821:
4820:
4816:
4811:
4809:
4805:
4800:
4791:
4787:
4780:
4777:
4768:
4762:
4753:
4711:
4677:
4676:
4643:
4627:
4606:
4596:
4592:
4561:
4560:
4518:
4517:
4482:
4470:
4463:
4440:
4433:
4424:
4417:
4414:
4410:
4384:
4379:
4378:
4341:
4340:
4300:
4299:
4280:
4279:
4258:
4229:
4228:
4194:
4179:
4159:
4130:
4120:
4107:
4097:
4093:
4072:
4071:
4044:
4039:
4038:
4017:
4012:
4011:
3992:
3991:
3960:
3959:
3939:
3938:
3922:
3905:
3895:
3882:
3856:
3850:
3849:
3833:
3823:
3803:
3793:
3790:
3789:
3773:
3759:
3733:
3723:
3714:
3713:
3666:
3665:
3648:
3647:
3601:
3584:
3569:
3534:
3531:
3530:
3484:
3467:
3452:
3413:
3412:
3376:
3375:
3354:
3346:
3331:
3322:
3318:
3311:
3310:
3289:
3281:
3266:
3258:
3251:
3250:
3210:
3209:
3187:
3186:
3185:with amplitude
3115:
3114:
3054:
3045:
3041:
3030:
3021:
3006:
2998:
2987:
2983:
2982:
2944:
2943:
2915:
2914:
2889:
2888:
2858:
2853:
2852:
2833:
2832:
2804:
2794:
2789:
2788:
2758:
2753:
2752:
2699:
2694:
2693:
2647:
2646:
2605:
2599:
2595:
2594:
2585:
2537:
2533:
2532:
2502:
2498:
2460:
2459:
2403:
2395:
2389:
2388:
2365:
2357:
2351:
2350:
2349:
2345:
2279:
2271:
2265:
2264:
2241:
2233:
2227:
2226:
2217:
2213:
2175:
2170:
2169:
2138:
2137:
2110:
2079:
2040:
1977:
1972:
1971:
1922:
1921:
1896:
1891:
1890:
1865:
1864:
1843:
1838:
1837:
1818:
1817:
1811:surface tension
1786:
1781:
1780:
1755:
1750:
1749:
1694:
1689:
1688:
1642:
1641:
1622:
1614:
1613:
1591:
1590:
1555:
1550:
1549:
1530:
1529:
1498:
1497:
1474:
1473:
1424:
1423:
1404:
1403:
1384:
1383:
1376:surface tension
1365:Richard Feynman
1361:
1328:
1323:
1322:
1301:
1296:
1295:
1258:
1251:
1231:
1216:
1200:
1175:
1165:
1140:
1135:
1134:
1110:
1105:
1104:
1083:
1078:
1077:
1073:
1049:
1048:
1047:Shorter (large
1045:
992:
991:
988:
982:
951:
926:
912:
911:
889:
875:
874:
851:
846:
845:
826:
825:
802:
801:
769:
758:
751:
729:
722:
713:
706:
703:
699:
673:
668:
667:
657:
655:
653:
651:
599:
598:
563:
555:
554:
529:
524:
523:
516:
485:
453:
448:
447:
417:
404:
403:
376:
375:
356:
351:
350:
327:
326:
323:surface tension
303:
302:
279:
278:
249:
229:
222:
204:
199:
198:
192:
172:
107:surface tension
48:
37:
30:
28:
23:
22:
15:
12:
11:
5:
5215:
5213:
5205:
5204:
5199:
5194:
5192:Fluid dynamics
5184:
5183:
5180:
5179:
5172:
5171:External links
5169:
5168:
5167:
5141:(4): 422â425.
5126:
5117:
5111:
5098:
5092:
5076:
5070:
5054:
5020:(1): 138â159.
5004:
5001:
4999:
4998:
4956:
4948:
4936:Whitham, G. B.
4924:
4915:
4906:
4894:
4876:
4867:
4858:
4839:
4832:
4814:
4803:
4788:
4786:
4783:
4782:
4781:
4778:
4771:
4769:
4766:water striders
4763:
4756:
4752:
4749:
4748:
4747:
4742:
4740:Two-phase flow
4737:
4732:
4727:
4722:
4717:
4710:
4707:
4704:
4703:
4690:
4687:
4684:
4667:
4666:
4655:
4650:
4646:
4640:
4634:
4630:
4626:
4623:
4620:
4616:
4612:
4609:
4605:
4602:
4599:
4595:
4588:
4585:
4580:
4574:
4571:
4545:
4541:
4537:
4534:
4531:
4528:
4525:
4511:
4510:
4499:
4495:
4489:
4485:
4476:
4473:
4469:
4466:
4462:
4457:
4454:
4446:
4443:
4439:
4436:
4430:
4427:
4423:
4420:
4413:
4408:
4404:
4400:
4396:
4391:
4387:
4363:
4360:
4357:
4354:
4351:
4348:
4339:. In our case
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4287:
4265:
4261:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4221:
4220:
4209:
4206:
4201:
4197:
4192:
4186:
4182:
4178:
4175:
4172:
4169:
4165:
4162:
4158:
4155:
4152:
4149:
4142:
4138:
4134:
4127:
4123:
4117:
4113:
4110:
4106:
4103:
4100:
4096:
4090:
4087:
4082:
4079:
4054:
4051:
4047:
4024:
4020:
3999:
3979:
3976:
3973:
3970:
3967:
3953:
3952:
3937:
3934:
3929:
3925:
3917:
3913:
3909:
3902:
3898:
3892:
3888:
3885:
3881:
3878:
3875:
3870:
3867:
3862:
3859:
3857:
3855:
3852:
3851:
3848:
3845:
3840:
3836:
3830:
3826:
3822:
3817:
3814:
3809:
3806:
3804:
3796:
3792:
3791:
3788:
3785:
3780:
3776:
3772:
3769:
3765:
3762:
3758:
3755:
3752:
3747:
3744:
3739:
3736:
3734:
3726:
3722:
3721:
3690:
3686:
3682:
3679:
3676:
3673:
3662:
3661:
3646:
3643:
3639:
3635:
3632:
3626:
3622:
3618:
3614:
3610:
3596:
3592:
3588:
3583:
3578:
3575:
3572:
3570:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3540:
3537:
3533:
3532:
3529:
3526:
3522:
3518:
3515:
3509:
3505:
3501:
3497:
3493:
3479:
3475:
3471:
3466:
3461:
3458:
3455:
3453:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3420:
3402:
3401:
3389:
3386:
3383:
3360:
3357:
3352:
3349:
3343:
3337:
3334:
3328:
3325:
3321:
3295:
3292:
3287:
3284:
3278:
3272:
3269:
3264:
3261:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3194:
3183:
3182:
3171:
3168:
3164:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3132:
3128:
3125:
3122:
3100:
3099:
3088:
3083:
3080:
3077:
3067:
3060:
3057:
3051:
3048:
3044:
3036:
3033:
3027:
3024:
3019:
3012:
3009:
3004:
3001:
2994:
2990:
2986:
2980:
2977:
2973:
2970:
2967:
2962:
2959:
2954:
2951:
2928:
2925:
2922:
2902:
2899:
2896:
2864:
2861:
2840:
2829:
2828:
2817:
2814:
2810:
2807:
2801:
2797:
2776:
2773:
2770:
2765:
2761:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2705:
2702:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2643:
2642:
2631:
2627:
2621:
2616:
2611:
2608:
2603:
2598:
2591:
2588:
2583:
2580:
2575:
2572:
2567:
2563:
2559:
2554:
2549:
2545:
2541:
2536:
2530:
2526:
2523:
2518:
2513:
2510:
2506:
2501:
2496:
2493:
2489:
2486:
2483:
2478:
2475:
2470:
2467:
2455:of the fluid:
2453:kinetic energy
2442:
2441:
2430:
2426:
2420:
2415:
2409:
2406:
2401:
2398:
2392:
2387:
2382:
2377:
2371:
2368:
2363:
2360:
2354:
2348:
2343:
2340:
2336:
2333:
2330:
2327:
2322:
2319:
2314:
2310:
2306:
2303:
2296:
2291:
2285:
2282:
2277:
2274:
2268:
2263:
2258:
2253:
2247:
2244:
2239:
2236:
2230:
2225:
2222:
2216:
2211:
2208:
2204:
2201:
2198:
2195:
2192:
2186:
2183:
2178:
2151:
2148:
2145:
2134:
2133:
2122:
2117:
2113:
2108:
2105:
2101:
2098:
2095:
2092:
2089:
2085:
2082:
2078:
2075:
2072:
2067:
2064:
2059:
2056:
2053:
2050:
2046:
2043:
2039:
2036:
2033:
2029:
2026:
2021:
2016:
2012:
2007:
2004:
2000:
1997:
1994:
1991:
1985:
1980:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1909:
1906:
1902:
1899:
1878:
1875:
1872:
1850:
1846:
1825:
1815:kinetic energy
1796:
1793:
1789:
1762:
1758:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1700:
1697:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1628:
1625:
1621:
1601:
1598:
1579:incompressible
1561:
1558:
1537:
1524:
1523:
1505:
1481:
1431:
1411:
1391:
1360:
1357:
1335:
1331:
1308:
1304:
1284:
1283:
1272:
1264:
1261:
1257:
1254:
1247:
1244:
1241:
1237:
1234:
1230:
1227:
1224:
1219:
1212:
1207:
1203:
1188:
1185:
1181:
1178:
1174:
1171:
1168:
1164:
1158:
1155:
1152:
1147:
1143:
1117:
1113:
1090:
1086:
1072:
1069:
1056:
1044:
1041:
1037:phase velocity
1033:group velocity
1016:
1012:
1008:
1005:
1002:
999:
981:
978:
961:
957:
954:
950:
947:
944:
940:
936:
932:
929:
925:
922:
919:
899:
895:
892:
888:
885:
882:
857:
854:
833:
809:
798:
797:
786:
782:
776:
772:
764:
761:
757:
754:
750:
745:
742:
735:
732:
728:
725:
719:
716:
712:
709:
702:
697:
693:
689:
685:
680:
676:
636:
633:
630:
627:
623:
619:
612:
609:
582:
577:
573:
569:
566:
542:
539:
535:
532:
515:
512:
511:
510:
499:
494:
489:
484:
480:
473:
470:
465:
460:
456:
432:
427:
423:
420:
414:
411:
383:
362:
359:
334:
310:
286:
275:
274:
263:
258:
253:
248:
244:
235:
232:
228:
225:
221:
216:
211:
207:
191:
188:
171:
168:
103:phase velocity
95:phase boundary
87:capillary wave
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5214:
5203:
5200:
5198:
5195:
5193:
5190:
5189:
5187:
5178:
5175:
5174:
5170:
5164:
5160:
5156:
5152:
5148:
5144:
5140:
5136:
5132:
5127:
5123:
5118:
5114:
5112:981-02-0427-2
5108:
5104:
5099:
5095:
5093:0-521-29801-6
5089:
5085:
5081:
5077:
5073:
5067:
5063:
5062:Hydrodynamics
5059:
5055:
5051:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5019:
5015:
5011:
5007:
5006:
5002:
4995:
4992:
4986:
4982:
4978:
4974:
4970:
4966:
4960:
4957:
4951:
4949:0-471-94090-9
4945:
4941:
4937:
4931:
4929:
4925:
4919:
4916:
4910:
4907:
4901:
4899:
4895:
4889:
4887:
4885:
4883:
4881:
4877:
4871:
4868:
4862:
4859:
4855:
4853:
4848:
4843:
4840:
4835:
4829:
4825:
4818:
4815:
4807:
4804:
4798:
4796:
4794:
4790:
4784:
4775:
4770:
4767:
4760:
4755:
4750:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4712:
4708:
4702:
4688:
4685:
4682:
4674:
4673:
4672:equipartition
4653:
4648:
4644:
4638:
4632:
4628:
4624:
4621:
4618:
4610:
4607:
4603:
4600:
4593:
4586:
4583:
4578:
4569:
4559:
4558:
4557:
4543:
4539:
4532:
4529:
4526:
4514:
4497:
4493:
4487:
4483:
4474:
4471:
4467:
4464:
4460:
4455:
4452:
4444:
4441:
4437:
4434:
4428:
4425:
4421:
4418:
4411:
4402:
4394:
4389:
4385:
4377:
4376:
4375:
4358:
4355:
4352:
4346:
4326:
4323:
4317:
4314:
4311:
4305:
4285:
4263:
4259:
4252:
4249:
4246:
4240:
4237:
4234:
4226:
4207:
4204:
4199:
4195:
4190:
4184:
4180:
4176:
4173:
4170:
4163:
4160:
4156:
4153:
4147:
4136:
4125:
4121:
4111:
4108:
4104:
4101:
4094:
4088:
4085:
4080:
4077:
4070:
4069:
4068:
4052:
4049:
4045:
4022:
4018:
3997:
3977:
3974:
3971:
3968:
3965:
3958:
3935:
3932:
3927:
3923:
3911:
3900:
3896:
3886:
3883:
3879:
3876:
3868:
3865:
3860:
3858:
3853:
3846:
3843:
3838:
3834:
3828:
3824:
3820:
3815:
3812:
3807:
3805:
3794:
3786:
3783:
3778:
3774:
3770:
3763:
3760:
3756:
3753:
3745:
3742:
3737:
3735:
3724:
3712:
3711:
3710:
3708:
3704:
3688:
3684:
3680:
3677:
3674:
3671:
3644:
3641:
3637:
3633:
3630:
3624:
3616:
3608:
3590:
3581:
3576:
3573:
3571:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3538:
3527:
3524:
3520:
3516:
3513:
3507:
3499:
3491:
3473:
3464:
3459:
3456:
3454:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3411:
3410:
3409:
3407:
3387:
3384:
3381:
3358:
3350:
3341:
3335:
3326:
3293:
3285:
3276:
3270:
3249:
3248:
3247:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3208:
3192:
3169:
3166:
3162:
3158:
3155:
3149:
3146:
3143:
3140:
3137:
3130:
3126:
3123:
3120:
3113:
3112:
3111:
3109:
3105:
3086:
3081:
3078:
3075:
3065:
3058:
3049:
3034:
3025:
3022:
3017:
3010:
2988:
2984:
2978:
2975:
2971:
2968:
2965:
2960:
2957:
2952:
2949:
2942:
2941:
2940:
2926:
2923:
2920:
2900:
2897:
2894:
2886:
2882:
2877:
2862:
2859:
2838:
2815:
2812:
2808:
2799:
2774:
2771:
2763:
2751:
2750:
2749:
2747:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2703:
2700:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2652:
2629:
2625:
2619:
2614:
2609:
2596:
2589:
2586:
2581:
2578:
2570:
2565:
2561:
2557:
2552:
2547:
2534:
2528:
2524:
2521:
2516:
2508:
2504:
2499:
2494:
2491:
2487:
2484:
2481:
2476:
2473:
2468:
2465:
2458:
2457:
2456:
2454:
2449:
2447:
2428:
2424:
2418:
2413:
2407:
2399:
2390:
2385:
2380:
2375:
2369:
2361:
2352:
2346:
2341:
2338:
2334:
2331:
2328:
2325:
2320:
2317:
2312:
2308:
2304:
2301:
2294:
2289:
2283:
2275:
2266:
2261:
2256:
2251:
2245:
2237:
2228:
2223:
2220:
2214:
2209:
2206:
2202:
2199:
2196:
2193:
2190:
2176:
2168:
2167:
2166:
2163:
2149:
2146:
2143:
2120:
2115:
2111:
2106:
2103:
2099:
2096:
2093:
2090:
2083:
2080:
2076:
2073:
2065:
2062:
2057:
2054:
2051:
2044:
2041:
2037:
2034:
2027:
2024:
2019:
2014:
2010:
2005:
2002:
1998:
1995:
1992:
1989:
1978:
1970:
1969:
1968:
1951:
1948:
1945:
1942:
1939:
1933:
1930:
1927:
1907:
1904:
1900:
1897:
1876:
1873:
1870:
1848:
1844:
1823:
1816:
1812:
1794:
1791:
1787:
1778:
1760:
1756:
1748:
1742:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1698:
1695:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1647:
1626:
1623:
1599:
1588:
1584:
1580:
1576:
1559:
1556:
1535:
1526:
1525:
1521:
1520:
1517:
1503:
1495:
1479:
1471:
1467:
1462:
1460:
1456:
1452:
1448:
1443:
1429:
1409:
1389:
1381:
1380:hydrodynamics
1377:
1372:
1370:
1366:
1358:
1356:
1354:
1349:
1333:
1329:
1306:
1302:
1293:
1289:
1270:
1262:
1259:
1255:
1252:
1245:
1242:
1235:
1232:
1228:
1225:
1217:
1210:
1205:
1201:
1186:
1179:
1176:
1172:
1169:
1162:
1156:
1153:
1150:
1145:
1141:
1133:
1132:
1131:
1115:
1111:
1088:
1084:
1070:
1068:
1054:
1042:
1040:
1038:
1034:
1030:
1029:gravity waves
1014:
1010:
1006:
1003:
1000:
997:
987:
979:
977:
975:
974:Atwood number
955:
952:
948:
945:
938:
930:
927:
923:
920:
910:. The factor
893:
890:
886:
883:
872:
855:
852:
831:
823:
807:
784:
780:
774:
770:
762:
759:
755:
752:
748:
743:
740:
733:
730:
726:
723:
717:
714:
710:
707:
700:
691:
683:
678:
674:
666:
665:
664:
631:
628:
621:
617:
610:
607:
580:
575:
571:
567:
564:
540:
537:
533:
530:
520:
513:
497:
492:
482:
471:
468:
463:
458:
454:
446:
445:
444:
430:
425:
421:
418:
412:
409:
401:
397:
381:
360:
357:
348:
332:
324:
308:
300:
284:
261:
256:
246:
233:
230:
226:
223:
219:
214:
209:
205:
197:
196:
195:
189:
187:
185:
181:
177:
169:
167:
165:
161:
157:
153:
148:
146:
145:gravity waves
142:
138:
134:
129:
127:
123:
119:
115:
110:
108:
104:
100:
96:
92:
88:
80:
75:
69:
65:
60:
52:
45:
41:
34:
19:
5138:
5134:
5121:
5102:
5083:
5061:
5017:
5013:
4993:
4990:
4976:
4972:
4959:
4939:
4918:
4909:
4870:
4861:
4850:
4847:R.P. Feynman
4842:
4823:
4817:
4806:
4670:
4668:
4515:
4512:
4222:
3954:
3706:
3702:
3663:
3403:
3184:
3110:âdirection:
3107:
3101:
2884:
2878:
2830:
2644:
2450:
2443:
2164:
2135:
1744:
1583:irrotational
1493:
1463:
1455:irrotational
1451:hydrodynamic
1444:
1373:
1368:
1362:
1350:
1285:
1074:
1046:
989:
871:mass density
799:
661:
276:
193:
173:
151:
149:
132:
130:
117:
111:
86:
84:
33:Rippled wave
5197:Water waves
2913:instead of
1809:due to the
1294:interface,
143:. Ordinary
126:phase speed
5186:Categories
5003:References
4725:Fluid pipe
3957:Lagrangian
3374: at
3104:sinusoidal
1359:Derivation
400:wavelength
396:wavenumber
180:wavelength
122:wavelength
5050:119740891
5042:1469-7645
4979:: 21â26.
4625:σ
4608:ρ
4604:−
4601:ρ
4573:¯
4544:λ
4472:ρ
4465:ρ
4461:σ
4442:ρ
4435:ρ
4426:ρ
4422:−
4419:ρ
4386:ω
4353:ω
4312:ω
4247:ω
4205:λ
4177:σ
4174:−
4161:ρ
4157:−
4154:ρ
4148:−
4122:ω
4109:ρ
4102:ρ
3975:−
3933:λ
3897:ω
3884:ρ
3877:ρ
3844:λ
3821:σ
3784:λ
3761:ρ
3757:−
3754:ρ
3681:π
3672:λ
3642:θ
3631:ω
3609:−
3577:−
3536:Φ
3525:θ
3514:ω
3423:Φ
3356:∂
3351:η
3348:∂
3333:∂
3324:Φ
3320:∂
3291:∂
3286:η
3283:∂
3268:∂
3263:Φ
3260:∂
3231:ω
3228:−
3216:θ
3205:and wave
3167:θ
3147:ω
3144:−
3121:η
3056:∂
3047:Φ
3043:∂
3032:Φ
3023:ρ
3018:−
3008:∂
3003:Φ
3000:∂
2993:Φ
2989:ρ
2966:∬
2953:≈
2927:η
2860:ϕ
2839:ϕ
2806:Φ
2796:∇
2769:Φ
2760:∇
2701:ϕ
2653:ϕ
2607:Φ
2602:∇
2587:ρ
2574:∞
2566:η
2562:∫
2544:Φ
2540:∇
2529:ρ
2517:η
2512:∞
2509:−
2505:∫
2482:∬
2405:∂
2400:η
2397:∂
2367:∂
2362:η
2359:∂
2329:∬
2326:σ
2313:≈
2302:−
2281:∂
2276:η
2273:∂
2243:∂
2238:η
2235:∂
2197:∬
2194:σ
2112:η
2094:∬
2081:ρ
2077:−
2074:ρ
2042:ρ
2038:−
2035:ρ
2020:η
2011:∫
1993:∬
1934:η
1898:ρ
1871:ρ
1696:ϕ
1648:ϕ
1624:ϕ
1620:∇
1600:ϕ
1597:∇
1587:potential
1557:ρ
1536:ρ
1459:potential
1410:σ
1378:, and to
1367:put it, "
1303:λ
1260:ρ
1253:ρ
1246:σ
1233:ρ
1229:−
1226:ρ
1177:ρ
1173:−
1170:ρ
1163:σ
1157:π
1142:λ
1085:λ
1035:half the
1015:λ
1007:π
953:ρ
946:ρ
928:ρ
924:−
921:ρ
891:ρ
884:ρ
853:ρ
832:ρ
760:ρ
753:ρ
749:σ
731:ρ
724:ρ
715:ρ
711:−
708:ρ
675:ω
629:ρ
618:σ
611:λ
576:ρ
568:σ
531:ρ
472:ρ
469:σ
455:ω
422:π
410:λ
358:ρ
333:ρ
309:σ
285:ω
231:ρ
224:ρ
220:σ
206:ω
184:frequency
152:cat's paw
5163:10040229
5082:(1977).
5060:(1994).
5058:Lamb, H.
4967:(1877).
4938:(1974).
4709:See also
4675:holds):
4611:′
4475:′
4445:′
4429:′
4164:′
4112:′
3887:′
3764:′
3539:′
3327:′
3072:at
3050:′
3035:′
3026:′
2863:′
2809:′
2704:′
2610:′
2590:′
2084:′
2045:′
1901:′
1813:and the
1699:′
1627:′
1575:inviscid
1560:′
1286:For the
1263:′
1236:′
1180:′
956:′
931:′
894:′
869:are the
856:′
763:′
734:′
718:′
534:′
361:′
234:′
99:dynamics
5143:Bibcode
5022:Bibcode
4751:Gallery
3990:, with
3701:in the
1777:gravity
1775:due to
1353:caustic
822:gravity
347:density
297:is the
141:inertia
137:gravity
120:. The
118:ripples
79:droplet
5161:
5109:
5090:
5068:
5048:
5040:
4946:
4830:
4556:, is:
2879:Using
800:where
398:. The
277:where
164:swells
114:nature
68:Norway
5046:S2CID
4785:Notes
3207:phase
2446:Monge
1292:water
1130:are:
89:is a
5159:PMID
5107:ISBN
5088:ISBN
5066:ISBN
5038:ISSN
4944:ISBN
4828:ISBN
2851:and
2692:and
1889:(or
1739:are
1687:and
1612:and
1577:and
1548:and
1402:and
887:>
844:and
402:is
394:the
345:the
321:the
182:and
174:The
162:and
160:seas
101:and
91:wave
5151:doi
5030:doi
4981:doi
3638:sin
3521:sin
3163:cos
3131:cos
1494:all
1363:As
1288:air
1196:and
5188::
5157:.
5149:.
5139:62
5137:.
5133:.
5044:.
5036:.
5028:.
5018:16
5016:.
4975:.
4971:.
4927:^
4897:^
4879:^
4792:^
4701:.
4067::
3799:st
2816:0.
2748::
2162:.
1967::
1743:.
976:.
824:,
325:,
301:,
109:.
85:A
66:,
5165:.
5153::
5145::
5115:.
5096:.
5074:.
5052:.
5032::
5024::
4994:1
4987:.
4983::
4977:9
4952:.
4854:.
4836:.
4689:V
4686:=
4683:T
4654:.
4649:2
4645:a
4639:]
4633:2
4629:k
4622:+
4619:g
4615:)
4598:(
4594:[
4587:2
4584:1
4579:=
4570:E
4540:/
4536:)
4533:V
4530:+
4527:T
4524:(
4498:,
4494:)
4488:2
4484:k
4468:+
4456:+
4453:g
4438:+
4412:(
4407:|
4403:k
4399:|
4395:=
4390:2
4362:)
4359:k
4356:,
4350:(
4347:D
4327:0
4324:=
4321:)
4318:k
4315:,
4309:(
4306:D
4286:a
4264:2
4260:a
4256:)
4253:k
4250:,
4244:(
4241:D
4238:=
4235:L
4208:.
4200:2
4196:a
4191:]
4185:2
4181:k
4171:g
4168:)
4151:(
4141:|
4137:k
4133:|
4126:2
4116:)
4105:+
4099:(
4095:[
4089:4
4086:1
4081:=
4078:L
4053:t
4050:s
4046:V
4023:g
4019:V
3998:V
3978:V
3972:T
3969:=
3966:L
3936:.
3928:2
3924:a
3916:|
3912:k
3908:|
3901:2
3891:)
3880:+
3874:(
3869:4
3866:1
3861:=
3854:T
3847:,
3839:2
3835:a
3829:2
3825:k
3816:4
3813:1
3808:=
3795:V
3787:,
3779:2
3775:a
3771:g
3768:)
3751:(
3746:4
3743:1
3738:=
3729:g
3725:V
3707:y
3703:x
3689:k
3685:/
3678:2
3675:=
3645:.
3634:a
3625:z
3621:|
3617:k
3613:|
3604:e
3595:|
3591:k
3587:|
3582:1
3574:=
3567:)
3564:t
3561:,
3558:z
3555:,
3552:y
3549:,
3546:x
3543:(
3528:,
3517:a
3508:z
3504:|
3500:k
3496:|
3492:+
3487:e
3478:|
3474:k
3470:|
3465:1
3460:+
3457:=
3450:)
3447:t
3444:,
3441:z
3438:,
3435:y
3432:,
3429:x
3426:(
3400:.
3388:0
3385:=
3382:z
3359:t
3342:=
3336:z
3294:t
3277:=
3271:z
3234:t
3225:x
3222:k
3219:=
3193:a
3170:,
3159:a
3156:=
3153:)
3150:t
3141:x
3138:k
3135:(
3127:a
3124:=
3108:x
3087:.
3082:0
3079:=
3076:z
3066:]
3059:z
3011:z
2985:[
2979:y
2976:d
2972:x
2969:d
2961:2
2958:1
2950:T
2924:=
2921:z
2901:0
2898:=
2895:z
2885:z
2813:=
2800:2
2775:0
2772:=
2764:2
2732:)
2729:t
2726:,
2723:z
2720:,
2717:y
2714:,
2711:x
2708:(
2680:)
2677:t
2674:,
2671:z
2668:,
2665:y
2662:,
2659:x
2656:(
2630:.
2626:]
2620:2
2615:|
2597:|
2582:z
2579:d
2571:+
2558:+
2553:2
2548:|
2535:|
2525:z
2522:d
2500:[
2495:y
2492:d
2488:x
2485:d
2477:2
2474:1
2469:=
2466:T
2429:,
2425:]
2419:2
2414:)
2408:y
2391:(
2386:+
2381:2
2376:)
2370:x
2353:(
2347:[
2342:y
2339:d
2335:x
2332:d
2321:2
2318:1
2309:]
2305:1
2295:2
2290:)
2284:y
2267:(
2262:+
2257:2
2252:)
2246:x
2229:(
2224:+
2221:1
2215:[
2210:y
2207:d
2203:x
2200:d
2191:=
2185:t
2182:s
2177:V
2150:0
2147:=
2144:z
2121:,
2116:2
2107:y
2104:d
2100:x
2097:d
2091:g
2088:)
2071:(
2066:2
2063:1
2058:=
2055:z
2052:g
2049:)
2032:(
2028:z
2025:d
2015:0
2006:y
2003:d
1999:x
1996:d
1990:=
1984:g
1979:V
1955:)
1952:t
1949:,
1946:y
1943:,
1940:x
1937:(
1931:=
1928:z
1908:z
1905:g
1877:z
1874:g
1849:g
1845:V
1824:T
1795:t
1792:s
1788:V
1761:g
1757:V
1727:)
1724:t
1721:,
1718:z
1715:,
1712:y
1709:,
1706:x
1703:(
1675:)
1672:t
1669:,
1666:z
1663:,
1660:y
1657:,
1654:x
1651:(
1504:k
1480:k
1430:g
1390:g
1334:m
1330:c
1307:m
1290:â
1271:.
1256:+
1243:g
1240:)
1223:(
1218:2
1211:=
1206:m
1202:c
1187:g
1184:)
1167:(
1154:2
1151:=
1146:m
1116:m
1112:c
1089:m
1055:k
1011:/
1004:2
1001:=
998:k
960:)
949:+
943:(
939:/
935:)
918:(
898:)
881:(
808:g
785:,
781:)
775:2
771:k
756:+
744:+
741:g
727:+
701:(
696:|
692:k
688:|
684:=
679:2
650:.
635:)
632:g
626:(
622:/
608:1
581:4
572:/
565:g
541:0
538:=
498:.
493:3
488:|
483:k
479:|
464:=
459:2
431:.
426:k
419:2
413:=
382:k
262:,
257:3
252:|
247:k
243:|
227:+
215:=
210:2
158:(
46:.
35:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
