4748:
4763:
40:
508:
63:
3939:
48:
3648:
1270:
3704:
1056:) 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.
2120:
1064:
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
3403:
4497:
784:
2428:
3934:{\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}}}
1431:(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.
1125:
3361:
3296:
1457:) 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
3086:
637:
261:
3643:{\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}}}
1516:
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,
1962:
3169:
497:
651:
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:
4653:
959:
2629:
584:
4207:
3709:
3408:
4369:
1358:
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.
658:
430:
2815:
3235:. 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:
897:
175:
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.
2731:
2160:
1726:
4265:
3233:
2774:
1627:
3688:
1014:
1442:
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
4543:
4326:
2679:
1954:
1674:
1907:
1265:{\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 '}}}.}
540:
4361:
1599:
1308:
1090:
2863:
1560:
856:
361:
2926:
1876:
1409:
308:
284:
4054:
1796:
4024:
3977:
2838:
1850:
1762:
1535:
1335:
1117:
831:
332:
4688:
3387:
2900:
2149:
1028:: 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.
4285:
3997:
3301:
3192:
1823:
1503:
1479:
1429:
1389:
1054:
807:
381:
3241:
4747:
1065:
gravityâcapillary waves as a function of wavelength (or wave number) has a minimum. Waves with wavelengths much smaller than this critical wavelength
4762:
2934:
589:
5058:
4820:
5190:
1340:
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
189:
2115:{\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},}
1461:.) On the other, its vertical component must match the motion of the surface. This contribution ends up being responsible for the extra
3105:
1371:. The first two are potential energies, and responsible for the two terms inside the parenthesis, as is clear from the appearance of
5099:
5080:
4936:
438:
143:
waves. Light breezes which stir up such small ripples are also sometimes referred to as cat's paws. On the open ocean, much larger
1092:
are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed
5119:
4840:
2634:
Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both
4551:
4492:{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}\,g+{\frac {\sigma }{\rho +\rho '}}\,k^{2}\right),}
2450:
545:
4062:
1411:. For gravity, an assumption is made of the density of the fluids being constant (i.e., incompressibility), and likewise
779:{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}g+{\frac {\sigma }{\rho +\rho '}}k^{2}\right),}
4998:
902:
394:
5094:. Advanced Series on Ocean Engineering. Vol. 13. World Scientific, Singapore. pp. 2 Parts, 967 pages.
4708:
2423:{\displaystyle V_{\mathrm {st} }=\sigma \iint dx\,dy\;\left\approx {\frac {1}{2}}\sigma \iint dx\,dy\;\left,}
5180:
3394:
32:
4723:
2779:
52:
4219:
3200:
2437:'s) representation, and the second applies for small values of the derivatives (surfaces not too rough).
4660:
2743:
1511:
Dispersion relation for gravityâcapillary waves on an interface between two semiâinfinite fluid domains
865:
3656:
982:
511:
Dispersion of gravityâcapillary waves on the surface of deep water (zero mass density of upper layer,
5185:
5131:
5068:
5010:
3945:
2684:
1679:
87:
4508:
4290:
2637:
2154:
An increase in area of the surface causes a proportional increase of energy due to surface tension:
1912:
1632:
4213:
2869:
1604:
164:
28:
4331:
1581:
1286:
1068:
5034:
4733:
4718:
2865:
must vanish well away from the surface (in the "deep water" case, which is the one we consider).
1458:
4713:
5165:
647: • Dash-dotted lines: dispersion relation valid for deep-water capillary waves.
5147:
5095:
5076:
5054:
5026:
4932:
4924:
4816:
1881:
514:
287:
2905:
1855:
1394:
432:
For the boundary between fluid and vacuum (free surface), the dispersion relation reduces to
293:
269:
5139:
5018:
4969:
4703:
4029:
2843:
2734:
1771:
1765:
1735:
1540:
1454:
1341:
974:
836:
810:
341:
125:
4002:
3950:
2823:
1828:
1740:
1520:
1313:
1095:
816:
317:
4835:
3356:{\displaystyle {\frac {\partial \Phi '}{\partial z}}={\frac {\partial \eta }{\partial t}}}
1799:
1364:
1353:
311:
95:
39:
4667:
3366:
3291:{\displaystyle {\frac {\partial \Phi }{\partial z}}={\frac {\partial \eta }{\partial t}}}
2879:
2128:
1852:
due to gravity is the simplest: integrating the potential energy density due to gravity,
5135:
5014:
4728:
4270:
3982:
3653:
Then the contributions to the wave energy, horizontally integrated over one wavelength
3177:
2441:
1808:
1803:
1729:
1575:
1567:
1505:, and high ones (except around the one value at which the two dispersions cancel out.)
1488:
1464:
1447:
1435:
1414:
1374:
1039:
1025:
1021:
792:
366:
91:
83:
62:
5174:
5038:
4953:
4754:
3195:
2434:
1439:
1368:
962:
152:
507:
4363:
is just the expression in the square brackets, so that the dispersion relation is:
1571:
1443:
1017:
859:
133:
21:
645: • Dashed lines: dispersion relation for deep-water gravity waves.
1360:" The derivation of the general dispersion relation is therefore quite involved.
641: • Blue lines (A): phase velocity, Red lines (B): group velocity.
5046:
643: • Drawn lines: dispersion relation for gravityâcapillary waves.
114:
5143:
5022:
4658:
As usual for linear wave motions, the potential and kinetic energy are equal (
3092:
388:
384:
168:
110:
47:
5030:
4973:
3081:{\displaystyle T\approx {\frac {1}{2}}\iint dx\,dy\;\left_{{\text{at }}z=0}.}
113:
of capillary waves on water is typically less than a few centimeters, with a
632:{\displaystyle \scriptstyle {\frac {1}{\lambda }}{\sqrt {\sigma /(\rho g)}}}
172:
148:
144:
5151:
2872:, and assuming the deviations of the surface elevation to be small (so the
1563:
1562:
for the lower and upper domain respectively. The fluid is assumed to be
139:
When generated by light wind in open water, a nautical name for them is
335:
129:
67:
1450:. These are typically also good approximations for common situations.
1578:, and the velocity in the lower and upper layer can be obtained from
256:{\displaystyle \omega ^{2}={\frac {\sigma }{\rho +\rho '}}\,|k|^{3},}
102:
56:
5001:(1963). "The generation of capillary waves by steep gravity waves".
4957:
2820:
These equations can be solved with the proper boundary conditions:
155:) may result from coalescence of smaller wind-caused ripple-waves.
1280:
506:
61:
46:
38:
5111:
Statistical thermodynamics of surfaces, interfaces, and membranes
3164:{\displaystyle \eta =a\,\cos \,(kx-\omega t)=a\,\cos \,\theta ,}
3091:
To find the dispersion relation, it is sufficient to consider a
1363:
There are three contributions to the energy, due to gravity, to
124:
which are influenced by both the effects of surface tension and
79:
4505:
As a result, the average wave energy per unit horizontal area,
1276:
492:{\displaystyle \omega ^{2}={\frac {\sigma }{\rho }}\,|k|^{3}.}
4267:, so that variation with respect to the only free parameter,
3393:
To tackle the problem of finding the potentials, one may try
4815:. Cambridge University Press. Section 3.1 and Exercise 3.3.
16:
Wave on the surface of a fluid, dominated by surface tension
1909:) from a reference height to the position of the surface,
1438:
of the fluids. It is the most complicated and calls for a
5118:
Tufillaro, N. B.; Ramshankar, R.; Gollub, J. P. (1989).
4648:{\displaystyle {\bar {E}}={\frac {1}{2}}\,\left\,a^{2}.}
120:
A longer wavelength on a fluid interface will result in
20:"Rippled waves" redirects here. Not to be confused with
4854:
See e.g. Safran (1994) for a more detailed description.
2876:âintegrations may be approximated by integrating up to
1507:
593:
549:
4670:
4554:
4511:
4372:
4334:
4293:
4273:
4222:
4065:
4032:
4005:
3985:
3953:
3944:
The dispersion relation can now be obtained from the
3707:
3659:
3406:
3369:
3304:
3244:
3203:
3180:
3108:
2937:
2908:
2882:
2846:
2826:
2782:
2746:
2687:
2640:
2453:
2163:
2131:
1965:
1915:
1884:
1858:
1831:
1811:
1774:
1743:
1682:
1635:
1607:
1584:
1543:
1523:
1491:
1467:
1417:
1397:
1377:
1316:
1289:
1128:
1098:
1071:
1042:
985:
905:
868:
839:
819:
795:
661:
592:
548:
517:
441:
397:
369:
344:
320:
296:
272:
192:
2624:{\displaystyle T={\frac {1}{2}}\iint dx\,dy\;\left.}
1734:
Three contributions to the energy are involved: the
579:{\displaystyle \scriptstyle {\sqrt{g\sigma /\rho }}}
4768:
Light breeze ripples in the surface water of a lake
4202:{\displaystyle L={\frac {1}{4}}\lefta^{2}\lambda .}
1453:The resulting equation for the potential (which is
4682:
4647:
4537:
4491:
4355:
4320:
4279:
4259:
4201:
4048:
4018:
3991:
3971:
3933:
3682:
3642:
3381:
3355:
3290:
3227:
3186:
3163:
3080:
2920:
2894:
2857:
2832:
2809:
2768:
2725:
2673:
2623:
2422:
2143:
2114:
1948:
1901:
1870:
1844:
1817:
1790:
1756:
1720:
1668:
1621:
1593:
1554:
1529:
1497:
1473:
1423:
1403:
1383:
1329:
1302:
1264:
1111:
1084:
1048:
1008:
953:
891:
850:
825:
801:
778:
631:
578:
534:
491:
424:
375:
355:
326:
302:
278:
255:
4212:For sinusoidal waves and linear wave theory, the
1344:which corresponds to the minimal group velocity.
5120:"Order-disorder transition in capillary ripples"
1485:regimes to be dispersive, both at low values of
1310:is found to be 1.7 cm (0.67 in), and
1016:), only the first term is relevant and one has
183:The dispersion relation for capillary waves is
70:impacts on the interface between water and air.
4962:Proceedings of the London Mathematical Society
4813:Fluid Mechanics, a short course for physicists
2433:where the first equality is the area in this (
3999:the sum of the potential energies by gravity
954:{\displaystyle (\rho -\rho ')/(\rho +\rho ')}
586:as a function of inverse relative wavelength
8:
5075:(2nd ed.). Cambridge University Press.
5053:(6th ed.). Cambridge University Press.
4799:Dingemans (1997), Section 2.1.1, p. 45.
425:{\displaystyle \lambda ={\frac {2\pi }{k}}.}
2125:assuming the mean interface position is at
5092:Water wave propagation over uneven bottoms
4919:
4917:
3095:wave on the interface, propagating in the
3009:
3005:
2970:
2573:
2516:
2486:
2333:
2201:
2098:
2019:
1997:
4889:
4887:
4877:
4875:
4873:
4871:
4869:
4801:Phillips (1977), Section 3.2, p. 37.
4669:
4636:
4631:
4620:
4606:
4580:
4570:
4556:
4555:
4553:
4527:
4510:
4475:
4470:
4447:
4440:
4404:
4394:
4386:
4377:
4371:
4333:
4292:
4272:
4251:
4221:
4187:
4172:
4128:
4120:
4113:
4107:
4072:
4064:
4037:
4031:
4010:
4004:
3984:
3952:
3915:
3903:
3895:
3888:
3882:
3852:
3826:
3816:
3799:
3786:
3766:
3729:
3716:
3708:
3706:
3694:âdirection, and over a unit width in the
3672:
3658:
3629:
3625:
3618:
3608:
3600:
3596:
3591:
3582:
3574:
3568:
3512:
3508:
3501:
3491:
3483:
3479:
3474:
3465:
3457:
3451:
3407:
3405:
3368:
3333:
3305:
3303:
3268:
3245:
3243:
3202:
3179:
3154:
3150:
3122:
3118:
3107:
3059:
3058:
3028:
3027:
3018:
2985:
2984:
2980:
2963:
2944:
2936:
2928:), the kinetic energy can be written as:
2907:
2881:
2845:
2825:
2787:
2781:
2751:
2745:
2686:
2639:
2607:
2589:
2582:
2558:
2553:
2540:
2527:
2520:
2504:
2496:
2479:
2460:
2452:
2406:
2382:
2368:
2344:
2326:
2304:
2282:
2258:
2244:
2220:
2207:
2194:
2169:
2168:
2162:
2130:
2103:
2091:
2049:
2007:
2002:
1990:
1971:
1970:
1964:
1914:
1883:
1857:
1836:
1830:
1810:
1779:
1773:
1748:
1742:
1681:
1634:
1606:
1583:
1542:
1522:
1490:
1466:
1416:
1396:
1376:
1321:
1315:
1294:
1288:
1209:
1202:
1193:
1183:
1148:
1133:
1127:
1103:
1097:
1076:
1070:
1041:
998:
984:
926:
904:
867:
838:
818:
794:
762:
735:
693:
683:
675:
666:
660:
609:
604:
594:
591:
568:
559:
550:
547:
516:
480:
475:
466:
465:
455:
446:
440:
404:
396:
368:
343:
319:
295:
271:
244:
239:
230:
229:
206:
197:
191:
3397:, when both fields can be expressed as:
4985:, MacMillan, 2nd revised edition, 1894.
4845:Addison-Wesley. Volume I, Chapter 51-4.
4778:
4743:
542:). Phase and group velocity divided by
4838:, R.B. Leighton, and M. Sands (1963).
4786:
4784:
4782:
1481:outside the parenthesis, which causes
363:the density of the lighter fluid and
7:
2810:{\displaystyle \nabla ^{2}\Phi '=0.}
1434:The third contribution involves the
4260:{\displaystyle L=D(\omega ,k)a^{2}}
3228:{\displaystyle \theta =kx-\omega t}
2440:The last contribution involves the
1337:is 0.23 m/s (0.75 ft/s).
167:describes the relationship between
117:in excess of 0.2â0.3 meter/second.
5166:Capillary waves entry at sklogwiki
3524:
3411:
3344:
3336:
3321:
3312:
3308:
3279:
3271:
3256:
3251:
3248:
3044:
3035:
3031:
3020:
2996:
2991:
2988:
2981:
2794:
2784:
2769:{\displaystyle \nabla ^{2}\Phi =0}
2757:
2748:
2595:
2590:
2562:
2532:
2528:
2500:
2393:
2385:
2355:
2347:
2269:
2261:
2231:
2223:
2173:
2170:
1972:
1608:
1585:
1020:. In this limit, the waves have a
43:Capillary waves (ripples) in water
14:
4287:, gives the dispersion relation
892:{\displaystyle (\rho >\rho ')}
4790:Lamb (1994), §267, page 458â460.
4761:
4746:
3683:{\displaystyle \lambda =2\pi /k}
1570:, and the flow is assumed to be
1009:{\displaystyle k=2\pi /\lambda }
136:have a still longer wavelength.
94:are dominated by the effects of
27:For ripples in electricity, see
5073:The dynamics of the upper ocean
4841:The Feynman Lectures on Physics
2726:{\displaystyle \phi '(x,y,z,t)}
1721:{\displaystyle \phi '(x,y,z,t)}
1188:
1182:
105:, and are often referred to as
4603:
4586:
4561:
4538:{\displaystyle (T+V)/\lambda }
4524:
4512:
4395:
4387:
4350:
4338:
4321:{\displaystyle D(\omega ,k)=0}
4309:
4297:
4244:
4232:
4156:
4139:
4129:
4121:
4104:
4087:
3904:
3896:
3879:
3862:
3756:
3739:
3609:
3601:
3583:
3575:
3555:
3531:
3492:
3484:
3466:
3458:
3438:
3414:
3141:
3123:
2720:
2696:
2674:{\displaystyle \phi (x,y,z,t)}
2668:
2644:
2076:
2059:
2037:
2020:
1949:{\displaystyle z=\eta (x,y,t)}
1943:
1925:
1715:
1691:
1669:{\displaystyle \phi (x,y,z,t)}
1663:
1639:
1228:
1211:
1172:
1155:
948:
931:
923:
906:
886:
869:
684:
676:
623:
614:
476:
467:
240:
231:
101:Capillary waves are common in
1:
1622:{\displaystyle \nabla \phi '}
979:For large wavelengths (small
4954:Lord Rayleigh (J. W. Strutt)
4753:Ripples on water created by
4356:{\displaystyle D(\omega ,k)}
1594:{\displaystyle \nabla \phi }
1303:{\displaystyle \lambda _{m}}
1085:{\displaystyle \lambda _{m}}
66:Capillary waves produced by
5191:Oceanographical terminology
4863:Lamb (1994), §174 and §230.
809:is the acceleration due to
5207:
5144:10.1103/PhysRevLett.62.422
5003:Journal of Fluid Mechanics
4978:Reprinted as Appendix in:
4929:Linear and nonlinear waves
972:
18:
5090:Dingemans, M. W. (1997).
5023:10.1017/S0022112063000641
4214:phaseâaveraged Lagrangian
961:in the first term is the
4709:Dispersion (water waves)
1902:{\displaystyle \rho 'gz}
535:{\displaystyle \rho '=0}
5124:Physical Review Letters
5109:Safran, Samuel (1994).
3395:separation of variables
2921:{\displaystyle z=\eta }
1871:{\displaystyle \rho gz}
1768:, the potential energy
1404:{\displaystyle \sigma }
503:Gravityâcapillary waves
338:of the heavier fluid,
303:{\displaystyle \sigma }
279:{\displaystyle \omega }
179:Capillary waves, proper
122:gravityâcapillary waves
33:Ripple (disambiguation)
4999:Longuet-Higgins, M. S.
4974:10.1112/plms/s1-9.1.21
4958:"On progressive waves"
4931:. Wiley-Interscience.
4811:Falkovich, G. (2011).
4724:Thermal capillary wave
4684:
4649:
4539:
4493:
4357:
4322:
4281:
4261:
4216:is always of the form
4203:
4050:
4049:{\displaystyle V_{st}}
4020:
3993:
3973:
3935:
3684:
3644:
3383:
3357:
3292:
3229:
3188:
3165:
3082:
2922:
2896:
2859:
2858:{\displaystyle \phi '}
2834:
2811:
2770:
2727:
2675:
2625:
2424:
2145:
2116:
1950:
1903:
1872:
1846:
1825:of the flow. The part
1819:
1792:
1791:{\displaystyle V_{st}}
1758:
1722:
1670:
1623:
1595:
1556:
1555:{\displaystyle \rho '}
1531:
1499:
1475:
1425:
1405:
1385:
1331:
1304:
1266:
1113:
1086:
1060:Phase velocity minimum
1050:
1010:
955:
893:
852:
851:{\displaystyle \rho '}
827:
803:
780:
648:
633:
580:
536:
493:
426:
377:
357:
356:{\displaystyle \rho '}
328:
304:
280:
257:
128:, as well as by fluid
71:
59:
51:Ripples on Lifjord in
44:
31:. For other uses, see
4685:
4650:
4540:
4494:
4358:
4323:
4282:
4262:
4204:
4051:
4021:
4019:{\displaystyle V_{g}}
3994:
3974:
3972:{\displaystyle L=T-V}
3936:
3685:
3645:
3384:
3358:
3293:
3230:
3189:
3166:
3083:
2923:
2897:
2860:
2835:
2833:{\displaystyle \phi }
2812:
2771:
2728:
2676:
2626:
2425:
2146:
2117:
1951:
1904:
1873:
1847:
1845:{\displaystyle V_{g}}
1820:
1793:
1759:
1757:{\displaystyle V_{g}}
1723:
1671:
1629:, respectively. Here
1624:
1596:
1574:. Then the flows are
1557:
1532:
1530:{\displaystyle \rho }
1500:
1476:
1426:
1406:
1386:
1332:
1330:{\displaystyle c_{m}}
1305:
1267:
1114:
1112:{\displaystyle c_{m}}
1087:
1051:
1032:Capillary wave regime
1011:
973:Further information:
956:
894:
853:
828:
826:{\displaystyle \rho }
804:
781:
634:
581:
537:
510:
494:
427:
378:
358:
329:
327:{\displaystyle \rho }
305:
281:
258:
65:
50:
42:
4668:
4552:
4509:
4370:
4332:
4291:
4271:
4220:
4063:
4030:
4026:and surface tension
4003:
3983:
3951:
3705:
3698:âdirection, become:
3657:
3404:
3367:
3302:
3242:
3201:
3178:
3106:
2935:
2906:
2880:
2844:
2824:
2780:
2744:
2685:
2638:
2451:
2161:
2129:
1963:
1913:
1882:
1856:
1829:
1809:
1772:
1741:
1680:
1633:
1605:
1582:
1541:
1521:
1489:
1465:
1415:
1395:
1375:
1314:
1287:
1126:
1096:
1069:
1040:
983:
903:
866:
837:
817:
793:
659:
590:
546:
515:
439:
395:
367:
342:
318:
294:
270:
190:
82:traveling along the
5136:1989PhRvL..62..422T
5015:1963JFM....16..138L
4683:{\displaystyle T=V}
4502:the same as above.
3382:{\displaystyle z=0}
2895:{\displaystyle z=0}
2566:
2509:
2144:{\displaystyle z=0}
2012:
1730:velocity potentials
1459:Ocean surface waves
1446:â the flow is then
969:Gravity wave regime
165:dispersion relation
159:Dispersion relation
145:ocean surface waves
53:Ăksnes Municipality
29:Ripple (electrical)
4911:Lamb (1994), §230.
4881:Lamb (1994), §266.
4734:Wave-formed ripple
4719:Ocean surface wave
4680:
4645:
4535:
4489:
4353:
4318:
4277:
4257:
4199:
4046:
4016:
3989:
3969:
3931:
3929:
3680:
3640:
3638:
3379:
3353:
3298: and
3288:
3225:
3184:
3161:
3078:
2918:
2892:
2855:
2830:
2807:
2776: and
2766:
2723:
2671:
2621:
2549:
2492:
2420:
2141:
2112:
1998:
1946:
1899:
1868:
1842:
1815:
1788:
1754:
1718:
1666:
1619:
1591:
1552:
1527:
1495:
1471:
1421:
1401:
1381:
1327:
1300:
1262:
1109:
1082:
1046:
1006:
951:
889:
862:of the two fluids
848:
823:
799:
776:
649:
629:
628:
576:
575:
532:
489:
422:
373:
353:
324:
300:
276:
253:
86:of a fluid, whose
72:
60:
45:
5113:. Addison-Wesley.
5060:978-0-521-45868-9
4943:See section 11.7.
4893:Lamb (1994), §61.
4822:978-1-107-00575-4
4695:
4694:
4578:
4564:
4468:
4438:
4280:{\displaystyle a}
4134:
4080:
3992:{\displaystyle V}
3909:
3860:
3807:
3789:
3737:
3719:
3594:
3588:
3477:
3471:
3351:
3328:
3286:
3263:
3187:{\displaystyle a}
3062:
3051:
3003:
2952:
2733:must satisfy the
2468:
2400:
2362:
2312:
2288:
2276:
2238:
2057:
1818:{\displaystyle T}
1498:{\displaystyle k}
1474:{\displaystyle k}
1424:{\displaystyle g}
1384:{\displaystyle g}
1257:
1256:
1237:
1186:
1180:
1179:
1049:{\displaystyle k}
802:{\displaystyle g}
756:
727:
626:
602:
573:
463:
417:
376:{\displaystyle k}
288:angular frequency
227:
5198:
5155:
5114:
5105:
5086:
5064:
5042:
4986:
4977:
4950:
4944:
4942:
4921:
4912:
4909:
4903:
4902:Lamb (1994), §20
4900:
4894:
4891:
4882:
4879:
4864:
4861:
4855:
4852:
4846:
4833:
4827:
4826:
4808:
4802:
4797:
4791:
4788:
4765:
4750:
4704:Capillary action
4689:
4687:
4686:
4681:
4654:
4652:
4651:
4646:
4641:
4640:
4630:
4626:
4625:
4624:
4602:
4579:
4571:
4566:
4565:
4557:
4544:
4542:
4541:
4536:
4531:
4498:
4496:
4495:
4490:
4485:
4481:
4480:
4479:
4469:
4467:
4466:
4448:
4439:
4437:
4436:
4421:
4420:
4405:
4398:
4390:
4382:
4381:
4362:
4360:
4359:
4354:
4327:
4325:
4324:
4319:
4286:
4284:
4283:
4278:
4266:
4264:
4263:
4258:
4256:
4255:
4208:
4206:
4205:
4200:
4192:
4191:
4182:
4178:
4177:
4176:
4155:
4135:
4133:
4132:
4124:
4118:
4117:
4108:
4103:
4081:
4073:
4055:
4053:
4052:
4047:
4045:
4044:
4025:
4023:
4022:
4017:
4015:
4014:
3998:
3996:
3995:
3990:
3978:
3976:
3975:
3970:
3940:
3938:
3937:
3932:
3930:
3920:
3919:
3910:
3908:
3907:
3899:
3893:
3892:
3883:
3878:
3861:
3853:
3831:
3830:
3821:
3820:
3808:
3800:
3791:
3790:
3787:
3771:
3770:
3755:
3738:
3730:
3721:
3720:
3717:
3689:
3687:
3686:
3681:
3676:
3649:
3647:
3646:
3641:
3639:
3617:
3616:
3612:
3604:
3595:
3592:
3589:
3587:
3586:
3578:
3569:
3530:
3500:
3499:
3495:
3487:
3478:
3475:
3472:
3470:
3469:
3461:
3452:
3388:
3386:
3385:
3380:
3362:
3360:
3359:
3354:
3352:
3350:
3342:
3334:
3329:
3327:
3319:
3318:
3306:
3297:
3295:
3294:
3289:
3287:
3285:
3277:
3269:
3264:
3262:
3254:
3246:
3234:
3232:
3231:
3226:
3193:
3191:
3190:
3185:
3170:
3168:
3167:
3162:
3087:
3085:
3084:
3079:
3074:
3073:
3063:
3060:
3057:
3053:
3052:
3050:
3042:
3041:
3029:
3026:
3017:
3004:
3002:
2994:
2986:
2953:
2945:
2927:
2925:
2924:
2919:
2901:
2899:
2898:
2893:
2870:Green's identity
2864:
2862:
2861:
2856:
2854:
2839:
2837:
2836:
2831:
2816:
2814:
2813:
2808:
2800:
2792:
2791:
2775:
2773:
2772:
2767:
2756:
2755:
2735:Laplace equation
2732:
2730:
2729:
2724:
2695:
2680:
2678:
2677:
2672:
2630:
2628:
2627:
2622:
2617:
2613:
2612:
2611:
2606:
2602:
2601:
2593:
2581:
2565:
2557:
2545:
2544:
2539:
2535:
2531:
2508:
2503:
2469:
2461:
2429:
2427:
2426:
2421:
2416:
2412:
2411:
2410:
2405:
2401:
2399:
2391:
2383:
2373:
2372:
2367:
2363:
2361:
2353:
2345:
2313:
2305:
2300:
2296:
2289:
2287:
2286:
2281:
2277:
2275:
2267:
2259:
2249:
2248:
2243:
2239:
2237:
2229:
2221:
2208:
2178:
2177:
2176:
2150:
2148:
2147:
2142:
2121:
2119:
2118:
2113:
2108:
2107:
2075:
2058:
2050:
2036:
2011:
2006:
1977:
1976:
1975:
1955:
1953:
1952:
1947:
1908:
1906:
1905:
1900:
1892:
1877:
1875:
1874:
1869:
1851:
1849:
1848:
1843:
1841:
1840:
1824:
1822:
1821:
1816:
1797:
1795:
1794:
1789:
1787:
1786:
1763:
1761:
1760:
1755:
1753:
1752:
1736:potential energy
1727:
1725:
1724:
1719:
1690:
1675:
1673:
1672:
1667:
1628:
1626:
1625:
1620:
1618:
1600:
1598:
1597:
1592:
1561:
1559:
1558:
1553:
1551:
1536:
1534:
1533:
1528:
1508:
1504:
1502:
1501:
1496:
1480:
1478:
1477:
1472:
1455:Laplace equation
1436:kinetic energies
1430:
1428:
1427:
1422:
1410:
1408:
1407:
1402:
1390:
1388:
1387:
1382:
1336:
1334:
1333:
1328:
1326:
1325:
1309:
1307:
1306:
1301:
1299:
1298:
1271:
1269:
1268:
1263:
1258:
1255:
1254:
1239:
1238:
1227:
1210:
1204:
1203:
1198:
1197:
1187:
1184:
1181:
1178:
1171:
1150:
1149:
1138:
1137:
1118:
1116:
1115:
1110:
1108:
1107:
1091:
1089:
1088:
1083:
1081:
1080:
1055:
1053:
1052:
1047:
1015:
1013:
1012:
1007:
1002:
975:Airy wave theory
960:
958:
957:
952:
947:
930:
922:
898:
896:
895:
890:
885:
857:
855:
854:
849:
847:
832:
830:
829:
824:
808:
806:
805:
800:
785:
783:
782:
777:
772:
768:
767:
766:
757:
755:
754:
736:
728:
726:
725:
710:
709:
694:
687:
679:
671:
670:
638:
636:
635:
630:
627:
613:
605:
603:
595:
585:
583:
582:
577:
574:
572:
567:
563:
551:
541:
539:
538:
533:
525:
498:
496:
495:
490:
485:
484:
479:
470:
464:
456:
451:
450:
431:
429:
428:
423:
418:
413:
405:
382:
380:
379:
374:
362:
360:
359:
354:
352:
333:
331:
330:
325:
309:
307:
306:
301:
285:
283:
282:
277:
262:
260:
259:
254:
249:
248:
243:
234:
228:
226:
225:
207:
202:
201:
36:
25:
5206:
5205:
5201:
5200:
5199:
5197:
5196:
5195:
5171:
5170:
5162:
5117:
5108:
5102:
5089:
5083:
5069:Phillips, O. M.
5067:
5061:
5045:
4997:
4994:
4989:
4980:Theory of Sound
4952:
4951:
4947:
4939:
4923:
4922:
4915:
4910:
4906:
4901:
4897:
4892:
4885:
4880:
4867:
4862:
4858:
4853:
4849:
4834:
4830:
4823:
4810:
4809:
4805:
4800:
4798:
4794:
4789:
4780:
4776:
4769:
4766:
4757:
4751:
4742:
4700:
4666:
4665:
4632:
4616:
4595:
4585:
4581:
4550:
4549:
4507:
4506:
4471:
4459:
4452:
4429:
4422:
4413:
4406:
4403:
4399:
4373:
4368:
4367:
4330:
4329:
4289:
4288:
4269:
4268:
4247:
4218:
4217:
4183:
4168:
4148:
4119:
4109:
4096:
4086:
4082:
4061:
4060:
4033:
4028:
4027:
4006:
4001:
4000:
3981:
3980:
3949:
3948:
3928:
3927:
3911:
3894:
3884:
3871:
3845:
3839:
3838:
3822:
3812:
3792:
3782:
3779:
3778:
3762:
3748:
3722:
3712:
3703:
3702:
3655:
3654:
3637:
3636:
3590:
3573:
3558:
3523:
3520:
3519:
3473:
3456:
3441:
3402:
3401:
3365:
3364:
3343:
3335:
3320:
3311:
3307:
3300:
3299:
3278:
3270:
3255:
3247:
3240:
3239:
3199:
3198:
3176:
3175:
3174:with amplitude
3104:
3103:
3043:
3034:
3030:
3019:
3010:
2995:
2987:
2976:
2972:
2971:
2933:
2932:
2904:
2903:
2878:
2877:
2847:
2842:
2841:
2822:
2821:
2793:
2783:
2778:
2777:
2747:
2742:
2741:
2688:
2683:
2682:
2636:
2635:
2594:
2588:
2584:
2583:
2574:
2526:
2522:
2521:
2491:
2487:
2449:
2448:
2392:
2384:
2378:
2377:
2354:
2346:
2340:
2339:
2338:
2334:
2268:
2260:
2254:
2253:
2230:
2222:
2216:
2215:
2206:
2202:
2164:
2159:
2158:
2127:
2126:
2099:
2068:
2029:
1966:
1961:
1960:
1911:
1910:
1885:
1880:
1879:
1854:
1853:
1832:
1827:
1826:
1807:
1806:
1800:surface tension
1775:
1770:
1769:
1744:
1739:
1738:
1683:
1678:
1677:
1631:
1630:
1611:
1603:
1602:
1580:
1579:
1544:
1539:
1538:
1519:
1518:
1487:
1486:
1463:
1462:
1413:
1412:
1393:
1392:
1373:
1372:
1365:surface tension
1354:Richard Feynman
1350:
1317:
1312:
1311:
1290:
1285:
1284:
1247:
1240:
1220:
1205:
1189:
1164:
1154:
1129:
1124:
1123:
1099:
1094:
1093:
1072:
1067:
1066:
1062:
1038:
1037:
1036:Shorter (large
1034:
981:
980:
977:
971:
940:
915:
901:
900:
878:
864:
863:
840:
835:
834:
815:
814:
791:
790:
758:
747:
740:
718:
711:
702:
695:
692:
688:
662:
657:
656:
646:
644:
642:
640:
588:
587:
552:
544:
543:
518:
513:
512:
505:
474:
442:
437:
436:
406:
393:
392:
365:
364:
345:
340:
339:
316:
315:
312:surface tension
292:
291:
268:
267:
238:
218:
211:
193:
188:
187:
181:
161:
96:surface tension
37:
26:
19:
17:
12:
11:
5:
5204:
5202:
5194:
5193:
5188:
5183:
5181:Fluid dynamics
5173:
5172:
5169:
5168:
5161:
5160:External links
5158:
5157:
5156:
5130:(4): 422â425.
5115:
5106:
5100:
5087:
5081:
5065:
5059:
5043:
5009:(1): 138â159.
4993:
4990:
4988:
4987:
4945:
4937:
4925:Whitham, G. B.
4913:
4904:
4895:
4883:
4865:
4856:
4847:
4828:
4821:
4803:
4792:
4777:
4775:
4772:
4771:
4770:
4767:
4760:
4758:
4755:water striders
4752:
4745:
4741:
4738:
4737:
4736:
4731:
4729:Two-phase flow
4726:
4721:
4716:
4711:
4706:
4699:
4696:
4693:
4692:
4679:
4676:
4673:
4656:
4655:
4644:
4639:
4635:
4629:
4623:
4619:
4615:
4612:
4609:
4605:
4601:
4598:
4594:
4591:
4588:
4584:
4577:
4574:
4569:
4563:
4560:
4534:
4530:
4526:
4523:
4520:
4517:
4514:
4500:
4499:
4488:
4484:
4478:
4474:
4465:
4462:
4458:
4455:
4451:
4446:
4443:
4435:
4432:
4428:
4425:
4419:
4416:
4412:
4409:
4402:
4397:
4393:
4389:
4385:
4380:
4376:
4352:
4349:
4346:
4343:
4340:
4337:
4328:. In our case
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4276:
4254:
4250:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4210:
4209:
4198:
4195:
4190:
4186:
4181:
4175:
4171:
4167:
4164:
4161:
4158:
4154:
4151:
4147:
4144:
4141:
4138:
4131:
4127:
4123:
4116:
4112:
4106:
4102:
4099:
4095:
4092:
4089:
4085:
4079:
4076:
4071:
4068:
4043:
4040:
4036:
4013:
4009:
3988:
3968:
3965:
3962:
3959:
3956:
3942:
3941:
3926:
3923:
3918:
3914:
3906:
3902:
3898:
3891:
3887:
3881:
3877:
3874:
3870:
3867:
3864:
3859:
3856:
3851:
3848:
3846:
3844:
3841:
3840:
3837:
3834:
3829:
3825:
3819:
3815:
3811:
3806:
3803:
3798:
3795:
3793:
3785:
3781:
3780:
3777:
3774:
3769:
3765:
3761:
3758:
3754:
3751:
3747:
3744:
3741:
3736:
3733:
3728:
3725:
3723:
3715:
3711:
3710:
3679:
3675:
3671:
3668:
3665:
3662:
3651:
3650:
3635:
3632:
3628:
3624:
3621:
3615:
3611:
3607:
3603:
3599:
3585:
3581:
3577:
3572:
3567:
3564:
3561:
3559:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3529:
3526:
3522:
3521:
3518:
3515:
3511:
3507:
3504:
3498:
3494:
3490:
3486:
3482:
3468:
3464:
3460:
3455:
3450:
3447:
3444:
3442:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3409:
3391:
3390:
3378:
3375:
3372:
3349:
3346:
3341:
3338:
3332:
3326:
3323:
3317:
3314:
3310:
3284:
3281:
3276:
3273:
3267:
3261:
3258:
3253:
3250:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3183:
3172:
3171:
3160:
3157:
3153:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3121:
3117:
3114:
3111:
3089:
3088:
3077:
3072:
3069:
3066:
3056:
3049:
3046:
3040:
3037:
3033:
3025:
3022:
3016:
3013:
3008:
3001:
2998:
2993:
2990:
2983:
2979:
2975:
2969:
2966:
2962:
2959:
2956:
2951:
2948:
2943:
2940:
2917:
2914:
2911:
2891:
2888:
2885:
2853:
2850:
2829:
2818:
2817:
2806:
2803:
2799:
2796:
2790:
2786:
2765:
2762:
2759:
2754:
2750:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2694:
2691:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2632:
2631:
2620:
2616:
2610:
2605:
2600:
2597:
2592:
2587:
2580:
2577:
2572:
2569:
2564:
2561:
2556:
2552:
2548:
2543:
2538:
2534:
2530:
2525:
2519:
2515:
2512:
2507:
2502:
2499:
2495:
2490:
2485:
2482:
2478:
2475:
2472:
2467:
2464:
2459:
2456:
2444:of the fluid:
2442:kinetic energy
2431:
2430:
2419:
2415:
2409:
2404:
2398:
2395:
2390:
2387:
2381:
2376:
2371:
2366:
2360:
2357:
2352:
2349:
2343:
2337:
2332:
2329:
2325:
2322:
2319:
2316:
2311:
2308:
2303:
2299:
2295:
2292:
2285:
2280:
2274:
2271:
2266:
2263:
2257:
2252:
2247:
2242:
2236:
2233:
2228:
2225:
2219:
2214:
2211:
2205:
2200:
2197:
2193:
2190:
2187:
2184:
2181:
2175:
2172:
2167:
2140:
2137:
2134:
2123:
2122:
2111:
2106:
2102:
2097:
2094:
2090:
2087:
2084:
2081:
2078:
2074:
2071:
2067:
2064:
2061:
2056:
2053:
2048:
2045:
2042:
2039:
2035:
2032:
2028:
2025:
2022:
2018:
2015:
2010:
2005:
2001:
1996:
1993:
1989:
1986:
1983:
1980:
1974:
1969:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1898:
1895:
1891:
1888:
1867:
1864:
1861:
1839:
1835:
1814:
1804:kinetic energy
1785:
1782:
1778:
1751:
1747:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1689:
1686:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1617:
1614:
1610:
1590:
1587:
1568:incompressible
1550:
1547:
1526:
1513:
1512:
1494:
1470:
1420:
1400:
1380:
1349:
1346:
1324:
1320:
1297:
1293:
1273:
1272:
1261:
1253:
1250:
1246:
1243:
1236:
1233:
1230:
1226:
1223:
1219:
1216:
1213:
1208:
1201:
1196:
1192:
1177:
1174:
1170:
1167:
1163:
1160:
1157:
1153:
1147:
1144:
1141:
1136:
1132:
1106:
1102:
1079:
1075:
1061:
1058:
1045:
1033:
1030:
1026:phase velocity
1022:group velocity
1005:
1001:
997:
994:
991:
988:
970:
967:
950:
946:
943:
939:
936:
933:
929:
925:
921:
918:
914:
911:
908:
888:
884:
881:
877:
874:
871:
846:
843:
822:
798:
787:
786:
775:
771:
765:
761:
753:
750:
746:
743:
739:
734:
731:
724:
721:
717:
714:
708:
705:
701:
698:
691:
686:
682:
678:
674:
669:
665:
625:
622:
619:
616:
612:
608:
601:
598:
571:
566:
562:
558:
555:
531:
528:
524:
521:
504:
501:
500:
499:
488:
483:
478:
473:
469:
462:
459:
454:
449:
445:
421:
416:
412:
409:
403:
400:
372:
351:
348:
323:
299:
275:
264:
263:
252:
247:
242:
237:
233:
224:
221:
217:
214:
210:
205:
200:
196:
180:
177:
160:
157:
92:phase velocity
84:phase boundary
76:capillary wave
15:
13:
10:
9:
6:
4:
3:
2:
5203:
5192:
5189:
5187:
5184:
5182:
5179:
5178:
5176:
5167:
5164:
5163:
5159:
5153:
5149:
5145:
5141:
5137:
5133:
5129:
5125:
5121:
5116:
5112:
5107:
5103:
5101:981-02-0427-2
5097:
5093:
5088:
5084:
5082:0-521-29801-6
5078:
5074:
5070:
5066:
5062:
5056:
5052:
5051:Hydrodynamics
5048:
5044:
5040:
5036:
5032:
5028:
5024:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4995:
4991:
4984:
4981:
4975:
4971:
4967:
4963:
4959:
4955:
4949:
4946:
4940:
4938:0-471-94090-9
4934:
4930:
4926:
4920:
4918:
4914:
4908:
4905:
4899:
4896:
4890:
4888:
4884:
4878:
4876:
4874:
4872:
4870:
4866:
4860:
4857:
4851:
4848:
4844:
4842:
4837:
4832:
4829:
4824:
4818:
4814:
4807:
4804:
4796:
4793:
4787:
4785:
4783:
4779:
4773:
4764:
4759:
4756:
4749:
4744:
4739:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4701:
4697:
4691:
4677:
4674:
4671:
4663:
4662:
4661:equipartition
4642:
4637:
4633:
4627:
4621:
4617:
4613:
4610:
4607:
4599:
4596:
4592:
4589:
4582:
4575:
4572:
4567:
4558:
4548:
4547:
4546:
4532:
4528:
4521:
4518:
4515:
4503:
4486:
4482:
4476:
4472:
4463:
4460:
4456:
4453:
4449:
4444:
4441:
4433:
4430:
4426:
4423:
4417:
4414:
4410:
4407:
4400:
4391:
4383:
4378:
4374:
4366:
4365:
4364:
4347:
4344:
4341:
4335:
4315:
4312:
4306:
4303:
4300:
4294:
4274:
4252:
4248:
4241:
4238:
4235:
4229:
4226:
4223:
4215:
4196:
4193:
4188:
4184:
4179:
4173:
4169:
4165:
4162:
4159:
4152:
4149:
4145:
4142:
4136:
4125:
4114:
4110:
4100:
4097:
4093:
4090:
4083:
4077:
4074:
4069:
4066:
4059:
4058:
4057:
4041:
4038:
4034:
4011:
4007:
3986:
3966:
3963:
3960:
3957:
3954:
3947:
3924:
3921:
3916:
3912:
3900:
3889:
3885:
3875:
3872:
3868:
3865:
3857:
3854:
3849:
3847:
3842:
3835:
3832:
3827:
3823:
3817:
3813:
3809:
3804:
3801:
3796:
3794:
3783:
3775:
3772:
3767:
3763:
3759:
3752:
3749:
3745:
3742:
3734:
3731:
3726:
3724:
3713:
3701:
3700:
3699:
3697:
3693:
3677:
3673:
3669:
3666:
3663:
3660:
3633:
3630:
3626:
3622:
3619:
3613:
3605:
3597:
3579:
3570:
3565:
3562:
3560:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3527:
3516:
3513:
3509:
3505:
3502:
3496:
3488:
3480:
3462:
3453:
3448:
3445:
3443:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3400:
3399:
3398:
3396:
3376:
3373:
3370:
3347:
3339:
3330:
3324:
3315:
3282:
3274:
3265:
3259:
3238:
3237:
3236:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3197:
3181:
3158:
3155:
3151:
3147:
3144:
3138:
3135:
3132:
3129:
3126:
3119:
3115:
3112:
3109:
3102:
3101:
3100:
3098:
3094:
3075:
3070:
3067:
3064:
3054:
3047:
3038:
3023:
3014:
3011:
3006:
2999:
2977:
2973:
2967:
2964:
2960:
2957:
2954:
2949:
2946:
2941:
2938:
2931:
2930:
2929:
2915:
2912:
2909:
2889:
2886:
2883:
2875:
2871:
2866:
2851:
2848:
2827:
2804:
2801:
2797:
2788:
2763:
2760:
2752:
2740:
2739:
2738:
2736:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2692:
2689:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2641:
2618:
2614:
2608:
2603:
2598:
2585:
2578:
2575:
2570:
2567:
2559:
2554:
2550:
2546:
2541:
2536:
2523:
2517:
2513:
2510:
2505:
2497:
2493:
2488:
2483:
2480:
2476:
2473:
2470:
2465:
2462:
2457:
2454:
2447:
2446:
2445:
2443:
2438:
2436:
2417:
2413:
2407:
2402:
2396:
2388:
2379:
2374:
2369:
2364:
2358:
2350:
2341:
2335:
2330:
2327:
2323:
2320:
2317:
2314:
2309:
2306:
2301:
2297:
2293:
2290:
2283:
2278:
2272:
2264:
2255:
2250:
2245:
2240:
2234:
2226:
2217:
2212:
2209:
2203:
2198:
2195:
2191:
2188:
2185:
2182:
2179:
2165:
2157:
2156:
2155:
2152:
2138:
2135:
2132:
2109:
2104:
2100:
2095:
2092:
2088:
2085:
2082:
2079:
2072:
2069:
2065:
2062:
2054:
2051:
2046:
2043:
2040:
2033:
2030:
2026:
2023:
2016:
2013:
2008:
2003:
1999:
1994:
1991:
1987:
1984:
1981:
1978:
1967:
1959:
1958:
1957:
1940:
1937:
1934:
1931:
1928:
1922:
1919:
1916:
1896:
1893:
1889:
1886:
1865:
1862:
1859:
1837:
1833:
1812:
1805:
1801:
1783:
1780:
1776:
1767:
1749:
1745:
1737:
1731:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1687:
1684:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1636:
1615:
1612:
1588:
1577:
1573:
1569:
1565:
1548:
1545:
1524:
1515:
1514:
1510:
1509:
1506:
1492:
1484:
1468:
1460:
1456:
1451:
1449:
1445:
1441:
1437:
1432:
1418:
1398:
1378:
1370:
1369:hydrodynamics
1366:
1361:
1359:
1355:
1347:
1345:
1343:
1338:
1322:
1318:
1295:
1291:
1282:
1278:
1259:
1251:
1248:
1244:
1241:
1234:
1231:
1224:
1221:
1217:
1214:
1206:
1199:
1194:
1190:
1175:
1168:
1165:
1161:
1158:
1151:
1145:
1142:
1139:
1134:
1130:
1122:
1121:
1120:
1104:
1100:
1077:
1073:
1059:
1057:
1043:
1031:
1029:
1027:
1023:
1019:
1018:gravity waves
1003:
999:
995:
992:
989:
986:
976:
968:
966:
964:
963:Atwood number
944:
941:
937:
934:
927:
919:
916:
912:
909:
899:. The factor
882:
879:
875:
872:
861:
844:
841:
820:
812:
796:
773:
769:
763:
759:
751:
748:
744:
741:
737:
732:
729:
722:
719:
715:
712:
706:
703:
699:
696:
689:
680:
672:
667:
663:
655:
654:
653:
620:
617:
610:
606:
599:
596:
569:
564:
560:
556:
553:
529:
526:
522:
519:
509:
502:
486:
481:
471:
460:
457:
452:
447:
443:
435:
434:
433:
419:
414:
410:
407:
401:
398:
390:
386:
370:
349:
346:
337:
321:
313:
297:
289:
273:
250:
245:
235:
222:
219:
215:
212:
208:
203:
198:
194:
186:
185:
184:
178:
176:
174:
170:
166:
158:
156:
154:
150:
146:
142:
137:
135:
134:gravity waves
131:
127:
123:
118:
116:
112:
108:
104:
99:
97:
93:
89:
85:
81:
77:
69:
64:
58:
54:
49:
41:
34:
30:
23:
5127:
5123:
5110:
5091:
5072:
5050:
5006:
5002:
4982:
4979:
4965:
4961:
4948:
4928:
4907:
4898:
4859:
4850:
4839:
4836:R.P. Feynman
4831:
4812:
4806:
4795:
4659:
4657:
4504:
4501:
4211:
3943:
3695:
3691:
3652:
3392:
3173:
3099:âdirection:
3096:
3090:
2873:
2867:
2819:
2633:
2439:
2432:
2153:
2124:
1733:
1572:irrotational
1482:
1452:
1444:irrotational
1440:hydrodynamic
1433:
1362:
1357:
1351:
1339:
1274:
1063:
1035:
978:
860:mass density
788:
650:
265:
182:
162:
140:
138:
121:
119:
106:
100:
75:
73:
22:Rippled wave
5186:Water waves
2902:instead of
1798:due to the
1283:interface,
132:. Ordinary
115:phase speed
5175:Categories
4992:References
4714:Fluid pipe
3946:Lagrangian
3363: at
3093:sinusoidal
1348:Derivation
389:wavelength
385:wavenumber
169:wavelength
111:wavelength
5039:119740891
5031:1469-7645
4968:: 21â26.
4614:σ
4597:ρ
4593:−
4590:ρ
4562:¯
4533:λ
4461:ρ
4454:ρ
4450:σ
4431:ρ
4424:ρ
4415:ρ
4411:−
4408:ρ
4375:ω
4342:ω
4301:ω
4236:ω
4194:λ
4166:σ
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3670:π
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3631:θ
3620:ω
3598:−
3566:−
3525:Φ
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3503:ω
3412:Φ
3345:∂
3340:η
3337:∂
3322:∂
3313:Φ
3309:∂
3280:∂
3275:η
3272:∂
3257:∂
3252:Φ
3249:∂
3220:ω
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3205:θ
3194:and wave
3156:θ
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3036:Φ
3032:∂
3021:Φ
3012:ρ
3007:−
2997:∂
2992:Φ
2989:∂
2982:Φ
2978:ρ
2955:∬
2942:≈
2916:η
2849:ϕ
2828:ϕ
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2785:∇
2758:Φ
2749:∇
2690:ϕ
2642:ϕ
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2591:∇
2576:ρ
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2555:η
2551:∫
2533:Φ
2529:∇
2518:ρ
2506:η
2501:∞
2498:−
2494:∫
2471:∬
2394:∂
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2356:∂
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2318:∬
2315:σ
2302:≈
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2232:∂
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2186:∬
2183:σ
2101:η
2083:∬
2070:ρ
2066:−
2063:ρ
2031:ρ
2027:−
2024:ρ
2009:η
2000:∫
1982:∬
1923:η
1887:ρ
1860:ρ
1685:ϕ
1637:ϕ
1613:ϕ
1609:∇
1589:ϕ
1586:∇
1576:potential
1546:ρ
1525:ρ
1448:potential
1399:σ
1367:, and to
1356:put it, "
1292:λ
1249:ρ
1242:ρ
1235:σ
1222:ρ
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1162:−
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1074:λ
1024:half the
1004:λ
996:π
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910:ρ
880:ρ
873:ρ
842:ρ
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720:ρ
713:ρ
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700:−
697:ρ
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618:ρ
607:σ
600:λ
565:ρ
557:σ
520:ρ
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458:σ
444:ω
411:π
399:λ
347:ρ
322:ρ
298:σ
274:ω
220:ρ
213:ρ
209:σ
195:ω
173:frequency
141:cat's paw
5152:10040229
5071:(1977).
5049:(1994).
5047:Lamb, H.
4956:(1877).
4927:(1974).
4698:See also
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1802:and the
1688:′
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1275:For the
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883:′
858:are the
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88:dynamics
5132:Bibcode
5011:Bibcode
4740:Gallery
3979:, with
3690:in the
1766:gravity
1764:due to
1342:caustic
811:gravity
336:density
286:is the
130:inertia
126:gravity
109:. The
107:ripples
68:droplet
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789:where
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266:where
153:swells
103:nature
57:Norway
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3196:phase
2435:Monge
1281:water
1119:are:
78:is a
5148:PMID
5096:ISBN
5077:ISBN
5055:ISBN
5027:ISSN
4933:ISBN
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2840:and
2681:and
1878:(or
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1676:and
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833:and
391:is
383:the
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310:the
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151:and
149:seas
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80:wave
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