Airy wave theory - Knowledge

9190: 8653: 9185:{\displaystyle {\begin{aligned}{\boldsymbol {S}}&={\begin{pmatrix}S_{xx}&S_{xy}\\S_{yx}&S_{yy}\end{pmatrix}}={\boldsymbol {I}}\left({\frac {c_{g}}{c_{p}}}-{\frac {1}{2}}\right)E+{\frac {1}{k^{2}}}{\begin{pmatrix}k_{x}k_{x}&k_{x}k_{y}\\k_{y}k_{x}&k_{y}k_{y}\end{pmatrix}}{\frac {c_{g}}{c_{p}}}E,\\{\boldsymbol {I}}&={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\\\nabla \mathbf {U} &={\begin{pmatrix}\displaystyle {\frac {\partial U_{x}}{\partial x}}&\displaystyle {\frac {\partial U_{y}}{\partial x}}\\\displaystyle {\frac {\partial U_{x}}{\partial y}}&\displaystyle {\frac {\partial U_{y}}{\partial y}}\end{pmatrix}},\end{aligned}}} 5395: 748: 2027: 12156: 4833: 132: 13044: 740: 124: 13065: 7764: 12145: 13054: 9932: 9448: 10210: 7454: 7165: 7477: 9754: 9267: 9985: 7252: 9510:, the second definition (S2) is more appropriate. These wave systems have zero mass flux and momentum when using the second definition. In contrast, according to Stokes' first definition (S1), there is a wave-induced mass flux in the wave propagation direction, which has to be balanced by a mean flow 760:

position. With the propagating wave motion, the fluid particles transfer energy in the wave propagation direction, without having a mean velocity. The diameter of the orbits reduces with depth below the free surface. In deep water, the orbit's diameter is reduced to 4% of its free-surface value at a

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In the table below, several second-order wave properties – as well as the dynamical equations they satisfy in case of slowly varying conditions in space and time – are given. More details on these can be found below. The table gives results for wave propagation in one horizontal spatial dimension.

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Underneath the surface, there is a fluid motion associated with the free surface motion. While the surface elevation shows a propagating wave, the fluid particles are in an orbital motion. Within the framework of Airy wave theory, the orbits are closed curves: circles in deep water and ellipses in

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Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains. As can be seen above, many wave quantities like surface elevation and orbital velocity are oscillatory in nature with zero mean (within the framework of linear theory). In water

6621: 2583: 6978: 2712: 5337: 7759:{\displaystyle {\begin{aligned}E_{\text{kin}}&={\overline {\int _{-h}^{0}{\tfrac {1}{2}}\rho \left\,\mathrm {d} z}}-\int _{-h}^{0}{\tfrac {1}{2}}\rho \left|\mathbf {U} \right|^{2}\,\mathrm {d} z\\&={\tfrac {1}{4}}\rho {\frac {\sigma ^{2}}{k\tanh kh}}a^{2},\end{aligned}}} 6740: 6855: 1373:

for an unsteady potential flow. The pressure above the free surface is assumed to be constant. This constant pressure is taken equal to zero, without loss of generality, since the level of such a constant pressure does not alter the flow. After linearisation, this gives the

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In the table below, several flow quantities and parameters according to Airy wave theory are given. The given quantities are for a bit more general situation as for the solution given above. Firstly, the waves may propagate in an arbitrary horizontal direction in the

987: 10632: 8539: 1336: 6962: 9927:{\displaystyle {\frac {\partial }{\partial t}}\left(\rho h{\tilde {\mathbf {U} }}\right)+\nabla \cdot \left(\rho h{\tilde {\mathbf {U} }}\otimes {\tilde {\mathbf {U} }}+{\tfrac {1}{2}}\rho gh^{2}{\boldsymbol {I}}+{\boldsymbol {S}}\right)=\rho gh\nabla d,} 214:) approach to describe the motion of gravity waves on a fluid surface. The use of (inviscid and irrotational) potential flow in water waves is remarkably successful, given its failure to describe many other fluid flows where it is often essential to take 1101: 9443:{\displaystyle {\begin{aligned}\mathbf {M} &={\overline {\int _{-h}^{\eta }\rho \left(\mathbf {U} +\mathbf {u} _{x}\right)\,\mathrm {d} z}}-\int _{-h}^{0}\rho \mathbf {U} \,\mathrm {d} z\\&={\frac {E}{c_{p}}}\mathbf {e} _{k},\end{aligned}}} 5655: 10347: 8169: 756:

finite depth—with the circles dying out before reaching the bottom of the fluid layer, and the ellipses becoming flatter near the bottom of the fluid layer. So while the wave propagates, the fluid particles just orbit (oscillate) around their

10205:{\displaystyle {\tilde {\mathbf {U} }}\otimes {\tilde {\mathbf {U} }}={\begin{pmatrix}{\tilde {U}}_{x}{\tilde {U}}_{x}&{\tilde {U}}_{x}{\tilde {U}}_{y}\\{\tilde {U}}_{y}{\tilde {U}}_{x}&{\tilde {U}}_{y}{\tilde {U}}_{y}\end{pmatrix}}.} 8057: 5746:, and mean-flow variations of currents and surface elevation. As well as in the description of the wave and mean-flow interactions due to time and space-variations in amplitude, frequency, wavelength and direction of the wave field itself. 6474: 7449:{\displaystyle {\begin{aligned}E_{\text{pot}}&={\overline {\int _{-h}^{\eta }\rho gz\,\mathrm {d} z}}-\int _{-h}^{0}\rho gz\,\mathrm {d} z\\&={\overline {{\tfrac {1}{2}}\rho g\eta ^{2}}}={\tfrac {1}{4}}\rho ga^{2}.\end{aligned}}} 1719: 1466: 3272: 10412: 5022: 3407: 9604: 3868: 8334: 3605: 4371: 1200: 6305: 7160:{\displaystyle {\frac {\partial }{\partial t}}\left(\rho h{\tilde {U}}\right)+{\frac {\partial }{\partial x}}\left(\rho h{\tilde {U}}^{2}+{\tfrac {1}{2}}\rho gh^{2}+S_{xx}\right)=\rho gh{\frac {\partial d}{\partial x}}} 6490: 5996: 7951: 6093: 2471: 4102: 10230:

The system of equations is closed through the description of the waves. Wave energy propagation is described through the wave-action conservation equation (without dissipation and nonlinear wave interactions):

4820: 3779: 2621: 8658: 8194:), but the total energy density – the sum of the energy density per unit area of the wave motion and the mean flow motion – is. However, there is for slowly varying wave trains, propagating in slowly varying 627: 5106:

in case of a water–air interface. For very short wavelengths – 2 mm or less, in case of the interface between air and water – gravity effects are negligible. Note that surface tension can be altered by

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are needed at the bed and the free surface in order to close the system of equations. For their formulation within the framework of linear theory, it is necessary to specify what the base state (or

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was a known function, this would be enough to solve the flow problem. However, the surface elevation is an extra unknown, for which an additional boundary condition is needed. This is provided by

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which is an exact result for periodic progressive water waves, also valid for nonlinear waves. However, its validity strongly depends on the way how wave momentum and mass flux are defined.

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In general, there can be an energy transfer between the wave motion and the mean fluid motion. This means, that the wave energy density is not in all cases a conserved quantity (neglecting

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oscillation underneath the free surface, with wave-induced pressure oscillations reducing with depth below the free surface – in the same way as for the orbital motion of fluid parcels.

4427: 2780: 1603: 5161:. This is opposite to the situation of surface gravity waves (with surface tension negligible compared to the effects of gravity) where the phase velocity exceeds the group velocity. 2912: 3924: 2866: 2385: 1972:

The table only gives the oscillatory parts of flow quantities – velocities, particle excursions and pressure – and not their mean value or drift. The oscillatory particle excursions

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The associated velocity potential, satisfying the Laplace equation (1) in the fluid interior, as well as the kinematic boundary conditions at the free surface (2), and bed (3), is:

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homogeneous fluids of different density, confined between horizontal rigid boundaries (at the top and bottom). The motion is forced by gravity. The upper layer has mean depth

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In the limiting cases of deep and shallow water, simplifying approximations to the solution can be made. While for intermediate depth, the full formulations have to be used.

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Orbital motion under linear waves. The yellow dots indicate the momentary position of fluid particles on their (orange) orbits. The black dots are the centres of the orbits.

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density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area

2817: 6358: 2609: 2459: 10470:), according to linear Airy wave theory, a first approximation gives closed elliptical orbits for water particles. However, for nonlinear waves, particles exhibit a 5904: 107:

This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the

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The overbar denotes the mean value (which in the present case of periodic waves can be taken either as a time average or an average over one wavelength in space).

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For the equations, solution and resulting approximations in deep and shallow water, see Dingemans (1997), Part 1, §2.1, pp. 38–45. Or: Phillips (1977), pp. 36–45.

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waves, the vertical motion of the flow has to be equal to the vertical velocity of the free surface. This leads to the kinematic free-surface boundary-condition:

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Airy wave theory is a linear theory for the propagation of waves on the surface of a potential flow and above a horizontal bottom. The free surface elevation

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As can be seen, the mean kinetic and potential energy densities are equal. This is a general property of energy densities of progressive linear waves in a

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valid in shallow water. Drawn lines: based on dispersion relation valid in arbitrary depth. Dashed lines: based on dispersion relation valid in deep water.

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into account. This is due to the fact that for the oscillatory part of the fluid motion, wave-induced vorticity is restricted to some thin oscillatory

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Further on in this section, more detailed descriptions and results are given for the general case of propagation in two-dimensional horizontal space.

12784: 6616:{\displaystyle {\frac {\partial E}{\partial t}}+{\frac {\partial }{\partial x}}{\bigl (}(U+c_{g})E{\bigr )}+S_{xx}{\frac {\partial U}{\partial x}}=0} 4288: 6223: 2578:{\displaystyle \left(\omega -\mathbf {k} \cdot \mathbf {U} \right)^{2}={\bigl (}\Omega (k){\bigr )}^{2}\quad {\text{ with }}\quad k=|\mathbf {k} |} 5913: 12774: 11833: 7876: 6037: 5102:

As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a few

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properties of surface gravity waves, and their propagation, can be estimated from its results. Airy wave theory is also a good approximation for

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So in general, there are quite some subtleties involved. Therefore also the term pseudo-momentum of the waves is used instead of wave momentum.

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is the source term describing the energy exchange of the wave motion with the mean flow. Only in the case that the mean shear-rate is zero,

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In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving

2707:{\displaystyle \quad \sigma ^{2}={\bigl (}\Omega (k){\bigr )}^{2}\quad {\text{ with }}\quad \sigma =\omega -\mathbf {k} \cdot \mathbf {U} } 4031: 4746: 3731: 6735:{\displaystyle {\frac {\partial {\mathcal {A}}}{\partial t}}+{\frac {\partial }{\partial x}}{\bigl (}(U+c_{g}){\mathcal {A}}{\bigr )}=0} 296:

was the first to publish the correct derivation and formulation in 1841. Soon after, in 1847, the linear theory of Airy was extended by

6850:{\displaystyle {\frac {\partial k}{\partial t}}+{\frac {\partial \omega }{\partial x}}=0\quad {\text{with}}\quad \omega =\Omega (k)+kU} 576: 676: 5041: 4528: 4228: 5031:

the surface tension in newtons per metre. All above equations for linear waves remain the same, if the gravitational acceleration

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Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to Airy wave theory

7783: 6128: 4113: 13057: 12105: 11873: 10670: 8411:. Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density: 360: 11805: 11725: 11694: 11664: 11629: 11598: 11571: 11440: 11415: 11116: 10780: 9720:{\displaystyle {\frac {\partial }{\partial t}}\left(\rho h\right)+\nabla \cdot \left(\rho h{\tilde {\mathbf {U} }}\right)=0,} 2396: 982:{\displaystyle u_{x}={\frac {\partial \Phi }{\partial x}}\quad {\text{and}}\quad u_{z}={\frac {\partial \Phi }{\partial z}}.} 751:

Particle motion in an ocean wave at deep (A) and shallow (B) depths. 1) Propagation direction. 2) Wave crest. 3) Wave trough.

10627:{\displaystyle {\bar {\mathbf {u} }}_{S}={\tfrac {1}{2}}\sigma ka^{2}{\frac {\cosh 2k(z+h)}{\sinh ^{2}kh}}\mathbf {e} _{k},} 4169: 12337: 1927:(so the mean velocity as observed from this reference frame is zero), the angular frequency is different. It is called the 1748:

also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitude

1370: 9535:, wave and mean-flow fields, the evolution of the mean flow can de described in terms of the mean mass-transport velocity 5332:{\displaystyle \Omega ^{2}(k)=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}g+{\frac {\gamma }{\rho +\rho '}}k^{2}\right),} 3980: 1484:

Because this is a linear theory, in both free-surface boundary conditions – the kinematic and the dynamic one, equations (

11826: 9472: 8534:{\displaystyle {\frac {\partial E}{\partial t}}+\nabla \cdot \left+{\boldsymbol {S}}:\left(\nabla \mathbf {U} \right)=0,} 7217: 1331:{\displaystyle {\frac {\partial \eta }{\partial t}}={\frac {\partial \Phi }{\partial z}}\quad {\text{ at }}z=\eta (x,t).} 12227: 11801: 6957:{\displaystyle {\frac {\partial }{\partial t}}(\rho h)+{\frac {\partial }{\partial x}}\left(\rho h{\tilde {U}}\right)=0} 5524:

above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship

3616: 12932: 12359: 12247: 11037: 3106: 12779: 12050: 11061: 8199: 6002: 2918: 2156:, where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory. 12237: 12197: 11755: 7229:

waves, the most used energy measure is the mean wave energy density per unit horizontal area. It is the sum of the

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so it varies as a function of elevation. The given formula is for Stokes first definition of wave celerity. When

5812: 3933: 12967: 11953: 5383: 4660: 5734:, can be derived directly from Airy wave theory. They are of importance in many practical applications, such as 4382: 13047: 12640: 11819: 10693: 7246:

of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:

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of the waves is hardly influenced by depth (this is the case for most wind waves on the sea and ocean surface),

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Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation

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waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed.

3879: 2829: 2353: 1837:– cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is an 1762: 823: 309: 11484: 3418: 1096:{\displaystyle {\frac {\partial ^{2}\Phi }{\partial x^{2}}}+{\frac {\partial ^{2}\Phi }{\partial z^{2}}}=0.} 12095: 4436: 2260: 5742:, second-order wave properties also find their applications in describing waves in case of slowly varying 8644: 4710: 4610: 782: 12155: 4836:

Dispersion of gravity–capillary waves on the surface of deep water. Phase and group velocity divided by

993: 313: 257:– is predicted well over not too long distances (in terms of wavelengths) and in not too shallow water. 12292: 10775:. Advanced Series on Ocean Engineering. Vol. 15. Singapore: World Scientific Publishing Company. 10342:{\displaystyle {\frac {\partial }{\partial t}}\left({\frac {E}{\sigma }}\right)+\nabla \cdot \left=0.} 2723: 1126:) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero. 12827: 12232: 12192: 11228: 8164:{\displaystyle E=E_{\text{pot}}+E_{\text{kin}}={\tfrac {1}{2}}\left(\rho g+\gamma k^{2}\right)a^{2},} 3463: 6318: 6009: 276:

Earlier attempts to describe surface gravity waves using potential flow were made by, among others,

12957: 12332: 12322: 12262: 11898: 11868: 10431: 8052:{\displaystyle E_{\text{pot}}=E_{\text{kin}}={\tfrac {1}{4}}\left(\rho g+\gamma k^{2}\right)a^{2},} 2786: 1753: 1537: 649: 135: 6190: 5686:

is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth

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for which a second-order expression can be derived from the results of Airy wave theory (see the

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with the free surface wave. The dispersion relation for this case is of a more complicated form.

11795: 7216:, including the splash-zone effects of the waves on horizontal mass transport, and not the mean 6469:{\displaystyle {\bar {u}}_{S}={\tfrac {1}{2}}\sigma ka^{2}{\frac {\cosh 2k(z+h)}{\sinh ^{2}kh}}} 2793: 281: 12989: 12927: 12354: 12040: 9517: 1222:

water depth, from a mathematical point of view – the flow velocities have to go to zero in the

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is one of the wave effects which can be described with Airy wave theory. Further, by using the

112: 12822: 12804: 12312: 12207: 11842: 11495: 9454: 5170: 2594: 2444: 2102:, the phase speed of the waves is only dependent on water depth, and no longer a function of 1714:{\displaystyle \Phi ={\frac {\omega }{k}}a{\frac {\cosh k(z+h)}{\sinh kh}}\sin(kx-\omega t),} 1461:{\displaystyle {\frac {\partial \Phi }{\partial t}}+g\eta =0\quad {\text{ at }}z=\eta (x,t).} 844:– a good approximation of the flow in the fluid interior for waves on a liquid surface – and 297: 285: 238: 231: 71: 5879: 13090: 13009: 12842: 12545: 12402: 12267: 11978: 11713: 11554: 11542: 11349: 11240: 10834: 10754: 10224: 5676: 277: 90: 1853:

can be chosen freely (but small enough for Airy wave theory to be a valid approximation).

8: 13004: 12889: 12884: 12610: 12282: 12242: 11958: 11476: 11337: 10698: 10435: 8404: 7845: 7185: 5723: 5650:{\displaystyle \Omega ^{2}(k)={\frac {gk(\rho -\rho ')}{\rho \coth kh+\rho '\coth kh'}},} 3267:{\displaystyle {\tfrac {1}{2}}{\sqrt {\frac {g}{k}}}={\tfrac {1}{2}}{\frac {g}{\sigma }}} 1223: 1123: 837: 293: 242: 94: 75: 64: 11747: 11546: 11353: 11244: 10838: 10407:{\displaystyle {\frac {\partial \mathbf {k} }{\partial t}}+\nabla \omega =\mathbf {0} ,} 12947: 12660: 12650: 12615: 12515: 12500: 12397: 11751: 11512: 11365: 11256: 10850: 10704: 10439: 6104: 5785: 5739: 5727: 5017:{\displaystyle \Omega ^{2}(k)=\left(g+{\frac {\gamma }{\rho }}k^{2}\right)k\,\tanh kh,} 3493: 3402:{\displaystyle {\tfrac {1}{2}}c_{p}\left(1+kh{\frac {1-\tanh ^{2}kh}{\tanh kh}}\right)} 1966: 1918: 1818: 1375: 1119: 849: 460: 262: 211: 5371:

For interfacial waves to exist, the lower layer has to be heavier than the upper one,

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layer. The theory assumes that the fluid layer has a uniform mean depth, and that the

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The error in the phase speed is less than 2% if wavelength effects are neglected for

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Visualization of deep and shallow water waves by relating wavelength to depth to bed.

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is the still water depth, that is, the bed underneath the fluid layer is located at

5394: 3863:{\displaystyle \mathbf {e} _{k}\sigma a{\frac {\cosh k(z+h)}{\sinh kh}}\cos \theta } 12984: 12952: 12922: 12731: 12716: 12585: 12520: 12412: 12327: 12257: 12182: 11963: 11933: 11863: 11743: 11593:. Advanced Series on Ocean Engineering. Vol. 13. Singapore: World Scientific. 11550: 11410:. Elsevier Oceanography Series. Vol. 20. Elsevier. pp. 85 & 110–111. 11357: 11248: 11190: 10842: 10750: 8576: 8329:{\displaystyle {\frac {\partial {\mathcal {A}}}{\partial t}}+\nabla \cdot \left=0,} 7234: 5864: 1001: 845: 747: 667: 512: 97: 36: 11566:. Advanced Series on Ocean Engineering. Vol. 2. Singapore: World Scientific. 11076: 7172:

The last four equations describe the evolution of slowly varying wave trains over

3600:{\displaystyle {\tfrac {1}{2}}\left(1+kh{\frac {1-\tanh ^{2}kh}{\tanh kh}}\right)} 12789: 12685: 12635: 12600: 12560: 12452: 12422: 12272: 12222: 12132: 12090: 12023: 11948: 11908: 11682: 11652: 11480: 10822: 10684: 9968: 9480: 8400: 8396: 8179: 6747: 5710:

mode where the opposite is the case – the interfacial wave is higher than and in

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The wave kinematics are described through the wave-crest conservation equation:

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of capillary waves – dominated by surface tension effects – is greater than the

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Also: "Trigonometry, On the Figure of the Earth, Tides and Waves", 396 pp.

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Originally published in 1879, the 6th extended edition appeared first in 1932.

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Second-order quantities and their dynamics, using results of Airy wave theory

4366:{\displaystyle -\mathbf {e} _{k}a{\frac {\cosh k(z+h)}{\sinh kh}}\sin \theta } 1195:{\displaystyle {\frac {\partial \Phi }{\partial z}}=0\quad {\text{ at }}z=-h.} 13084: 13069: 12917: 12837: 12726: 12645: 12620: 12555: 12485: 12392: 12287: 12164: 12085: 12045: 12018: 11928: 11878: 11771: 11194: 11174: 8408: 7209:. Note that the mean-flow velocity in the mass and momentum equations is the 6300:{\displaystyle {\tilde {U}}=U+{\frac {M}{\rho h}}=U+{\frac {E}{\rho hc_{p}}}} 4652: 1902: 1543: 1237: 868: 266: 32: 11674: 11639: 11608: 11581: 10790: 5991:{\displaystyle S_{xx}=\left(2{\frac {c_{g}}{c_{p}}}-{\tfrac {1}{2}}\right)E} 648:

are not independent), but are coupled. Surface gravity waves on a fluid are

13024: 12972: 12912: 12863: 12741: 12736: 12711: 12695: 12670: 12387: 12277: 12217: 12003: 11913: 11888: 11735: 11704: 11252: 10689: 10471: 10455: 10443: 9507: 7946:{\displaystyle E=E_{\text{pot}}+E_{\text{kin}}={\tfrac {1}{2}}\rho ga^{2}.} 6311: 6088:{\displaystyle {\mathcal {A}}={\frac {E}{\sigma }}={\frac {E}{\omega -kU}}} 1838: 321: 60: 40: 11126: 7777:. Using the dispersion relation, the result for surface gravity waves is: 13014: 12746: 12675: 12540: 12480: 12447: 12437: 12432: 12317: 12252: 12212: 12202: 12177: 12060: 12033: 12013: 11973: 11938: 11617: 11006: 10710: 9506:

However, for waves perpendicular to a coast line or in closed laboratory

8191: 5735: 5468: 3150: 2973: 2068: 656: 561: 305: 258: 108: 11811: 11775: 11231:; Garrett, C. J. R. (1968). "Wavetrains in inhomogeneous moving media". 12832: 12680: 12655: 12550: 12530: 12457: 12442: 12427: 12417: 12382: 12302: 12122: 12117: 12080: 12075: 12070: 11968: 11648: 10825:(1986). "Fundamentals concerning wave loading on offshore structures". 10716: 10422: 9532: 8592: 8195: 7173: 5743: 5707: 5699: 5108: 4832: 2039: 1878: 1130: 468: 343: 301: 270: 223: 131: 86: 48: 13064: 10219:

is only conserved if the sea bed is horizontal (the still-water depth

1897:, in the horizontal direction and uniform over (independent of) depth 739: 123: 12904: 12766: 12751: 12665: 12510: 12349: 12344: 12127: 12055: 11983: 11903: 11893: 11850: 9254: 8185: 7177: 5711: 5703: 5399: 5103: 519: 440: 219: 215: 82: 11178: 5706:

is large compared with the amplitude of the interfacial wave, and a

4097:{\displaystyle \sigma a{\frac {\sinh k(z+h)}{\sinh kh}}\sin \theta } 12999: 12721: 12580: 12472: 12462: 12407: 11883: 10652: 10216: 9244: 5868: 4815:{\displaystyle \rho ga{\frac {\cosh k(z+h)}{\cosh kh}}\cos \theta } 4602: 3774:{\displaystyle \mathbf {e} _{k}{\sqrt {\frac {g}{h}}}a\cos \theta } 1995: 1219: 765: 254: 52: 10489:, which is the particle drift after one wave cycle divided by the 6099:

mean mass-flux due to the wave motion or the wave pseudo-momentum

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of the wave-crest conservation, it can be seen that an initially

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and mean-flow fields, a similar and conserved wave quantity, the

5174: 757: 622:{\displaystyle c_{p}={\frac {\omega }{k}}={\frac {\lambda }{T}}.} 101: 10701:– refraction and diffraction of surface waves over varying depth 1531: 12575: 11988: 9958: 9040: 8832: 8579: 2822: 2614: 2464: 2389: 811:

the vertical coordinate (positive in the upward direction) and

726:{\displaystyle H=2a\quad {\text{and}}\quad a={\tfrac {1}{2}}H,} 464: 450: 246: 79: 10692:– nonlinear periodic waves in shallow water, solutions of the 5169:

Surface waves are a special case of interfacial waves, on the

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velocity (for example, as measured with a fixed flow meter).

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In this case the dispersion relation allows for two modes: a

5693: 5389: 5092:{\displaystyle {\tilde {g}}=g+{\frac {\gamma }{\rho }}k^{2}.} 4591:{\displaystyle a{\frac {\sinh k(z+h)}{\sinh kh}}\cos \theta } 4277:{\displaystyle -\mathbf {e} _{k}{\frac {1}{kh}}a\sin \theta } 1905:

in the dispersion relations. At an Earth-fixed location, the

44: 11533:

Craik, A. D. D. (2004). "The origins of water wave theory".

11173: 3720:{\displaystyle \mathbf {e} _{k}\sigma a\,e^{kz}\cos \theta } 3197:{\displaystyle c_{g}={\frac {\partial \Omega }{\partial k}}} 253:, the evolution of the wave statistics – including the wave 115:), and wave height to wavelength (for waves in deep water). 93:

of high-enough accuracy for many purposes. Further, several

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is the mean water depth, slowly varying in space and time.

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is valid within the framework of Stokes' first definition.

8360: 5871: 3067:{\displaystyle {\sqrt {\frac {g}{k}}}={\frac {g}{\sigma }}} 10475: 7834:{\displaystyle E_{\text{kin}}={\tfrac {1}{4}}\rho ga^{2}.} 6174:{\displaystyle M={\frac {E}{c_{p}}}=k{\frac {E}{\sigma }}} 4155:{\displaystyle {\boldsymbol {\xi }}_{x}(\mathbf {x} ,z,t)} 7462:

The mean kinetic energy density per unit horizontal area

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are not independent parameters (and thus also wavelength

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in the wave steepness. Even before Airy's linear theory,

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Fluid dynamics theory on the propagation of gravity waves

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The error in the phase speed is less than 0.2% if depth

10741:

Craik, Alex (2004). "The Origins of Water Wave Theory".

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For two homogeneous layers of fluids, of mean thickness

104:

waves in the ocean, before they steepen near the coast.

10493:, can be estimated using the results of linear theory: 9253:

induced by the wave motion – and also the wave-induced

5671:

are the densities below and above the interface, while

2425:{\displaystyle \mathbf {k} \cdot \mathbf {x} -\omega t} 2077:– for a water depth smaller than 5% of the wavelength, 1969:, and have no Doppler shift (for monochromatic waves). 1891:. Secondly, allowance is made for a mean flow velocity 63:. This theory was first published, in correct form, by 10655:

over depth, the expression for the mean wave momentum

10528: 10460:

When following a single particle in pure wave motion (

10028: 9858: 8982: 8678: 8105: 7993: 7913: 7801: 7694: 7640: 7524: 7412: 7375: 7077: 6385: 5969: 5823: 4217:{\displaystyle -\mathbf {e} _{k}a\,e^{kz}\sin \theta } 3519: 3468: 3311: 3243: 3216: 788:, and a fluid domain bound above by a free surface at 781:

The waves propagate in the horizontal direction, with

706: 10812:

Phillips (1977), §3.2, pp. 37–43 and §3.6, pp. 60–69.

10502: 10361: 10240: 9988: 9757: 9637: 9550: 9270: 9138: 9107: 9074: 9043: 8656: 8420: 8395:

vector. Action conservation forms the basis for many

8236: 8071: 7965: 7879: 7786: 7480: 7255: 6981: 6874: 6763: 6640: 6493: 6361: 6321: 6226: 6193: 6131: 6107: 6040: 6012: 5916: 5882: 5815: 5788: 5544: 5205: 5044: 4936: 4749: 4713: 4663: 4613: 4531: 4483: 4439: 4385: 4291: 4231: 4172: 4116: 4034: 3983: 3936: 3882: 3788: 3734: 3675: 3619: 3517: 3496: 3466: 3421: 3309: 3281: 3214: 3160: 3109: 3081: 3038: 2983: 2921: 2875: 2832: 2796: 2759: 2726: 2624: 2597: 2474: 2447: 2399: 2356: 2304: 2263: 1765: 1617: 1555: 1389: 1251: 1144: 1015: 902: 679: 579: 560:

The waves propagate along the water surface with the

363: 11489:. Mixed Sciences. Vol. 3 (published 1817–1845). 11233:

Proceedings of the Royal Society of London, Series A

9748:

Similarly, the mean horizontal momentum evolves as:

4020:{\displaystyle \sigma a{\frac {z+h}{h}}\sin \theta } 1849:

satisfy the dispersion relation, the wave amplitude

736:

valid in the present case of linear periodic waves.

11500:

Transactions of the Cambridge Philosophical Society

9238: 7866:, the mean energy density per unit horizontal area 7180:, and can be derived from a variational principle: 5398:Wave motion on the interface between two layers of 1226:as the vertical coordinate goes to minus infinity: 10626: 10434:. For this to be possible, the wave field must be 10406: 10341: 10204: 9926: 9719: 9598: 9489:(S2) – with the mean mass transport equal to zero. 9442: 9184: 8533: 8328: 8186:Wave action, wave energy flux and radiation stress 8163: 8051: 7945: 7833: 7758: 7448: 7188:method. In the mean horizontal-momentum equation, 7159: 6956: 6849: 6734: 6615: 6468: 6343: 6299: 6208: 6173: 6113: 6087: 6022: 5990: 5898: 5852: 5794: 5780:mean wave-energy density per unit horizontal area 5649: 5331: 5091: 5016: 4814: 4734: 4698: 4642: 4590: 4516: 4468: 4421: 4365: 4276: 4216: 4154: 4096: 4019: 3968: 3918: 3862: 3773: 3719: 3658:{\displaystyle \mathbf {u} _{x}(\mathbf {x} ,z,t)} 3657: 3599: 3502: 3481: 3448: 3401: 3294: 3266: 3196: 3138: 3094: 3066: 3020: 2961: 2906: 2860: 2811: 2774: 2741: 2706: 2603: 2577: 2453: 2424: 2379: 2336: 2286: 1802: 1713: 1597: 1460: 1330: 1194: 1095: 981: 725: 621: 421: 12785:North West Shelf Operational Oceanographic System 11564:Water wave mechanics for engineers and scientists 11383: 11381: 11379: 11227: 10719:– using ocean and sea waves for power generation. 1542:For a propagating wave of a single frequency – a 822:corresponds with the mean surface elevation. The 13082: 3139:{\displaystyle {\sqrt {{\frac {g}{k}}\tanh kh}}} 12775:Deep-ocean Assessment and Reporting of Tsunamis 11742: 11647: 10713:– nonlinear periodic waves in non-shallow water 10707:– real water waves as seen in the ocean and sea 9457:already identified two possible definitions of 8599:In this equation in non-conservation form, the 5717: 5180: 2962:{\displaystyle \Omega (k)={\sqrt {gk\tanh kh}}} 11498:(1847). "On the theory of oscillatory waves". 11454: 11452: 11376: 11214: 11212: 11183:Proceedings of the London Mathematical Society 11160: 11158: 11148: 11146: 11099: 11097: 11095: 11093: 11091: 10975: 10973: 9609:Note that for deep water, when the mean depth 9229:the components in of the mean velocity vector 7848:. Adding potential and kinetic contributions, 2718:unit vector in the wave propagation direction 2023:Water depth is classified into three regimes: 11827: 11561: 11303: 10773:Random Seas and Design of Maritime Structures 9613:goes to infinity, the mean Eulerian velocity 7774: 7471:of the wave motion is similarly found to be: 6721: 6685: 6566: 6534: 5382:. Otherwise, the interface is unstable and a 4851:as a function of inverse relative wavelength 4517:{\displaystyle a{\frac {z+h}{h}}\cos \theta } 3021:{\displaystyle c_{p}={\frac {\Omega (k)}{k}}} 2661: 2641: 2536: 2516: 2337:{\displaystyle a\cos \theta (\mathbf {x} ,t)} 1546:wave – the surface elevation is of the form: 1532:Solution for a progressive monochromatic wave 11720:(2nd ed.). Cambridge University Press. 11624:(6th ed.). Cambridge University Press. 11405: 10991: 10866: 10864: 5853:{\displaystyle E={\tfrac {1}{2}}\rho ga^{2}} 5353:are the densities of the two fluids, below ( 3969:{\displaystyle \sigma a\,e^{kz}\sin \theta } 11449: 11435:. Cambridge University Press. p. 105. 11285: 11209: 11155: 11143: 11088: 10970: 10736: 10734: 10732: 10476:table above on second-order wave properties 4699:{\displaystyle \rho ga\,e^{kz}\cos \theta } 1856: 776: 772:Mathematical formulation of the wave motion 11834: 11820: 11591:Water wave propagation over uneven bottoms 11514:Mathematical and Physical Papers, Volume I 11032:, Cambridge University Press, p. 18, 10882: 10880: 10878: 10876: 5694:Two layers bounded above by a free surface 5390:Two layers between horizontal rigid planes 4827: 4422:{\displaystyle \xi _{z}(\mathbf {x} ,z,t)} 1887:, and is perpendicular to the cams of the 1218:In case of deep water – by which is meant 996:for an incompressible flow, the potential 11841: 11681: 11588: 10861: 10821: 9941:the still-water depth (the sea bed is at 9493:The above relation between wave momentum 9487:Stokes second definition of wave celerity 9385: 9341: 7674: 7603: 7352: 7307: 5368:is the surface tension on the interface. 4998: 4673: 4443: 4191: 3943: 3694: 2775:{\displaystyle {\frac {\mathbf {k} }{k}}} 2038:– for a water depth larger than half the 1598:{\displaystyle \eta =a\cos(kx-\omega t).} 11712: 11336: 10729: 9209:the components of the wavenumber vector 6185:mean horizontal mass-transport velocity 5393: 4831: 2907:{\displaystyle \Omega (k)=k{\sqrt {gh}}} 2025: 1129:The bed being impermeable, leads to the 746: 738: 130: 122: 11517:. Cambridge University Press. pp. 11399: 11340:(1978). "On the 'wave-momentum' myth". 11206:, MacMillan, 2nd revised edition, 1894. 11103: 11018:Dingemans (1997), Section 2.1.1, p. 45. 10873: 9894: 9886: 9628:The equation for mass conservation is: 9516:in the opposite direction – called the 8966: 8753: 8662: 8497: 5416:, while the lower layer has mean depth 5364:) the interface, respectively. Further 4119: 3919:{\displaystyle u_{z}(\mathbf {x} ,z,t)} 2861:{\displaystyle \Omega (k)={\sqrt {gk}}} 2380:{\displaystyle \theta (\mathbf {x} ,t)} 1803:{\displaystyle \omega ^{2}=gk\tanh kh,} 346:, as a function of horizontal position 234:at the boundaries of the fluid domain. 13083: 12106:one-dimensional Saint-Venant equations 11562:Dean, R. G.; Dalrymple, R. A. (1991). 11555:10.1146/annurev.fluid.36.050802.122118 11510: 11494: 11027: 10766: 10764: 10755:10.1146/annurev.fluid.36.050802.122118 10671:Boussinesq approximation (water waves) 7223: 5750:Table of second-order wave properties 4927:, the dispersion relation changes to: 3449:{\displaystyle {\frac {c_{g}}{c_{p}}}} 848:can be used to describe the flow. The 764:In a similar fashion, there is also a 662:– the difference in elevation between 511:in radians per second, related to the 11815: 11532: 11430: 10895:Dean & Dalrymple (1991) pp. 64–65 10740: 10446:wavenumber field stays irrotational. 9527:Mass and momentum evolution equations 8567:is the mean wave energy density flux, 5463:), the gravitational acceleration by 4905:Dash-dot lines: pure capillary waves. 4901:Drawn lines: gravity–capillary waves. 4469:{\displaystyle a\,e^{kz}\cos \theta } 2287:{\displaystyle \eta (\mathbf {x} ,t)} 836:. Further, the flow is assumed to be 826:bed underneath the fluid layer is at 320:theory in 1802, which however is not 138:of gravity waves on a fluid surface. 70:Airy wave theory is often applied in 13053: 11796:Linear theory of ocean surface waves 11616: 11475: 11309:Phillips (1977), pp. 39–40 & 61. 11056:, Academic Press, pp. 231–239, 11051: 10770: 10683:– surface waves under the action of 5164: 1380: 1242: 1135: 1006: 11406:LeBlond, P.H.; Mysak, L.A. (1981). 10761: 6968:mean horizontal-momentum evolution 4735:{\displaystyle \rho ga\cos \theta } 4643:{\displaystyle p(\mathbf {x} ,z,t)} 1998:of the oscillatory flow velocities 85:– giving a description of the wave 13: 12933:National Oceanographic Data Center 12360:World Ocean Circulation Experiment 12248:Global Ocean Data Analysis Project 11526: 10387: 10375: 10365: 10277: 10247: 10243: 10223:is a constant), in agreement with 9915: 9806: 9764: 9760: 9672: 9644: 9640: 9387: 9343: 9157: 9142: 9126: 9111: 9093: 9078: 9062: 9047: 9020: 8509: 8444: 8432: 8424: 8307: 8264: 8252: 8245: 8240: 7676: 7605: 7354: 7309: 7148: 7140: 7034: 7030: 6988: 6984: 6911: 6907: 6881: 6877: 6826: 6798: 6790: 6775: 6767: 6714: 6674: 6670: 6656: 6649: 6644: 6598: 6590: 6523: 6519: 6505: 6497: 6043: 6015: 5546: 5535:for gravity waves is provided by: 5207: 4938: 3185: 3180: 3177: 3000: 2922: 2876: 2833: 2797: 2646: 2521: 1618: 1401: 1396: 1393: 1286: 1281: 1278: 1263: 1255: 1156: 1151: 1148: 1071: 1066: 1057: 1034: 1029: 1020: 967: 962: 959: 927: 922: 919: 237:Airy wave theory is often used in 14: 13107: 12780:Global Sea Level Observing System 11789: 11766:, Springer Verlag: 653–667, §27, 11433:Wave interactions and fluid flows 9475:equal to zero for all elevations 7773:the intrinsic frequency, see the 4913:is surface tension in this graph. 1378:free-surface boundary condition: 13063: 13052: 13043: 13042: 12238:Geochemical Ocean Sections Study 12154: 12143: 11535:Annual Review of Fluid Mechanics 10743:Annual Review of Fluid Mechanics 10673:– nonlinear theory for waves in 10611: 10508: 10397: 10369: 10303: 10294: 10010: 9993: 9844: 9827: 9788: 9693: 9579: 9569: 9555: 9466:Stokes first definition of wave 9423: 9381: 9326: 9317: 9276: 9239:Wave mass flux and wave momentum 9024: 8513: 8470: 8461: 8403:models. It is also the basis of 8290: 8281: 7660: 7559: 7550: 5173:between two fluids of different 4892:, Red lines (B): group velocity 4621: 4400: 4297: 4237: 4178: 4133: 3897: 3791: 3737: 3678: 3636: 3622: 2763: 2742:{\displaystyle \mathbf {e} _{k}} 2729: 2700: 2692: 2566: 2496: 2488: 2409: 2401: 2364: 2321: 2271: 43:on the surface of a homogeneous 12968:Ocean thermal energy conversion 12691:Vine–Matthews–Morley hypothesis 11718:The dynamics of the upper ocean 11424: 11390: 11330: 11321: 11312: 11294: 11276: 11267: 11221: 11167: 11134: 11069: 11045: 11021: 11012: 11000: 10982: 10936: 10449: 8626:, the mean wave energy density 6819: 6813: 5726:wave properties, ones that are 5428:, the wavelength is denoted by 4885:Blue lines (A): phase velocity 3482:{\displaystyle {\tfrac {1}{2}}} 2678: 2672: 2625: 2553: 2547: 1939:. So in pure wave motion, with 1425: 1295: 1171: 942: 936: 698: 692: 11689:. Cambridge University Press. 11479:(1841). "Tides and waves". In 10898: 10889: 10815: 10806: 10797: 10582: 10570: 10512: 10179: 10160: 10139: 10120: 10097: 10078: 10057: 10038: 10014: 9997: 9848: 9831: 9792: 9697: 9559: 9461:for periodic nonlinear waves: 8630:is conserved. The two tensors 8407:models for the computation of 7061: 7014: 6937: 6899: 6890: 6835: 6829: 6709: 6690: 6558: 6539: 6439: 6427: 6369: 6344:{\displaystyle {\bar {u}}_{S}} 6329: 6233: 6200: 6023:{\displaystyle {\mathcal {A}}} 5593: 5576: 5561: 5555: 5237: 5229: 5222: 5216: 5051: 4953: 4947: 4783: 4771: 4637: 4617: 4559: 4547: 4416: 4396: 4334: 4322: 4149: 4129: 4108:horizontal particle excursion 4065: 4053: 3913: 3893: 3831: 3819: 3652: 3632: 3009: 3003: 2931: 2925: 2885: 2879: 2842: 2836: 2806: 2800: 2655: 2649: 2571: 2561: 2530: 2524: 2374: 2360: 2331: 2317: 2281: 2267: 1921:moving with the mean velocity 1705: 1687: 1661: 1649: 1589: 1571: 1452: 1440: 1354:If the free surface elevation 1322: 1310: 379: 367: 118: 1: 11746:& Laitone, E. V. (1960), 11469: 11464: 11273:Phillips (1977), pp. 179–183. 10908:is taken to be infinite, for 10478:). The Stokes drift velocity 9957:is the wave radiation-stress 9619:and mean transport velocity 655:Note that in engineering the 111:to water depth (for waves in 12228:El Niño–Southern Oscillation 12198:Craik–Leibovich vortex force 11954:Luke's variational principle 11175:Lord Rayleigh (J. W. Strutt) 11085:'s National Weather service. 10803:Dean & Dalrymple (1991). 10421:a function of the (angular) 9351: 7613: 7403: 7317: 6209:{\displaystyle {\tilde {U}}} 5738:of wave conditions. Using a 5718:Second-order wave properties 5702:mode where the free surface 5181:Two layers of infinite depth 4903:Dashed lines: gravity waves. 4377:vertical particle excursion 3295:{\displaystyle {\sqrt {gh}}} 3095:{\displaystyle {\sqrt {gh}}} 2589:intrinsic angular frequency 1740:function, respectively. But 761:depth of half a wavelength. 7: 11387:Phillips (1977), pp. 61–63. 11282:Phillips (1977), pp. 70–74. 11054:Principles of ocean physics 11009:(1994), §267, page 458–460. 10664: 10417:with the angular frequency 8645:Cartesian coordinate system 5432:(related to the wavenumber 5384:Rayleigh–Taylor instability 1957:are equal. The wave number 1929:intrinsic angular frequency 1492: 1486: 1474: 1344: 1208: 1109: 191:: phase and group velocity 10: 13112: 12293:Ocean dynamical thermostat 12141: 11486:Encyclopædia Metropolitana 11342:Journal of Fluid Mechanics 11199:Reprinted as Appendix in: 11140:Phillips (1977), p. 23–25. 11109:Linear and nonlinear waves 11030:Buoyancy effects in fluids 10694:Korteweg–de Vries equation 10453: 10215:Note that mean horizontal 4916: 2812:{\displaystyle \Omega (k)} 1933:relative angular frequency 1917:. On the other hand, in a 1911:absolute angular frequency 1907:observed angular frequency 1535: 467:per metre, related to the 13038: 12877: 12851: 12828:Ocean acoustic tomography 12813: 12765: 12704: 12641:Mohorovičić discontinuity 12599: 12471: 12368: 12233:General circulation model 12163: 11869:Benjamin–Feir instability 11849: 11589:Dingemans, M. W. (1997). 11362:10.1017/S0022112081001626 10988:Lighthill (1978), p. 223. 10847:10.1017/S0022112086001313 6627:wave action conservation 5760: 2753: 2618: 2468: 2393: 2298: 2165: 1965:) are independent of the 1829:– or equivalently period 1236:At the free surface, for 342:of one wave component is 12958:Ocean surface topography 12333:Thermohaline circulation 12323:Subsurface ocean current 12263:Hydrothermal circulation 12096:Wave–current interaction 11874:Boussinesq approximation 11760:Encyclopaedia of Physics 10997:Phillips (1977), p. 175. 10723: 9499:and wave energy density 7775:table of wave quantities 7176:in interaction with the 6480:wave-energy propagation 5517:below the interface and 5424:. The wave amplitude is 1857:Table of wave quantities 1538:Dispersion (water waves) 1521:at the fixed mean level 1133:bed boundary-condition: 777:Flow problem formulation 249:waves, sometimes called 206:Airy wave theory uses a 12995:Sea surface temperature 12978:Outline of oceanography 12173:Atmospheric circulation 12111:shallow water equations 12101:Waves and shallow water 11994:Significant wave height 11458:Phillips (1977), p. 44. 11396:Phillips (1977), p. 23. 11327:Phillips (1977), p. 70. 11318:Phillips (1977), p. 40. 11300:Phillips (1977), p. 68. 11291:Phillips (1977), p. 66. 11218:Phillips (1977), p. 26. 11164:Phillips (1977), p. 38. 11152:Phillips (1977), p. 39. 10979:Phillips (1977), p. 37. 8601:Frobenius inner product 7870:of the wave motion is: 7211:mass transport velocity 6861:mean mass conservation 5874:due to the wave motion 5679:function. For the case 4828:Surface tension effects 2604:{\displaystyle \sigma } 2454:{\displaystyle \omega } 1821:. So angular frequency 632:The angular wavenumber 304:wave motion – known as 12990:Sea surface microlayer 12355:Wind generated current 11511:Stokes, G. G. (1880). 11483:; et al. (eds.). 11431:Craik, A.D.D. (1988). 11253:10.1098/rspa.1968.0034 11195:10.1112/plms/s1-9.1.21 11179:"On progressive waves" 11111:. Wiley-Interscience. 11028:Turner, J. S. (1979), 10628: 10430:, related through the 10408: 10343: 10206: 9928: 9721: 9600: 9473:Eulerian flow velocity 9444: 9186: 8535: 8330: 8165: 8053: 7947: 7835: 7760: 7450: 7161: 6958: 6851: 6736: 6617: 6470: 6345: 6301: 6210: 6175: 6115: 6089: 6024: 5992: 5900: 5899:{\displaystyle S_{xx}} 5854: 5796: 5730:in the wave amplitude 5651: 5510: 5333: 5093: 5018: 4914: 4816: 4736: 4700: 4644: 4592: 4518: 4470: 4423: 4367: 4278: 4218: 4156: 4098: 4021: 3970: 3920: 3864: 3775: 3721: 3659: 3601: 3504: 3483: 3450: 3403: 3296: 3268: 3198: 3140: 3096: 3068: 3022: 2963: 2908: 2862: 2813: 2776: 2743: 2708: 2605: 2579: 2455: 2426: 2381: 2338: 2288: 2031: 1804: 1715: 1599: 1462: 1332: 1196: 1097: 983: 815:being time. The level 752: 744: 727: 623: 423: 232:Stokes boundary layers 203: 128: 27:(often referred to as 12823:Deep scattering layer 12805:World Geodetic System 12313:Princeton Ocean Model 12193:Coriolis–Stokes force 11843:Physical oceanography 11613:Two parts, 967 pages. 11075:See for example: the 10629: 10409: 10344: 10207: 9929: 9722: 9601: 9471:(S1) – with the mean 9445: 9187: 8591:is the mean-velocity 8536: 8331: 8166: 8054: 7948: 7836: 7761: 7451: 7162: 6959: 6852: 6737: 6618: 6471: 6346: 6302: 6211: 6176: 6116: 6090: 6025: 5993: 5901: 5867:or excess horizontal 5855: 5797: 5652: 5397: 5334: 5094: 5019: 4835: 4817: 4737: 4701: 4645: 4593: 4519: 4471: 4424: 4368: 4279: 4219: 4157: 4099: 4022: 3971: 3921: 3865: 3776: 3722: 3660: 3602: 3505: 3484: 3451: 3404: 3297: 3269: 3199: 3141: 3097: 3069: 3023: 2964: 2909: 2863: 2814: 2777: 2744: 2709: 2606: 2580: 2456: 2427: 2382: 2339: 2289: 2029: 1805: 1716: 1600: 1463: 1333: 1197: 1124:zeroth-order solution 1098: 984: 750: 742: 728: 624: 424: 134: 127:Wave characteristics. 126: 78:for the modelling of 67:in the 19th century. 12843:Underwater acoustics 12403:Perigean spring tide 12268:Langmuir circulation 11979:Rossby-gravity waves 11052:Apel, J. R. (1987), 10500: 10359: 10238: 9986: 9755: 9635: 9548: 9268: 9243:The mean horizontal 8654: 8418: 8234: 8069: 7963: 7877: 7784: 7478: 7253: 6979: 6872: 6761: 6638: 6491: 6359: 6319: 6224: 6191: 6129: 6105: 6038: 6010: 5914: 5880: 5813: 5786: 5677:hyperbolic cotangent 5542: 5203: 5042: 4934: 4747: 4711: 4661: 4611: 4529: 4481: 4437: 4383: 4289: 4229: 4170: 4114: 4032: 3981: 3934: 3880: 3786: 3732: 3673: 3617: 3611:horizontal velocity 3515: 3494: 3464: 3419: 3307: 3279: 3212: 3158: 3107: 3079: 3036: 2981: 2919: 2873: 2830: 2794: 2757: 2724: 2622: 2595: 2472: 2445: 2397: 2354: 2302: 2261: 1901:. This introduces a 1763: 1615: 1553: 1387: 1371:Bernoulli's equation 1249: 1142: 1013: 900: 677: 577: 361: 316:derived a nonlinear 13005:Science On a Sphere 12611:Convergent boundary 12283:Modular Ocean Model 12243:Geostrophic current 11959:Mild-slope equation 11547:2004AnRFM..36....1C 11354:1981JFM...106..331M 11245:1968RSPSA.302..529B 11078:High seas forecasts 10839:1986JFM...173..667L 10699:Mild-slope equation 10432:dispersion relation 9531:For slowly varying 9376: 9307: 8405:coastal engineering 8192:dissipative effects 7846:conservative system 7638: 7597: 7522: 7342: 7297: 7224:Wave energy density 7186:averaged Lagrangian 2787:dispersion relation 2112:– all other cases, 1949:, both frequencies 1754:dispersion relation 1752:only if the linear 1120:Boundary conditions 1000:has to satisfy the 994:continuity equation 885:in the horizontal ( 306:Stokes' wave theory 243:coastal engineering 76:coastal engineering 65:George Biddell Airy 35:description of the 12661:Seafloor spreading 12651:Outer trench swell 12616:Divergent boundary 12516:Continental margin 12501:Carbonate platform 12398:Lunitidal interval 11408:Waves in the Ocean 10705:Ocean surface wave 10624: 10537: 10404: 10339: 10202: 10193: 9924: 9867: 9717: 9596: 9440: 9438: 9359: 9290: 9182: 9180: 9169: 9166: 9135: 9102: 9071: 9007: 8925: 8743: 8531: 8326: 8161: 8114: 8049: 8002: 7943: 7922: 7831: 7810: 7756: 7754: 7703: 7649: 7621: 7583: 7533: 7505: 7446: 7444: 7421: 7384: 7325: 7280: 7157: 7086: 6954: 6847: 6732: 6613: 6466: 6394: 6341: 6297: 6206: 6171: 6111: 6085: 6020: 5988: 5978: 5896: 5850: 5832: 5792: 5740:WKBJ approximation 5647: 5511: 5329: 5089: 5014: 4915: 4812: 4732: 4696: 4640: 4588: 4514: 4466: 4419: 4363: 4274: 4214: 4152: 4094: 4017: 3966: 3916: 3874:vertical velocity 3860: 3771: 3717: 3655: 3597: 3528: 3500: 3479: 3477: 3446: 3399: 3320: 3292: 3264: 3252: 3225: 3194: 3136: 3092: 3064: 3018: 2959: 2904: 2858: 2809: 2772: 2739: 2704: 2601: 2575: 2451: 2422: 2377: 2334: 2284: 2255:surface elevation 2240:intermediate depth 2110:intermediate depth 2106:or wavelength; and 2032: 1967:frame of reference 1919:frame of reference 1819:hyperbolic tangent 1800: 1711: 1595: 1458: 1328: 1192: 1093: 979: 867:is related to the 850:velocity potential 753: 745: 723: 715: 619: 461:angular wavenumber 419: 273:can be predicted. 263:WKBJ approximation 212:velocity potential 204: 187:: group velocity, 183:: phase velocity, 129: 29:linear wave theory 13078: 13077: 13070:Oceans portal 13030:World Ocean Atlas 13020:Underwater glider 12963:Ocean temperature 12626:Hydrothermal vent 12591:Submarine volcano 12526:Continental shelf 12506:Coastal geography 12496:Bathymetric chart 12378:Amphidromic point 12066:Wave nonlinearity 11924:Infragravity wave 11727:978-0-521-29801-8 11696:978-0-521-29233-7 11666:978-0-08-033932-0 11631:978-0-521-45868-9 11600:978-981-02-0427-3 11573:978-981-02-0420-4 11442:978-0-521-36829-2 11417:978-0-444-41926-2 11239:(1471): 529–554. 11229:Bretherton, F. P. 11118:978-0-471-94090-6 10782:978-981-02-3256-6 10771:Goda, Y. (2000). 10607: 10536: 10515: 10382: 10326: 10268: 10254: 10225:Noether's theorem 10182: 10163: 10142: 10123: 10100: 10081: 10060: 10041: 10017: 10000: 9866: 9851: 9834: 9795: 9771: 9700: 9651: 9591: 9562: 9479:' below the wave 9419: 9354: 9164: 9133: 9100: 9069: 8953: 8825: 8797: 8784: 8439: 8259: 8113: 8098: 8085: 8001: 7986: 7973: 7921: 7906: 7893: 7809: 7794: 7737: 7702: 7648: 7616: 7532: 7492: 7420: 7406: 7383: 7320: 7267: 7170: 7169: 7155: 7085: 7064: 7041: 7017: 6995: 6940: 6918: 6888: 6817: 6805: 6782: 6681: 6663: 6605: 6530: 6512: 6464: 6393: 6372: 6332: 6295: 6261: 6236: 6203: 6169: 6153: 6114:{\displaystyle M} 6083: 6059: 5977: 5963: 5831: 5795:{\displaystyle E} 5642: 5309: 5280: 5196:is given through 5165:Interfacial waves 5074: 5054: 4978: 4825: 4824: 4801: 4577: 4503: 4352: 4260: 4083: 4006: 3849: 3757: 3756: 3590: 3527: 3503:{\displaystyle 1} 3476: 3444: 3392: 3319: 3290: 3262: 3251: 3237: 3236: 3224: 3192: 3134: 3120: 3090: 3062: 3049: 3048: 3016: 2957: 2902: 2856: 2770: 2676: 2551: 2438:angular frequency 1738:hyperbolic cosine 1679: 1632: 1496:) – the value of 1482: 1481: 1429: 1408: 1352: 1351: 1299: 1293: 1270: 1216: 1215: 1175: 1163: 1117: 1116: 1085: 1048: 992:Then, due to the 974: 940: 934: 893:) directions by: 714: 696: 670:– is often used: 614: 601: 509:angular frequency 245:. Especially for 239:ocean engineering 157:as a function of 72:ocean engineering 13103: 13068: 13067: 13056: 13055: 13046: 13045: 12985:Pelagic sediment 12923:Marine pollution 12717:Deep ocean water 12586:Submarine canyon 12521:Continental rise 12413:Rule of twelfths 12328:Sverdrup balance 12258:Humboldt Current 12183:Boundary current 12158: 12147: 11964:Radiation stress 11934:Iribarren number 11909:Equatorial waves 11864:Ballantine scale 11859:Airy wave theory 11836: 11829: 11822: 11813: 11812: 11785: 11784: 11783: 11774:, archived from 11739: 11708: 11683:Lighthill, M. J. 11678: 11653:Lifschitz, E. M. 11643: 11612: 11585: 11558: 11522: 11507: 11490: 11459: 11456: 11447: 11446: 11428: 11422: 11421: 11403: 11397: 11394: 11388: 11385: 11374: 11373: 11334: 11328: 11325: 11319: 11316: 11310: 11307: 11301: 11298: 11292: 11289: 11283: 11280: 11274: 11271: 11265: 11264: 11225: 11219: 11216: 11207: 11198: 11171: 11165: 11162: 11153: 11150: 11141: 11138: 11132: 11130: 11101: 11086: 11073: 11067: 11066: 11049: 11043: 11042: 11025: 11019: 11016: 11010: 11004: 10998: 10995: 10989: 10986: 10980: 10977: 10968: 10966: 10962: 10960: 10959: 10956: 10953: 10940: 10934: 10932: 10928: 10926: 10925: 10922: 10919: 10907: 10902: 10896: 10893: 10887: 10884: 10871: 10868: 10859: 10858: 10823:Lighthill, M. J. 10819: 10813: 10810: 10804: 10801: 10795: 10794: 10768: 10759: 10758: 10738: 10660: 10650: 10633: 10631: 10630: 10625: 10620: 10619: 10614: 10608: 10606: 10596: 10595: 10585: 10556: 10554: 10553: 10538: 10529: 10523: 10522: 10517: 10516: 10511: 10506: 10488: 10469: 10438:. By taking the 10429: 10420: 10413: 10411: 10410: 10405: 10400: 10383: 10381: 10373: 10372: 10363: 10348: 10346: 10345: 10340: 10332: 10328: 10327: 10319: 10317: 10313: 10312: 10311: 10306: 10297: 10273: 10269: 10261: 10255: 10253: 10242: 10222: 10211: 10209: 10208: 10203: 10198: 10197: 10190: 10189: 10184: 10183: 10175: 10171: 10170: 10165: 10164: 10156: 10150: 10149: 10144: 10143: 10135: 10131: 10130: 10125: 10124: 10116: 10108: 10107: 10102: 10101: 10093: 10089: 10088: 10083: 10082: 10074: 10068: 10067: 10062: 10061: 10053: 10049: 10048: 10043: 10042: 10034: 10019: 10018: 10013: 10008: 10002: 10001: 9996: 9991: 9974: 9966: 9956: 9950: 9940: 9933: 9931: 9930: 9925: 9902: 9898: 9897: 9889: 9884: 9883: 9868: 9859: 9853: 9852: 9847: 9842: 9836: 9835: 9830: 9825: 9802: 9798: 9797: 9796: 9791: 9786: 9772: 9770: 9759: 9744: 9726: 9724: 9723: 9718: 9707: 9703: 9702: 9701: 9696: 9691: 9668: 9664: 9652: 9650: 9639: 9624: 9618: 9612: 9605: 9603: 9602: 9597: 9592: 9590: 9582: 9577: 9572: 9564: 9563: 9558: 9553: 9540: 9515: 9502: 9498: 9478: 9449: 9447: 9446: 9441: 9439: 9432: 9431: 9426: 9420: 9418: 9417: 9405: 9397: 9390: 9384: 9375: 9370: 9355: 9350: 9346: 9340: 9336: 9335: 9334: 9329: 9320: 9306: 9301: 9288: 9279: 9252: 9234: 9228: 9221: 9214: 9208: 9201: 9191: 9189: 9188: 9183: 9181: 9174: 9173: 9165: 9163: 9155: 9154: 9153: 9140: 9134: 9132: 9124: 9123: 9122: 9109: 9101: 9099: 9091: 9090: 9089: 9076: 9070: 9068: 9060: 9059: 9058: 9045: 9027: 9012: 9011: 8969: 8954: 8952: 8951: 8942: 8941: 8932: 8930: 8929: 8922: 8921: 8912: 8911: 8900: 8899: 8890: 8889: 8876: 8875: 8866: 8865: 8854: 8853: 8844: 8843: 8826: 8824: 8823: 8811: 8803: 8799: 8798: 8790: 8785: 8783: 8782: 8773: 8772: 8763: 8756: 8748: 8747: 8740: 8739: 8725: 8724: 8708: 8707: 8693: 8692: 8665: 8642: 8635: 8629: 8625: 8614: 8590: 8577:radiation stress 8574: 8566: 8540: 8538: 8537: 8532: 8521: 8517: 8516: 8500: 8492: 8488: 8484: 8480: 8479: 8478: 8473: 8464: 8440: 8438: 8430: 8422: 8397:wind wave models 8390: 8358: 8357: 8335: 8333: 8332: 8327: 8316: 8312: 8311: 8310: 8304: 8300: 8299: 8298: 8293: 8284: 8260: 8258: 8250: 8249: 8248: 8238: 8226: 8225: 8223: 8222: 8217: 8214: 8205: 8177: 8170: 8168: 8167: 8162: 8157: 8156: 8147: 8143: 8142: 8141: 8115: 8106: 8100: 8099: 8096: 8087: 8086: 8083: 8058: 8056: 8055: 8050: 8045: 8044: 8035: 8031: 8030: 8029: 8003: 7994: 7988: 7987: 7984: 7975: 7974: 7971: 7952: 7950: 7949: 7944: 7939: 7938: 7923: 7914: 7908: 7907: 7904: 7895: 7894: 7891: 7869: 7865: 7856: 7840: 7838: 7837: 7832: 7827: 7826: 7811: 7802: 7796: 7795: 7792: 7772: 7765: 7763: 7762: 7757: 7755: 7748: 7747: 7738: 7736: 7719: 7718: 7709: 7704: 7695: 7686: 7679: 7673: 7672: 7667: 7663: 7650: 7641: 7637: 7632: 7617: 7612: 7608: 7602: 7598: 7596: 7591: 7579: 7578: 7573: 7569: 7568: 7567: 7562: 7553: 7534: 7525: 7521: 7516: 7503: 7494: 7493: 7490: 7470: 7455: 7453: 7452: 7447: 7445: 7438: 7437: 7422: 7413: 7407: 7402: 7401: 7400: 7385: 7376: 7372: 7364: 7357: 7341: 7336: 7321: 7316: 7312: 7296: 7291: 7278: 7269: 7268: 7265: 7245: 7235:potential energy 7215: 7208: 7198: 7166: 7164: 7163: 7158: 7156: 7154: 7146: 7138: 7124: 7120: 7119: 7118: 7103: 7102: 7087: 7078: 7072: 7071: 7066: 7065: 7057: 7042: 7040: 7029: 7024: 7020: 7019: 7018: 7010: 6996: 6994: 6983: 6963: 6961: 6960: 6955: 6947: 6943: 6942: 6941: 6933: 6919: 6917: 6906: 6889: 6887: 6876: 6856: 6854: 6853: 6848: 6818: 6815: 6806: 6804: 6796: 6788: 6783: 6781: 6773: 6765: 6741: 6739: 6738: 6733: 6725: 6724: 6718: 6717: 6708: 6707: 6689: 6688: 6682: 6680: 6669: 6664: 6662: 6654: 6653: 6652: 6642: 6622: 6620: 6619: 6614: 6606: 6604: 6596: 6588: 6586: 6585: 6570: 6569: 6557: 6556: 6538: 6537: 6531: 6529: 6518: 6513: 6511: 6503: 6495: 6475: 6473: 6472: 6467: 6465: 6463: 6453: 6452: 6442: 6413: 6411: 6410: 6395: 6386: 6380: 6379: 6374: 6373: 6365: 6350: 6348: 6347: 6342: 6340: 6339: 6334: 6333: 6325: 6306: 6304: 6303: 6298: 6296: 6294: 6293: 6292: 6273: 6262: 6260: 6249: 6238: 6237: 6229: 6215: 6213: 6212: 6207: 6205: 6204: 6196: 6180: 6178: 6177: 6172: 6170: 6162: 6154: 6152: 6151: 6139: 6120: 6118: 6117: 6112: 6094: 6092: 6091: 6086: 6084: 6082: 6065: 6060: 6052: 6047: 6046: 6029: 6027: 6026: 6021: 6019: 6018: 5997: 5995: 5994: 5989: 5984: 5980: 5979: 5970: 5964: 5962: 5961: 5952: 5951: 5942: 5929: 5928: 5905: 5903: 5902: 5897: 5895: 5894: 5865:radiation stress 5859: 5857: 5856: 5851: 5849: 5848: 5833: 5824: 5801: 5799: 5798: 5793: 5758: 5757: 5733: 5689: 5685: 5674: 5670: 5663: 5656: 5654: 5653: 5648: 5643: 5641: 5640: 5623: 5596: 5592: 5568: 5554: 5553: 5534: 5523: 5516: 5508: 5507: 5505: 5504: 5499: 5496: 5477: 5466: 5462: 5461: 5459: 5458: 5453: 5450: 5435: 5431: 5427: 5423: 5419: 5415: 5408: 5381: 5363: 5356: 5352: 5345: 5338: 5336: 5335: 5330: 5325: 5321: 5320: 5319: 5310: 5308: 5307: 5289: 5281: 5279: 5278: 5263: 5262: 5247: 5240: 5232: 5215: 5214: 5195: 5160: 5159: 5157: 5156: 5151: 5148: 5137: 5136: 5134: 5133: 5127: 5124: 5098: 5096: 5095: 5090: 5085: 5084: 5075: 5067: 5056: 5055: 5047: 5034: 5030: 5023: 5021: 5020: 5015: 4994: 4990: 4989: 4988: 4979: 4971: 4946: 4945: 4912: 4898: 4891: 4882: 4881: 4880: 4869: 4867: 4866: 4861: 4858: 4850: 4849: 4848: 4821: 4819: 4818: 4813: 4802: 4800: 4786: 4760: 4741: 4739: 4738: 4733: 4705: 4703: 4702: 4697: 4686: 4685: 4649: 4647: 4646: 4641: 4624: 4597: 4595: 4594: 4589: 4578: 4576: 4562: 4536: 4523: 4521: 4520: 4515: 4504: 4499: 4488: 4475: 4473: 4472: 4467: 4456: 4455: 4428: 4426: 4425: 4420: 4403: 4395: 4394: 4372: 4370: 4369: 4364: 4353: 4351: 4337: 4311: 4306: 4305: 4300: 4283: 4281: 4280: 4275: 4261: 4259: 4248: 4246: 4245: 4240: 4223: 4221: 4220: 4215: 4204: 4203: 4187: 4186: 4181: 4161: 4159: 4158: 4153: 4136: 4128: 4127: 4122: 4103: 4101: 4100: 4095: 4084: 4082: 4068: 4042: 4026: 4024: 4023: 4018: 4007: 4002: 3991: 3975: 3973: 3972: 3967: 3956: 3955: 3925: 3923: 3922: 3917: 3900: 3892: 3891: 3869: 3867: 3866: 3861: 3850: 3848: 3834: 3808: 3800: 3799: 3794: 3780: 3778: 3777: 3772: 3758: 3749: 3748: 3746: 3745: 3740: 3726: 3724: 3723: 3718: 3707: 3706: 3687: 3686: 3681: 3664: 3662: 3661: 3656: 3639: 3631: 3630: 3625: 3606: 3604: 3603: 3598: 3596: 3592: 3591: 3589: 3575: 3565: 3564: 3548: 3529: 3520: 3509: 3507: 3506: 3501: 3488: 3486: 3485: 3480: 3478: 3469: 3455: 3453: 3452: 3447: 3445: 3443: 3442: 3433: 3432: 3423: 3408: 3406: 3405: 3400: 3398: 3394: 3393: 3391: 3377: 3367: 3366: 3350: 3331: 3330: 3321: 3312: 3301: 3299: 3298: 3293: 3291: 3283: 3273: 3271: 3270: 3265: 3263: 3255: 3253: 3244: 3238: 3229: 3228: 3226: 3217: 3203: 3201: 3200: 3195: 3193: 3191: 3183: 3175: 3170: 3169: 3145: 3143: 3142: 3137: 3135: 3121: 3113: 3111: 3101: 3099: 3098: 3093: 3091: 3083: 3073: 3071: 3070: 3065: 3063: 3055: 3050: 3041: 3040: 3027: 3025: 3024: 3019: 3017: 3012: 2998: 2993: 2992: 2968: 2966: 2965: 2960: 2958: 2938: 2913: 2911: 2910: 2905: 2903: 2895: 2867: 2865: 2864: 2859: 2857: 2849: 2818: 2816: 2815: 2810: 2781: 2779: 2778: 2773: 2771: 2766: 2761: 2748: 2746: 2745: 2740: 2738: 2737: 2732: 2713: 2711: 2710: 2705: 2703: 2695: 2677: 2675: with 2674: 2671: 2670: 2665: 2664: 2645: 2644: 2635: 2634: 2610: 2608: 2607: 2602: 2584: 2582: 2581: 2576: 2574: 2569: 2564: 2552: 2550: with 2549: 2546: 2545: 2540: 2539: 2520: 2519: 2510: 2509: 2504: 2500: 2499: 2491: 2460: 2458: 2457: 2452: 2431: 2429: 2428: 2423: 2412: 2404: 2386: 2384: 2383: 2378: 2367: 2343: 2341: 2340: 2335: 2324: 2293: 2291: 2290: 2285: 2274: 2249: 2245: 2236: 2232: 2230: 2229: 2226: 2223: 2206: 2202: 2200: 2199: 2196: 2193: 2163: 2162: 2155: 2151: 2149: 2148: 2145: 2142: 2128: 2126: 2125: 2122: 2119: 2101: 2097: 2095: 2094: 2091: 2088: 2066: 2062: 2060: 2059: 2056: 2053: 2019: 2008: 1993: 1982: 1964: 1961:(and wavelength 1960: 1956: 1952: 1948: 1938: 1926: 1916: 1900: 1896: 1886: 1876: 1852: 1848: 1844: 1836: 1832: 1828: 1824: 1816: 1809: 1807: 1806: 1801: 1775: 1774: 1751: 1747: 1743: 1731: 1727: 1720: 1718: 1717: 1712: 1680: 1678: 1664: 1638: 1633: 1625: 1604: 1602: 1601: 1596: 1527: 1520: 1519: 1517: 1516: 1510: 1507: 1499: 1476: 1467: 1465: 1464: 1459: 1430: 1427: 1409: 1407: 1399: 1391: 1381: 1368: 1346: 1337: 1335: 1334: 1329: 1300: 1297: 1294: 1292: 1284: 1276: 1271: 1269: 1261: 1253: 1243: 1232: 1210: 1201: 1199: 1198: 1193: 1176: 1173: 1164: 1162: 1154: 1146: 1136: 1111: 1102: 1100: 1099: 1094: 1086: 1084: 1083: 1082: 1069: 1065: 1064: 1054: 1049: 1047: 1046: 1045: 1032: 1028: 1027: 1017: 1007: 1002:Laplace equation 999: 988: 986: 985: 980: 975: 973: 965: 957: 952: 951: 941: 938: 935: 933: 925: 917: 912: 911: 892: 889:) and vertical ( 888: 884: 877: 866: 846:potential theory 835: 821: 814: 810: 806: 787: 732: 730: 729: 724: 716: 707: 697: 694: 661: 647: 643: 639: 635: 628: 626: 625: 620: 615: 607: 602: 594: 589: 588: 569: 555: 550: 548: 547: 542: 539: 524: 517: 506: 500: 499: 497: 496: 491: 488: 473: 458: 448: 438: 428: 426: 425: 420: 418: 414: 353: 349: 341: 308:– correct up to 201: 200: 199: 178: 177: 175: 174: 169: 166: 156: 155: 154: 25:Airy wave theory 13111: 13110: 13106: 13105: 13104: 13102: 13101: 13100: 13081: 13080: 13079: 13074: 13062: 13034: 12873: 12847: 12809: 12790:Sea-level curve 12761: 12700: 12686:Transform fault 12636:Mid-ocean ridge 12602: 12595: 12561:Oceanic plateau 12467: 12453:Tidal resonance 12423:Theory of tides 12364: 12273:Longshore drift 12223:Ekman transport 12159: 12153: 12152: 12151: 12150: 12149: 12148: 12139: 12091:Wave turbulence 12024:Trochoidal wave 11949:Longshore drift 11845: 11840: 11792: 11781: 11779: 11756:"Surface Waves" 11744:Wehausen, J. V. 11728: 11714:Phillips, O. M. 11697: 11687:Waves in fluids 11667: 11657:Fluid mechanics 11632: 11601: 11574: 11529: 11527:Further reading 11508: 11481:Hugh James Rose 11472: 11467: 11462: 11457: 11450: 11443: 11429: 11425: 11418: 11404: 11400: 11395: 11391: 11386: 11377: 11338:McIntyre, M. E. 11335: 11331: 11326: 11322: 11317: 11313: 11308: 11304: 11299: 11295: 11290: 11286: 11281: 11277: 11272: 11268: 11226: 11222: 11217: 11210: 11201:Theory of Sound 11172: 11168: 11163: 11156: 11151: 11144: 11139: 11135: 11119: 11102: 11089: 11074: 11070: 11064: 11050: 11046: 11040: 11026: 11022: 11017: 11013: 11005: 11001: 10996: 10992: 10987: 10983: 10978: 10971: 10957: 10954: 10951: 10950: 10948: 10943: 10941: 10937: 10923: 10920: 10917: 10916: 10914: 10909: 10905: 10903: 10899: 10894: 10890: 10885: 10874: 10869: 10862: 10820: 10816: 10811: 10807: 10802: 10798: 10783: 10769: 10762: 10739: 10730: 10726: 10685:surface tension 10667: 10656: 10649: 10638: 10615: 10610: 10609: 10591: 10587: 10586: 10557: 10555: 10549: 10545: 10527: 10518: 10507: 10505: 10504: 10503: 10501: 10498: 10497: 10487: 10479: 10461: 10458: 10452: 10425: 10418: 10396: 10374: 10368: 10364: 10362: 10360: 10357: 10356: 10318: 10307: 10302: 10301: 10293: 10292: 10288: 10287: 10283: 10260: 10256: 10246: 10241: 10239: 10236: 10235: 10220: 10192: 10191: 10185: 10174: 10173: 10172: 10166: 10155: 10154: 10153: 10151: 10145: 10134: 10133: 10132: 10126: 10115: 10114: 10113: 10110: 10109: 10103: 10092: 10091: 10090: 10084: 10073: 10072: 10071: 10069: 10063: 10052: 10051: 10050: 10044: 10033: 10032: 10031: 10024: 10023: 10009: 10007: 10006: 9992: 9990: 9989: 9987: 9984: 9983: 9972: 9969:identity matrix 9962: 9952: 9942: 9938: 9893: 9885: 9879: 9875: 9857: 9843: 9841: 9840: 9826: 9824: 9823: 9816: 9812: 9787: 9785: 9784: 9777: 9773: 9763: 9758: 9756: 9753: 9752: 9731: 9692: 9690: 9689: 9682: 9678: 9657: 9653: 9643: 9638: 9636: 9633: 9632: 9620: 9614: 9610: 9583: 9578: 9576: 9568: 9554: 9552: 9551: 9549: 9546: 9545: 9536: 9529: 9511: 9500: 9494: 9476: 9437: 9436: 9427: 9422: 9421: 9413: 9409: 9404: 9395: 9394: 9386: 9380: 9371: 9363: 9342: 9330: 9325: 9324: 9316: 9315: 9311: 9302: 9294: 9289: 9287: 9280: 9275: 9271: 9269: 9266: 9265: 9248: 9241: 9230: 9227: 9223: 9220: 9216: 9210: 9207: 9203: 9200: 9196: 9179: 9178: 9168: 9167: 9156: 9149: 9145: 9141: 9139: 9136: 9125: 9118: 9114: 9110: 9108: 9104: 9103: 9092: 9085: 9081: 9077: 9075: 9072: 9061: 9054: 9050: 9046: 9044: 9036: 9035: 9028: 9023: 9017: 9016: 9006: 9005: 9000: 8994: 8993: 8988: 8978: 8977: 8970: 8965: 8962: 8961: 8947: 8943: 8937: 8933: 8931: 8924: 8923: 8917: 8913: 8907: 8903: 8901: 8895: 8891: 8885: 8881: 8878: 8877: 8871: 8867: 8861: 8857: 8855: 8849: 8845: 8839: 8835: 8828: 8827: 8819: 8815: 8810: 8789: 8778: 8774: 8768: 8764: 8762: 8761: 8757: 8752: 8742: 8741: 8732: 8728: 8726: 8717: 8713: 8710: 8709: 8700: 8696: 8694: 8685: 8681: 8674: 8673: 8666: 8661: 8657: 8655: 8652: 8651: 8637: 8631: 8627: 8616: 8603: 8585: 8570: 8561: 8548: 8512: 8508: 8504: 8496: 8474: 8469: 8468: 8460: 8459: 8455: 8454: 8450: 8431: 8423: 8421: 8419: 8416: 8415: 8401:wave turbulence 8389: 8381: 8372: 8364: 8355: 8353: 8340: 8306: 8305: 8294: 8289: 8288: 8280: 8279: 8275: 8274: 8270: 8251: 8244: 8243: 8239: 8237: 8235: 8232: 8231: 8218: 8215: 8210: 8209: 8207: 8203: 8202: 8188: 8180:surface tension 8175: 8152: 8148: 8137: 8133: 8120: 8116: 8104: 8095: 8091: 8082: 8078: 8070: 8067: 8066: 8040: 8036: 8025: 8021: 8008: 8004: 7992: 7983: 7979: 7970: 7966: 7964: 7961: 7960: 7934: 7930: 7912: 7903: 7899: 7890: 7886: 7878: 7875: 7874: 7867: 7864: 7858: 7855: 7849: 7822: 7818: 7800: 7791: 7787: 7785: 7782: 7781: 7770: 7753: 7752: 7743: 7739: 7720: 7714: 7710: 7708: 7693: 7684: 7683: 7675: 7668: 7659: 7655: 7654: 7639: 7633: 7625: 7604: 7592: 7587: 7574: 7563: 7558: 7557: 7549: 7548: 7544: 7543: 7542: 7538: 7523: 7517: 7509: 7504: 7502: 7495: 7489: 7485: 7481: 7479: 7476: 7475: 7469: 7463: 7443: 7442: 7433: 7429: 7411: 7396: 7392: 7374: 7373: 7371: 7362: 7361: 7353: 7337: 7329: 7308: 7292: 7284: 7279: 7277: 7270: 7264: 7260: 7256: 7254: 7251: 7250: 7244: 7238: 7226: 7213: 7200: 7189: 7147: 7139: 7137: 7111: 7107: 7098: 7094: 7076: 7067: 7056: 7055: 7054: 7047: 7043: 7033: 7028: 7009: 7008: 7001: 6997: 6987: 6982: 6980: 6977: 6976: 6932: 6931: 6924: 6920: 6910: 6905: 6880: 6875: 6873: 6870: 6869: 6814: 6797: 6789: 6787: 6774: 6766: 6764: 6762: 6759: 6758: 6720: 6719: 6713: 6712: 6703: 6699: 6684: 6683: 6673: 6668: 6655: 6648: 6647: 6643: 6641: 6639: 6636: 6635: 6597: 6589: 6587: 6578: 6574: 6565: 6564: 6552: 6548: 6533: 6532: 6522: 6517: 6504: 6496: 6494: 6492: 6489: 6488: 6448: 6444: 6443: 6414: 6412: 6406: 6402: 6384: 6375: 6364: 6363: 6362: 6360: 6357: 6356: 6335: 6324: 6323: 6322: 6320: 6317: 6316: 6288: 6284: 6277: 6272: 6253: 6248: 6228: 6227: 6225: 6222: 6221: 6195: 6194: 6192: 6189: 6188: 6161: 6147: 6143: 6138: 6130: 6127: 6126: 6106: 6103: 6102: 6069: 6064: 6051: 6042: 6041: 6039: 6036: 6035: 6014: 6013: 6011: 6008: 6007: 5968: 5957: 5953: 5947: 5943: 5941: 5937: 5933: 5921: 5917: 5915: 5912: 5911: 5887: 5883: 5881: 5878: 5877: 5844: 5840: 5822: 5814: 5811: 5810: 5787: 5784: 5783: 5752: 5731: 5720: 5696: 5687: 5680: 5672: 5665: 5661: 5633: 5616: 5597: 5585: 5569: 5567: 5549: 5545: 5543: 5540: 5539: 5525: 5518: 5514: 5500: 5497: 5490: 5489: 5487: 5484: 5479: 5476: 5472: 5464: 5454: 5451: 5445: 5444: 5442: 5437: 5433: 5429: 5425: 5421: 5417: 5410: 5403: 5392: 5372: 5358: 5354: 5347: 5343: 5315: 5311: 5300: 5293: 5288: 5271: 5264: 5255: 5248: 5246: 5245: 5241: 5236: 5228: 5210: 5206: 5204: 5201: 5200: 5186: 5183: 5167: 5152: 5149: 5146: 5145: 5143: 5142: 5128: 5125: 5122: 5121: 5119: 5118: 5080: 5076: 5066: 5046: 5045: 5043: 5040: 5039: 5035:is replaced by 5032: 5028: 4984: 4980: 4970: 4963: 4959: 4941: 4937: 4935: 4932: 4931: 4925:surface tension 4921: 4908: 4906: 4904: 4902: 4900: 4897: 4893: 4890: 4886: 4884: 4872: 4870: 4862: 4859: 4856: 4855: 4853: 4852: 4840: 4838: 4837: 4830: 4787: 4761: 4759: 4748: 4745: 4744: 4712: 4709: 4708: 4678: 4674: 4662: 4659: 4658: 4620: 4612: 4609: 4608: 4563: 4537: 4535: 4530: 4527: 4526: 4489: 4487: 4482: 4479: 4478: 4448: 4444: 4438: 4435: 4434: 4399: 4390: 4386: 4384: 4381: 4380: 4338: 4312: 4310: 4301: 4296: 4295: 4290: 4287: 4286: 4252: 4247: 4241: 4236: 4235: 4230: 4227: 4226: 4196: 4192: 4182: 4177: 4176: 4171: 4168: 4167: 4132: 4123: 4118: 4117: 4115: 4112: 4111: 4069: 4043: 4041: 4033: 4030: 4029: 3992: 3990: 3982: 3979: 3978: 3948: 3944: 3935: 3932: 3931: 3896: 3887: 3883: 3881: 3878: 3877: 3835: 3809: 3807: 3795: 3790: 3789: 3787: 3784: 3783: 3747: 3741: 3736: 3735: 3733: 3730: 3729: 3699: 3695: 3682: 3677: 3676: 3674: 3671: 3670: 3635: 3626: 3621: 3620: 3618: 3615: 3614: 3576: 3560: 3556: 3549: 3547: 3534: 3530: 3518: 3516: 3513: 3512: 3495: 3492: 3491: 3467: 3465: 3462: 3461: 3438: 3434: 3428: 3424: 3422: 3420: 3417: 3416: 3378: 3362: 3358: 3351: 3349: 3336: 3332: 3326: 3322: 3310: 3308: 3305: 3304: 3282: 3280: 3277: 3276: 3254: 3242: 3227: 3215: 3213: 3210: 3209: 3184: 3176: 3174: 3165: 3161: 3159: 3156: 3155: 3112: 3110: 3108: 3105: 3104: 3082: 3080: 3077: 3076: 3054: 3039: 3037: 3034: 3033: 2999: 2997: 2988: 2984: 2982: 2979: 2978: 2937: 2920: 2917: 2916: 2894: 2874: 2871: 2870: 2848: 2831: 2828: 2827: 2795: 2792: 2791: 2762: 2760: 2758: 2755: 2754: 2733: 2728: 2727: 2725: 2722: 2721: 2699: 2691: 2673: 2666: 2660: 2659: 2658: 2640: 2639: 2630: 2626: 2623: 2620: 2619: 2596: 2593: 2592: 2570: 2565: 2560: 2548: 2541: 2535: 2534: 2533: 2515: 2514: 2505: 2495: 2487: 2480: 2476: 2475: 2473: 2470: 2469: 2446: 2443: 2442: 2408: 2400: 2398: 2395: 2394: 2363: 2355: 2352: 2351: 2320: 2303: 2300: 2299: 2270: 2262: 2259: 2258: 2247: 2243: 2241: 2227: 2224: 2221: 2220: 2218: 2213: 2211: 2197: 2194: 2191: 2190: 2188: 2183: 2181: 2146: 2143: 2140: 2139: 2137: 2123: 2120: 2117: 2116: 2114: 2113: 2092: 2089: 2086: 2085: 2083: 2078: 2057: 2054: 2051: 2050: 2048: 2043: 2018: 2010: 2007: 1999: 1992: 1984: 1981: 1973: 1962: 1958: 1954: 1950: 1940: 1936: 1922: 1914: 1898: 1892: 1882: 1863: 1859: 1850: 1846: 1842: 1834: 1833:and wavelength 1830: 1826: 1825:and wavenumber 1822: 1814: 1770: 1766: 1764: 1761: 1760: 1749: 1745: 1741: 1734:hyperbolic sine 1729: 1725: 1665: 1639: 1637: 1624: 1616: 1613: 1612: 1554: 1551: 1550: 1540: 1534: 1522: 1511: 1508: 1505: 1504: 1502: 1501: 1497: 1426: 1400: 1392: 1390: 1388: 1385: 1384: 1355: 1296: 1285: 1277: 1275: 1262: 1254: 1252: 1250: 1247: 1246: 1227: 1172: 1155: 1147: 1145: 1143: 1140: 1139: 1078: 1074: 1070: 1060: 1056: 1055: 1053: 1041: 1037: 1033: 1023: 1019: 1018: 1016: 1014: 1011: 1010: 997: 966: 958: 956: 947: 943: 937: 926: 918: 916: 907: 903: 901: 898: 897: 890: 886: 883: 879: 876: 872: 852: 827: 816: 812: 808: 789: 785: 779: 774: 705: 693: 678: 675: 674: 659: 645: 641: 637: 633: 606: 593: 584: 580: 578: 575: 574: 568: 564: 543: 540: 534: 533: 531: 526: 522: 515: 504: 492: 489: 483: 482: 480: 475: 471: 456: 446: 436: 398: 394: 362: 359: 358: 351: 347: 328: 318:trochoidal wave 251:wave turbulence 228:flow separation 195: 193: 192: 170: 167: 162: 161: 159: 158: 150: 148: 147: 121: 17: 12: 11: 5: 13109: 13099: 13098: 13096:Wave mechanics 13093: 13076: 13075: 13073: 13072: 13060: 13050: 13039: 13036: 13035: 13033: 13032: 13027: 13022: 13017: 13012: 13010:Stratification 13007: 13002: 12997: 12992: 12987: 12982: 12981: 12980: 12970: 12965: 12960: 12955: 12950: 12945: 12940: 12935: 12930: 12925: 12920: 12915: 12910: 12902: 12900:Color of water 12897: 12895:Benthic lander 12892: 12887: 12881: 12879: 12875: 12874: 12872: 12871: 12866: 12861: 12855: 12853: 12849: 12848: 12846: 12845: 12840: 12835: 12830: 12825: 12819: 12817: 12811: 12810: 12808: 12807: 12802: 12800:Sea level rise 12797: 12795:Sea level drop 12792: 12787: 12782: 12777: 12771: 12769: 12763: 12762: 12760: 12759: 12754: 12749: 12744: 12739: 12734: 12729: 12724: 12719: 12714: 12708: 12706: 12702: 12701: 12699: 12698: 12693: 12688: 12683: 12678: 12673: 12668: 12663: 12658: 12653: 12648: 12643: 12638: 12633: 12631:Marine geology 12628: 12623: 12618: 12613: 12607: 12605: 12597: 12596: 12594: 12593: 12588: 12583: 12578: 12573: 12571:Passive margin 12568: 12566:Oceanic trench 12563: 12558: 12553: 12548: 12543: 12538: 12533: 12528: 12523: 12518: 12513: 12508: 12503: 12498: 12493: 12488: 12483: 12477: 12475: 12469: 12468: 12466: 12465: 12460: 12455: 12450: 12445: 12440: 12435: 12430: 12425: 12420: 12415: 12410: 12405: 12400: 12395: 12390: 12385: 12380: 12374: 12372: 12366: 12365: 12363: 12362: 12357: 12352: 12347: 12342: 12341: 12340: 12330: 12325: 12320: 12315: 12310: 12305: 12300: 12298:Ocean dynamics 12295: 12290: 12285: 12280: 12275: 12270: 12265: 12260: 12255: 12250: 12245: 12240: 12235: 12230: 12225: 12220: 12215: 12210: 12205: 12200: 12195: 12190: 12188:Coriolis force 12185: 12180: 12175: 12169: 12167: 12161: 12160: 12142: 12140: 12138: 12137: 12136: 12135: 12125: 12120: 12115: 12114: 12113: 12108: 12098: 12093: 12088: 12083: 12078: 12073: 12068: 12063: 12058: 12053: 12048: 12043: 12038: 12037: 12036: 12026: 12021: 12016: 12011: 12009:Stokes problem 12006: 12001: 11996: 11991: 11986: 11981: 11976: 11971: 11966: 11961: 11956: 11951: 11946: 11944:Kinematic wave 11941: 11936: 11931: 11926: 11921: 11916: 11911: 11906: 11901: 11896: 11891: 11886: 11881: 11876: 11871: 11866: 11861: 11855: 11853: 11847: 11846: 11839: 11838: 11831: 11824: 11816: 11810: 11809: 11799: 11791: 11790:External links 11788: 11787: 11786: 11740: 11726: 11710: 11695: 11679: 11665: 11645: 11630: 11614: 11599: 11586: 11572: 11559: 11528: 11525: 11524: 11523: 11509:Reprinted in: 11492: 11471: 11468: 11466: 11463: 11461: 11460: 11448: 11441: 11423: 11416: 11398: 11389: 11375: 11329: 11320: 11311: 11302: 11293: 11284: 11275: 11266: 11220: 11208: 11166: 11154: 11142: 11133: 11117: 11087: 11068: 11062: 11044: 11039:978-0521297264 11038: 11020: 11011: 10999: 10990: 10981: 10969: 10935: 10897: 10888: 10872: 10870:Stokes (1847). 10860: 10814: 10805: 10796: 10781: 10760: 10727: 10725: 10722: 10721: 10720: 10714: 10708: 10702: 10696: 10687: 10681:Capillary wave 10678: 10666: 10663: 10661:is recovered. 10645: 10635: 10634: 10623: 10618: 10613: 10605: 10602: 10599: 10594: 10590: 10584: 10581: 10578: 10575: 10572: 10569: 10566: 10563: 10560: 10552: 10548: 10544: 10541: 10535: 10532: 10526: 10521: 10514: 10510: 10483: 10454:Main article: 10451: 10448: 10415: 10414: 10403: 10399: 10395: 10392: 10389: 10386: 10380: 10377: 10371: 10367: 10350: 10349: 10338: 10335: 10331: 10325: 10322: 10316: 10310: 10305: 10300: 10296: 10291: 10286: 10282: 10279: 10276: 10272: 10267: 10264: 10259: 10252: 10249: 10245: 10213: 10212: 10201: 10196: 10188: 10181: 10178: 10169: 10162: 10159: 10152: 10148: 10141: 10138: 10129: 10122: 10119: 10112: 10111: 10106: 10099: 10096: 10087: 10080: 10077: 10070: 10066: 10059: 10056: 10047: 10040: 10037: 10030: 10029: 10027: 10022: 10016: 10012: 10005: 9999: 9995: 9977:dyadic product 9935: 9934: 9923: 9920: 9917: 9914: 9911: 9908: 9905: 9901: 9896: 9892: 9888: 9882: 9878: 9874: 9871: 9865: 9862: 9856: 9850: 9846: 9839: 9833: 9829: 9822: 9819: 9815: 9811: 9808: 9805: 9801: 9794: 9790: 9783: 9780: 9776: 9769: 9766: 9762: 9728: 9727: 9716: 9713: 9710: 9706: 9699: 9695: 9688: 9685: 9681: 9677: 9674: 9671: 9667: 9663: 9660: 9656: 9649: 9646: 9642: 9625:become equal. 9607: 9606: 9595: 9589: 9586: 9581: 9575: 9571: 9567: 9561: 9557: 9528: 9525: 9491: 9490: 9484: 9459:phase velocity 9451: 9450: 9435: 9430: 9425: 9416: 9412: 9408: 9403: 9400: 9398: 9396: 9393: 9389: 9383: 9379: 9374: 9369: 9366: 9362: 9358: 9353: 9349: 9345: 9339: 9333: 9328: 9323: 9319: 9314: 9310: 9305: 9300: 9297: 9293: 9286: 9283: 9281: 9278: 9274: 9273: 9247:per unit area 9240: 9237: 9225: 9218: 9215:and similarly 9205: 9198: 9193: 9192: 9177: 9172: 9162: 9159: 9152: 9148: 9144: 9137: 9131: 9128: 9121: 9117: 9113: 9106: 9105: 9098: 9095: 9088: 9084: 9080: 9073: 9067: 9064: 9057: 9053: 9049: 9042: 9041: 9039: 9034: 9031: 9029: 9026: 9022: 9019: 9018: 9015: 9010: 9004: 9001: 8999: 8996: 8995: 8992: 8989: 8987: 8984: 8983: 8981: 8976: 8973: 8971: 8968: 8964: 8963: 8960: 8957: 8950: 8946: 8940: 8936: 8928: 8920: 8916: 8910: 8906: 8902: 8898: 8894: 8888: 8884: 8880: 8879: 8874: 8870: 8864: 8860: 8856: 8852: 8848: 8842: 8838: 8834: 8833: 8831: 8822: 8818: 8814: 8809: 8806: 8802: 8796: 8793: 8788: 8781: 8777: 8771: 8767: 8760: 8755: 8751: 8746: 8738: 8735: 8731: 8727: 8723: 8720: 8716: 8712: 8711: 8706: 8703: 8699: 8695: 8691: 8688: 8684: 8680: 8679: 8677: 8672: 8669: 8667: 8664: 8660: 8659: 8597: 8596: 8583: 8568: 8557: 8542: 8541: 8530: 8527: 8524: 8520: 8515: 8511: 8507: 8503: 8499: 8495: 8491: 8487: 8483: 8477: 8472: 8467: 8463: 8458: 8453: 8449: 8446: 8443: 8437: 8434: 8429: 8426: 8393:group velocity 8385: 8377: 8368: 8349: 8337: 8336: 8325: 8322: 8319: 8315: 8309: 8303: 8297: 8292: 8287: 8283: 8278: 8273: 8269: 8266: 8263: 8257: 8254: 8247: 8242: 8187: 8184: 8172: 8171: 8160: 8155: 8151: 8146: 8140: 8136: 8132: 8129: 8126: 8123: 8119: 8112: 8109: 8103: 8094: 8090: 8081: 8077: 8074: 8060: 8059: 8048: 8043: 8039: 8034: 8028: 8024: 8020: 8017: 8014: 8011: 8007: 8000: 7997: 7991: 7982: 7978: 7969: 7954: 7953: 7942: 7937: 7933: 7929: 7926: 7920: 7917: 7911: 7902: 7898: 7889: 7885: 7882: 7862: 7853: 7842: 7841: 7830: 7825: 7821: 7817: 7814: 7808: 7805: 7799: 7790: 7767: 7766: 7751: 7746: 7742: 7735: 7732: 7729: 7726: 7723: 7717: 7713: 7707: 7701: 7698: 7692: 7689: 7687: 7685: 7682: 7678: 7671: 7666: 7662: 7658: 7653: 7647: 7644: 7636: 7631: 7628: 7624: 7620: 7615: 7611: 7607: 7601: 7595: 7590: 7586: 7582: 7577: 7572: 7566: 7561: 7556: 7552: 7547: 7541: 7537: 7531: 7528: 7520: 7515: 7512: 7508: 7501: 7498: 7496: 7488: 7484: 7483: 7467: 7457: 7456: 7441: 7436: 7432: 7428: 7425: 7419: 7416: 7410: 7405: 7399: 7395: 7391: 7388: 7382: 7379: 7370: 7367: 7365: 7363: 7360: 7356: 7351: 7348: 7345: 7340: 7335: 7332: 7328: 7324: 7319: 7315: 7311: 7306: 7303: 7300: 7295: 7290: 7287: 7283: 7276: 7273: 7271: 7263: 7259: 7258: 7242: 7225: 7222: 7168: 7167: 7153: 7150: 7145: 7142: 7136: 7133: 7130: 7127: 7123: 7117: 7114: 7110: 7106: 7101: 7097: 7093: 7090: 7084: 7081: 7075: 7070: 7063: 7060: 7053: 7050: 7046: 7039: 7036: 7032: 7027: 7023: 7016: 7013: 7007: 7004: 7000: 6993: 6990: 6986: 6974: 6971: 6969: 6965: 6964: 6953: 6950: 6946: 6939: 6936: 6930: 6927: 6923: 6916: 6913: 6909: 6904: 6901: 6898: 6895: 6892: 6886: 6883: 6879: 6867: 6864: 6862: 6858: 6857: 6846: 6843: 6840: 6837: 6834: 6831: 6828: 6825: 6822: 6812: 6809: 6803: 6800: 6795: 6792: 6786: 6780: 6777: 6772: 6769: 6756: 6753: 6751: 6743: 6742: 6731: 6728: 6723: 6716: 6711: 6706: 6702: 6698: 6695: 6692: 6687: 6679: 6676: 6672: 6667: 6661: 6658: 6651: 6646: 6633: 6630: 6628: 6624: 6623: 6612: 6609: 6603: 6600: 6595: 6592: 6584: 6581: 6577: 6573: 6568: 6563: 6560: 6555: 6551: 6547: 6544: 6541: 6536: 6528: 6525: 6521: 6516: 6510: 6507: 6502: 6499: 6486: 6483: 6481: 6477: 6476: 6462: 6459: 6456: 6451: 6447: 6441: 6438: 6435: 6432: 6429: 6426: 6423: 6420: 6417: 6409: 6405: 6401: 6398: 6392: 6389: 6383: 6378: 6371: 6368: 6354: 6351: 6338: 6331: 6328: 6314: 6308: 6307: 6291: 6287: 6283: 6280: 6276: 6271: 6268: 6265: 6259: 6256: 6252: 6247: 6244: 6241: 6235: 6232: 6219: 6216: 6202: 6199: 6186: 6182: 6181: 6168: 6165: 6160: 6157: 6150: 6146: 6142: 6137: 6134: 6124: 6121: 6110: 6100: 6096: 6095: 6081: 6078: 6075: 6072: 6068: 6063: 6058: 6055: 6050: 6045: 6033: 6030: 6017: 6005: 5999: 5998: 5987: 5983: 5976: 5973: 5967: 5960: 5956: 5950: 5946: 5940: 5936: 5932: 5927: 5924: 5920: 5909: 5906: 5893: 5890: 5886: 5875: 5861: 5860: 5847: 5843: 5839: 5836: 5830: 5827: 5821: 5818: 5808: 5802: 5791: 5781: 5777: 5776: 5773: 5770: 5767: 5763: 5762: 5751: 5748: 5719: 5716: 5695: 5692: 5658: 5657: 5646: 5639: 5636: 5632: 5629: 5626: 5622: 5619: 5615: 5612: 5609: 5606: 5603: 5600: 5595: 5591: 5588: 5584: 5581: 5578: 5575: 5572: 5566: 5563: 5560: 5557: 5552: 5548: 5482: 5474: 5391: 5388: 5340: 5339: 5328: 5324: 5318: 5314: 5306: 5303: 5299: 5296: 5292: 5287: 5284: 5277: 5274: 5270: 5267: 5261: 5258: 5254: 5251: 5244: 5239: 5235: 5231: 5227: 5224: 5221: 5218: 5213: 5209: 5182: 5179: 5166: 5163: 5140:phase velocity 5116:group velocity 5100: 5099: 5088: 5083: 5079: 5073: 5070: 5065: 5062: 5059: 5053: 5050: 5025: 5024: 5013: 5010: 5007: 5004: 5001: 4997: 4993: 4987: 4983: 4977: 4974: 4969: 4966: 4962: 4958: 4955: 4952: 4949: 4944: 4940: 4919:Capillary wave 4917:Main article: 4895: 4888: 4829: 4826: 4823: 4822: 4811: 4808: 4805: 4799: 4796: 4793: 4790: 4785: 4782: 4779: 4776: 4773: 4770: 4767: 4764: 4758: 4755: 4752: 4742: 4731: 4728: 4725: 4722: 4719: 4716: 4706: 4695: 4692: 4689: 4684: 4681: 4677: 4672: 4669: 4666: 4656: 4650: 4639: 4636: 4633: 4630: 4627: 4623: 4619: 4616: 4606: 4599: 4598: 4587: 4584: 4581: 4575: 4572: 4569: 4566: 4561: 4558: 4555: 4552: 4549: 4546: 4543: 4540: 4534: 4524: 4513: 4510: 4507: 4502: 4498: 4495: 4492: 4486: 4476: 4465: 4462: 4459: 4454: 4451: 4447: 4442: 4432: 4429: 4418: 4415: 4412: 4409: 4406: 4402: 4398: 4393: 4389: 4378: 4374: 4373: 4362: 4359: 4356: 4350: 4347: 4344: 4341: 4336: 4333: 4330: 4327: 4324: 4321: 4318: 4315: 4309: 4304: 4299: 4294: 4284: 4273: 4270: 4267: 4264: 4258: 4255: 4251: 4244: 4239: 4234: 4224: 4213: 4210: 4207: 4202: 4199: 4195: 4190: 4185: 4180: 4175: 4165: 4162: 4151: 4148: 4145: 4142: 4139: 4135: 4131: 4126: 4121: 4109: 4105: 4104: 4093: 4090: 4087: 4081: 4078: 4075: 4072: 4067: 4064: 4061: 4058: 4055: 4052: 4049: 4046: 4040: 4037: 4027: 4016: 4013: 4010: 4005: 4001: 3998: 3995: 3989: 3986: 3976: 3965: 3962: 3959: 3954: 3951: 3947: 3942: 3939: 3929: 3926: 3915: 3912: 3909: 3906: 3903: 3899: 3895: 3890: 3886: 3875: 3871: 3870: 3859: 3856: 3853: 3847: 3844: 3841: 3838: 3833: 3830: 3827: 3824: 3821: 3818: 3815: 3812: 3806: 3803: 3798: 3793: 3781: 3770: 3767: 3764: 3761: 3755: 3752: 3744: 3739: 3727: 3716: 3713: 3710: 3705: 3702: 3698: 3693: 3690: 3685: 3680: 3668: 3665: 3654: 3651: 3648: 3645: 3642: 3638: 3634: 3629: 3624: 3612: 3608: 3607: 3595: 3588: 3585: 3582: 3579: 3574: 3571: 3568: 3563: 3559: 3555: 3552: 3546: 3543: 3540: 3537: 3533: 3526: 3523: 3510: 3499: 3489: 3475: 3472: 3459: 3456: 3441: 3437: 3431: 3427: 3414: 3410: 3409: 3397: 3390: 3387: 3384: 3381: 3376: 3373: 3370: 3365: 3361: 3357: 3354: 3348: 3345: 3342: 3339: 3335: 3329: 3325: 3318: 3315: 3302: 3289: 3286: 3274: 3261: 3258: 3250: 3247: 3241: 3235: 3232: 3223: 3220: 3207: 3204: 3190: 3187: 3182: 3179: 3173: 3168: 3164: 3153: 3147: 3146: 3133: 3130: 3127: 3124: 3119: 3116: 3102: 3089: 3086: 3074: 3061: 3058: 3053: 3047: 3044: 3031: 3028: 3015: 3011: 3008: 3005: 3002: 2996: 2991: 2987: 2976: 2970: 2969: 2956: 2953: 2950: 2947: 2944: 2941: 2936: 2933: 2930: 2927: 2924: 2914: 2901: 2898: 2893: 2890: 2887: 2884: 2881: 2878: 2868: 2855: 2852: 2847: 2844: 2841: 2838: 2835: 2825: 2819: 2808: 2805: 2802: 2799: 2789: 2783: 2782: 2769: 2765: 2752: 2749: 2736: 2731: 2719: 2715: 2714: 2702: 2698: 2694: 2690: 2687: 2684: 2681: 2669: 2663: 2657: 2654: 2651: 2648: 2643: 2638: 2633: 2629: 2617: 2611: 2600: 2590: 2586: 2585: 2573: 2568: 2563: 2559: 2556: 2544: 2538: 2532: 2529: 2526: 2523: 2518: 2513: 2508: 2503: 2498: 2494: 2490: 2486: 2483: 2479: 2467: 2461: 2450: 2440: 2433: 2432: 2421: 2418: 2415: 2411: 2407: 2403: 2392: 2387: 2376: 2373: 2370: 2366: 2362: 2359: 2349: 2345: 2344: 2333: 2330: 2327: 2323: 2319: 2316: 2313: 2310: 2307: 2297: 2294: 2283: 2280: 2277: 2273: 2269: 2266: 2256: 2252: 2251: 2238: 2208: 2178: 2175: 2172: 2168: 2167: 2158: 2157: 2107: 2072: 2020:respectively. 2014: 2003: 1988: 1977: 1858: 1855: 1811: 1810: 1799: 1796: 1793: 1790: 1787: 1784: 1781: 1778: 1773: 1769: 1756:is satisfied: 1722: 1721: 1710: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1686: 1683: 1677: 1674: 1671: 1668: 1663: 1660: 1657: 1654: 1651: 1648: 1645: 1642: 1636: 1631: 1628: 1623: 1620: 1606: 1605: 1594: 1591: 1588: 1585: 1582: 1579: 1576: 1573: 1570: 1567: 1564: 1561: 1558: 1533: 1530: 1480: 1479: 1470: 1468: 1457: 1454: 1451: 1448: 1445: 1442: 1439: 1436: 1433: 1428: at 1424: 1421: 1418: 1415: 1412: 1406: 1403: 1398: 1395: 1350: 1349: 1340: 1338: 1327: 1324: 1321: 1318: 1315: 1312: 1309: 1306: 1303: 1298: at 1291: 1288: 1283: 1280: 1274: 1268: 1265: 1260: 1257: 1214: 1213: 1204: 1202: 1191: 1188: 1185: 1182: 1179: 1174: at 1170: 1167: 1161: 1158: 1153: 1150: 1115: 1114: 1105: 1103: 1092: 1089: 1081: 1077: 1073: 1068: 1063: 1059: 1052: 1044: 1040: 1036: 1031: 1026: 1022: 990: 989: 978: 972: 969: 964: 961: 955: 950: 946: 932: 929: 924: 921: 915: 910: 906: 881: 874: 838:incompressible 778: 775: 773: 770: 734: 733: 722: 719: 713: 710: 704: 701: 691: 688: 685: 682: 636:and frequency 630: 629: 618: 613: 610: 605: 600: 597: 592: 587: 583: 566: 558: 557: 502: 454: 444: 430: 429: 417: 413: 410: 407: 404: 401: 397: 393: 390: 387: 384: 381: 378: 375: 372: 369: 366: 208:potential flow 144:group velocity 120: 117: 57:incompressible 21:fluid dynamics 15: 9: 6: 4: 3: 2: 13108: 13097: 13094: 13092: 13089: 13088: 13086: 13071: 13066: 13061: 13059: 13051: 13049: 13041: 13040: 13037: 13031: 13028: 13026: 13023: 13021: 13018: 13016: 13013: 13011: 13008: 13006: 13003: 13001: 12998: 12996: 12993: 12991: 12988: 12986: 12983: 12979: 12976: 12975: 12974: 12971: 12969: 12966: 12964: 12961: 12959: 12956: 12954: 12951: 12949: 12946: 12944: 12941: 12939: 12936: 12934: 12931: 12929: 12926: 12924: 12921: 12919: 12918:Marine energy 12916: 12914: 12911: 12909: 12908: 12903: 12901: 12898: 12896: 12893: 12891: 12888: 12886: 12885:Acidification 12883: 12882: 12880: 12876: 12870: 12867: 12865: 12862: 12860: 12857: 12856: 12854: 12850: 12844: 12841: 12839: 12838:SOFAR channel 12836: 12834: 12831: 12829: 12826: 12824: 12821: 12820: 12818: 12816: 12812: 12806: 12803: 12801: 12798: 12796: 12793: 12791: 12788: 12786: 12783: 12781: 12778: 12776: 12773: 12772: 12770: 12768: 12764: 12758: 12755: 12753: 12750: 12748: 12745: 12743: 12740: 12738: 12735: 12733: 12730: 12728: 12725: 12723: 12720: 12718: 12715: 12713: 12710: 12709: 12707: 12703: 12697: 12694: 12692: 12689: 12687: 12684: 12682: 12679: 12677: 12674: 12672: 12669: 12667: 12664: 12662: 12659: 12657: 12654: 12652: 12649: 12647: 12646:Oceanic crust 12644: 12642: 12639: 12637: 12634: 12632: 12629: 12627: 12624: 12622: 12621:Fracture zone 12619: 12617: 12614: 12612: 12609: 12608: 12606: 12604: 12598: 12592: 12589: 12587: 12584: 12582: 12579: 12577: 12574: 12572: 12569: 12567: 12564: 12562: 12559: 12557: 12556:Oceanic basin 12554: 12552: 12549: 12547: 12544: 12542: 12539: 12537: 12534: 12532: 12529: 12527: 12524: 12522: 12519: 12517: 12514: 12512: 12509: 12507: 12504: 12502: 12499: 12497: 12494: 12492: 12489: 12487: 12486:Abyssal plain 12484: 12482: 12479: 12478: 12476: 12474: 12470: 12464: 12461: 12459: 12456: 12454: 12451: 12449: 12446: 12444: 12441: 12439: 12436: 12434: 12431: 12429: 12426: 12424: 12421: 12419: 12416: 12414: 12411: 12409: 12406: 12404: 12401: 12399: 12396: 12394: 12393:Internal tide 12391: 12389: 12386: 12384: 12381: 12379: 12376: 12375: 12373: 12371: 12367: 12361: 12358: 12356: 12353: 12351: 12348: 12346: 12343: 12339: 12336: 12335: 12334: 12331: 12329: 12326: 12324: 12321: 12319: 12316: 12314: 12311: 12309: 12306: 12304: 12301: 12299: 12296: 12294: 12291: 12289: 12288:Ocean current 12286: 12284: 12281: 12279: 12276: 12274: 12271: 12269: 12266: 12264: 12261: 12259: 12256: 12254: 12251: 12249: 12246: 12244: 12241: 12239: 12236: 12234: 12231: 12229: 12226: 12224: 12221: 12219: 12216: 12214: 12211: 12209: 12206: 12204: 12201: 12199: 12196: 12194: 12191: 12189: 12186: 12184: 12181: 12179: 12176: 12174: 12171: 12170: 12168: 12166: 12162: 12157: 12146: 12134: 12131: 12130: 12129: 12126: 12124: 12121: 12119: 12116: 12112: 12109: 12107: 12104: 12103: 12102: 12099: 12097: 12094: 12092: 12089: 12087: 12086:Wave shoaling 12084: 12082: 12079: 12077: 12074: 12072: 12069: 12067: 12064: 12062: 12059: 12057: 12054: 12052: 12049: 12047: 12046:Ursell number 12044: 12042: 12039: 12035: 12032: 12031: 12030: 12027: 12025: 12022: 12020: 12017: 12015: 12012: 12010: 12007: 12005: 12002: 12000: 11997: 11995: 11992: 11990: 11987: 11985: 11982: 11980: 11977: 11975: 11972: 11970: 11967: 11965: 11962: 11960: 11957: 11955: 11952: 11950: 11947: 11945: 11942: 11940: 11937: 11935: 11932: 11930: 11929:Internal wave 11927: 11925: 11922: 11920: 11917: 11915: 11912: 11910: 11907: 11905: 11902: 11900: 11897: 11895: 11892: 11890: 11887: 11885: 11882: 11880: 11879:Breaking wave 11877: 11875: 11872: 11870: 11867: 11865: 11862: 11860: 11857: 11856: 11854: 11852: 11848: 11844: 11837: 11832: 11830: 11825: 11823: 11818: 11817: 11814: 11807: 11803: 11800: 11798:on WikiWaves. 11797: 11794: 11793: 11778:on 2013-05-21 11777: 11773: 11769: 11765: 11761: 11757: 11753: 11752:Truesdell, C. 11749: 11745: 11741: 11737: 11733: 11729: 11723: 11719: 11715: 11711: 11706: 11702: 11698: 11692: 11688: 11684: 11680: 11676: 11672: 11668: 11662: 11658: 11654: 11650: 11649:Landau, L. D. 11646: 11641: 11637: 11633: 11627: 11623: 11622:Hydrodynamics 11619: 11615: 11610: 11606: 11602: 11596: 11592: 11587: 11583: 11579: 11575: 11569: 11565: 11560: 11556: 11552: 11548: 11544: 11540: 11536: 11531: 11530: 11520: 11516: 11515: 11505: 11501: 11497: 11496:Stokes, G. G. 11493: 11488: 11487: 11482: 11478: 11474: 11473: 11455: 11453: 11444: 11438: 11434: 11427: 11419: 11413: 11409: 11402: 11393: 11384: 11382: 11380: 11371: 11367: 11363: 11359: 11355: 11351: 11347: 11343: 11339: 11333: 11324: 11315: 11306: 11297: 11288: 11279: 11270: 11262: 11258: 11254: 11250: 11246: 11242: 11238: 11234: 11230: 11224: 11215: 11213: 11205: 11202: 11196: 11192: 11188: 11184: 11180: 11176: 11170: 11161: 11159: 11149: 11147: 11137: 11128: 11124: 11120: 11114: 11110: 11106: 11105:Whitham, G.B. 11100: 11098: 11096: 11094: 11092: 11084: 11080: 11079: 11072: 11065: 11063:9780080570747 11059: 11055: 11048: 11041: 11035: 11031: 11024: 11015: 11008: 11003: 10994: 10985: 10976: 10974: 10965: 10946: 10939: 10931: 10912: 10901: 10892: 10883: 10881: 10879: 10877: 10867: 10865: 10856: 10852: 10848: 10844: 10840: 10836: 10832: 10828: 10827:J. Fluid Mech 10824: 10818: 10809: 10800: 10792: 10788: 10784: 10778: 10774: 10767: 10765: 10756: 10752: 10748: 10744: 10737: 10735: 10733: 10728: 10718: 10715: 10712: 10709: 10706: 10703: 10700: 10697: 10695: 10691: 10688: 10686: 10682: 10679: 10676: 10675:shallow water 10672: 10669: 10668: 10662: 10659: 10654: 10648: 10644: 10641: 10621: 10616: 10603: 10600: 10597: 10592: 10588: 10579: 10576: 10573: 10567: 10564: 10561: 10558: 10550: 10546: 10542: 10539: 10533: 10530: 10524: 10519: 10496: 10495: 10494: 10492: 10486: 10482: 10477: 10473: 10468: 10464: 10457: 10447: 10445: 10441: 10437: 10433: 10428: 10424: 10401: 10393: 10390: 10384: 10378: 10355: 10354: 10353: 10336: 10333: 10329: 10323: 10320: 10314: 10308: 10298: 10289: 10284: 10280: 10274: 10270: 10265: 10262: 10257: 10250: 10234: 10233: 10232: 10228: 10226: 10218: 10199: 10194: 10186: 10176: 10167: 10157: 10146: 10136: 10127: 10117: 10104: 10094: 10085: 10075: 10064: 10054: 10045: 10035: 10025: 10020: 10003: 9982: 9981: 9980: 9978: 9970: 9965: 9960: 9955: 9949: 9945: 9921: 9918: 9912: 9909: 9906: 9903: 9899: 9890: 9880: 9876: 9872: 9869: 9863: 9860: 9854: 9837: 9820: 9817: 9813: 9809: 9803: 9799: 9781: 9778: 9774: 9767: 9751: 9750: 9749: 9746: 9742: 9738: 9734: 9714: 9711: 9708: 9704: 9686: 9683: 9679: 9675: 9669: 9665: 9661: 9658: 9654: 9647: 9631: 9630: 9629: 9626: 9623: 9617: 9593: 9587: 9584: 9573: 9565: 9544: 9543: 9542: 9539: 9534: 9524: 9521: 9519: 9514: 9509: 9504: 9497: 9488: 9485: 9482: 9474: 9470: 9469: 9464: 9463: 9462: 9460: 9456: 9433: 9428: 9414: 9410: 9406: 9401: 9399: 9391: 9377: 9372: 9367: 9364: 9360: 9356: 9347: 9337: 9331: 9321: 9312: 9308: 9303: 9298: 9295: 9291: 9284: 9282: 9264: 9263: 9262: 9260: 9256: 9251: 9246: 9236: 9233: 9213: 9175: 9170: 9160: 9150: 9146: 9129: 9119: 9115: 9096: 9086: 9082: 9065: 9055: 9051: 9037: 9032: 9030: 9013: 9008: 9002: 8997: 8990: 8985: 8979: 8974: 8972: 8958: 8955: 8948: 8944: 8938: 8934: 8926: 8918: 8914: 8908: 8904: 8896: 8892: 8886: 8882: 8872: 8868: 8862: 8858: 8850: 8846: 8840: 8836: 8829: 8820: 8816: 8812: 8807: 8804: 8800: 8794: 8791: 8786: 8779: 8775: 8769: 8765: 8758: 8749: 8744: 8736: 8733: 8729: 8721: 8718: 8714: 8704: 8701: 8697: 8689: 8686: 8682: 8675: 8670: 8668: 8650: 8649: 8648: 8647:of the form: 8646: 8641: 8634: 8624: 8620: 8612: 8608: 8607: 8602: 8594: 8589: 8584: 8581: 8578: 8573: 8569: 8565: 8560: 8556: 8552: 8547: 8546: 8545: 8528: 8525: 8522: 8518: 8505: 8501: 8493: 8489: 8485: 8481: 8475: 8465: 8456: 8451: 8447: 8441: 8435: 8427: 8414: 8413: 8412: 8410: 8409:wave shoaling 8406: 8402: 8398: 8394: 8388: 8384: 8380: 8376: 8371: 8367: 8362: 8352: 8348: 8344: 8323: 8320: 8317: 8313: 8301: 8295: 8285: 8276: 8271: 8267: 8261: 8255: 8230: 8229: 8228: 8221: 8213: 8201: 8197: 8193: 8183: 8181: 8158: 8153: 8149: 8144: 8138: 8134: 8130: 8127: 8124: 8121: 8117: 8110: 8107: 8101: 8092: 8088: 8079: 8075: 8072: 8065: 8064: 8063: 8046: 8041: 8037: 8032: 8026: 8022: 8018: 8015: 8012: 8009: 8005: 7998: 7995: 7989: 7980: 7976: 7967: 7959: 7958: 7957: 7940: 7935: 7931: 7927: 7924: 7918: 7915: 7909: 7900: 7896: 7887: 7883: 7880: 7873: 7872: 7871: 7861: 7852: 7847: 7828: 7823: 7819: 7815: 7812: 7806: 7803: 7797: 7788: 7780: 7779: 7778: 7776: 7749: 7744: 7740: 7733: 7730: 7727: 7724: 7721: 7715: 7711: 7705: 7699: 7696: 7690: 7688: 7680: 7669: 7664: 7656: 7651: 7645: 7642: 7634: 7629: 7626: 7622: 7618: 7609: 7599: 7593: 7588: 7584: 7580: 7575: 7570: 7564: 7554: 7545: 7539: 7535: 7529: 7526: 7518: 7513: 7510: 7506: 7499: 7497: 7486: 7474: 7473: 7472: 7466: 7460: 7439: 7434: 7430: 7426: 7423: 7417: 7414: 7408: 7397: 7393: 7389: 7386: 7380: 7377: 7368: 7366: 7358: 7349: 7346: 7343: 7338: 7333: 7330: 7326: 7322: 7313: 7304: 7301: 7298: 7293: 7288: 7285: 7281: 7274: 7272: 7261: 7249: 7248: 7247: 7241: 7236: 7232: 7221: 7219: 7212: 7207: 7203: 7196: 7192: 7187: 7183: 7179: 7175: 7151: 7143: 7134: 7131: 7128: 7125: 7121: 7115: 7112: 7108: 7104: 7099: 7095: 7091: 7088: 7082: 7079: 7073: 7068: 7058: 7051: 7048: 7044: 7037: 7025: 7021: 7011: 7005: 7002: 6998: 6991: 6975: 6972: 6970: 6967: 6966: 6951: 6948: 6944: 6934: 6928: 6925: 6921: 6914: 6902: 6896: 6893: 6884: 6868: 6865: 6863: 6860: 6859: 6844: 6841: 6838: 6832: 6823: 6820: 6810: 6807: 6801: 6793: 6784: 6778: 6770: 6757: 6754: 6752: 6750:conservation 6749: 6745: 6744: 6729: 6726: 6704: 6700: 6696: 6693: 6677: 6665: 6659: 6634: 6631: 6629: 6626: 6625: 6610: 6607: 6601: 6593: 6582: 6579: 6575: 6571: 6561: 6553: 6549: 6545: 6542: 6526: 6514: 6508: 6500: 6487: 6484: 6482: 6479: 6478: 6460: 6457: 6454: 6449: 6445: 6436: 6433: 6430: 6424: 6421: 6418: 6415: 6407: 6403: 6399: 6396: 6390: 6387: 6381: 6376: 6366: 6355: 6352: 6336: 6326: 6315: 6313: 6310: 6309: 6289: 6285: 6281: 6278: 6274: 6269: 6266: 6263: 6257: 6254: 6250: 6245: 6242: 6239: 6230: 6220: 6217: 6197: 6187: 6184: 6183: 6166: 6163: 6158: 6155: 6148: 6144: 6140: 6135: 6132: 6125: 6122: 6108: 6101: 6098: 6097: 6079: 6076: 6073: 6070: 6066: 6061: 6056: 6053: 6048: 6034: 6031: 6006: 6004: 6001: 6000: 5985: 5981: 5974: 5971: 5965: 5958: 5954: 5948: 5944: 5938: 5934: 5930: 5925: 5922: 5918: 5910: 5907: 5891: 5888: 5884: 5876: 5873: 5870: 5866: 5863: 5862: 5845: 5841: 5837: 5834: 5828: 5825: 5819: 5816: 5809: 5806: 5803: 5789: 5782: 5779: 5778: 5774: 5771: 5768: 5765: 5764: 5759: 5756: 5747: 5745: 5741: 5737: 5729: 5725: 5715: 5713: 5709: 5705: 5701: 5691: 5683: 5678: 5668: 5644: 5637: 5634: 5630: 5627: 5624: 5620: 5617: 5613: 5610: 5607: 5604: 5601: 5598: 5589: 5586: 5582: 5579: 5573: 5570: 5564: 5558: 5550: 5538: 5537: 5536: 5532: 5528: 5521: 5503: 5494: 5485: 5470: 5457: 5449: 5440: 5413: 5406: 5401: 5396: 5387: 5385: 5379: 5375: 5369: 5367: 5361: 5357:) and above ( 5350: 5326: 5322: 5316: 5312: 5304: 5301: 5297: 5294: 5290: 5285: 5282: 5275: 5272: 5268: 5265: 5259: 5256: 5252: 5249: 5242: 5233: 5225: 5219: 5211: 5199: 5198: 5197: 5193: 5189: 5178: 5176: 5172: 5162: 5155: 5141: 5132: 5117: 5112: 5110: 5105: 5086: 5081: 5077: 5071: 5068: 5063: 5060: 5057: 5048: 5038: 5037: 5036: 5011: 5008: 5005: 5002: 4999: 4995: 4991: 4985: 4981: 4975: 4972: 4967: 4964: 4960: 4956: 4950: 4942: 4930: 4929: 4928: 4926: 4920: 4911: 4879: 4875: 4865: 4847: 4843: 4834: 4809: 4806: 4803: 4797: 4794: 4791: 4788: 4780: 4777: 4774: 4768: 4765: 4762: 4756: 4753: 4750: 4743: 4729: 4726: 4723: 4720: 4717: 4714: 4707: 4693: 4690: 4687: 4682: 4679: 4675: 4670: 4667: 4664: 4657: 4654: 4651: 4634: 4631: 4628: 4625: 4614: 4607: 4604: 4601: 4600: 4585: 4582: 4579: 4573: 4570: 4567: 4564: 4556: 4553: 4550: 4544: 4541: 4538: 4532: 4525: 4511: 4508: 4505: 4500: 4496: 4493: 4490: 4484: 4477: 4463: 4460: 4457: 4452: 4449: 4445: 4440: 4433: 4430: 4413: 4410: 4407: 4404: 4391: 4387: 4379: 4376: 4375: 4360: 4357: 4354: 4348: 4345: 4342: 4339: 4331: 4328: 4325: 4319: 4316: 4313: 4307: 4302: 4292: 4285: 4271: 4268: 4265: 4262: 4256: 4253: 4249: 4242: 4232: 4225: 4211: 4208: 4205: 4200: 4197: 4193: 4188: 4183: 4173: 4166: 4163: 4146: 4143: 4140: 4137: 4124: 4110: 4107: 4106: 4091: 4088: 4085: 4079: 4076: 4073: 4070: 4062: 4059: 4056: 4050: 4047: 4044: 4038: 4035: 4028: 4014: 4011: 4008: 4003: 3999: 3996: 3993: 3987: 3984: 3977: 3963: 3960: 3957: 3952: 3949: 3945: 3940: 3937: 3930: 3927: 3910: 3907: 3904: 3901: 3888: 3884: 3876: 3873: 3872: 3857: 3854: 3851: 3845: 3842: 3839: 3836: 3828: 3825: 3822: 3816: 3813: 3810: 3804: 3801: 3796: 3782: 3768: 3765: 3762: 3759: 3753: 3750: 3742: 3728: 3714: 3711: 3708: 3703: 3700: 3696: 3691: 3688: 3683: 3669: 3666: 3649: 3646: 3643: 3640: 3627: 3613: 3610: 3609: 3593: 3586: 3583: 3580: 3577: 3572: 3569: 3566: 3561: 3557: 3553: 3550: 3544: 3541: 3538: 3535: 3531: 3524: 3521: 3511: 3497: 3490: 3473: 3470: 3460: 3457: 3439: 3435: 3429: 3425: 3415: 3412: 3411: 3395: 3388: 3385: 3382: 3379: 3374: 3371: 3368: 3363: 3359: 3355: 3352: 3346: 3343: 3340: 3337: 3333: 3327: 3323: 3316: 3313: 3303: 3287: 3284: 3275: 3259: 3256: 3248: 3245: 3239: 3233: 3230: 3221: 3218: 3208: 3205: 3188: 3171: 3166: 3162: 3154: 3152: 3149: 3148: 3131: 3128: 3125: 3122: 3117: 3114: 3103: 3087: 3084: 3075: 3059: 3056: 3051: 3045: 3042: 3032: 3029: 3013: 3006: 2994: 2989: 2985: 2977: 2975: 2972: 2971: 2954: 2951: 2948: 2945: 2942: 2939: 2934: 2928: 2915: 2899: 2896: 2891: 2888: 2882: 2869: 2853: 2850: 2845: 2839: 2826: 2824: 2820: 2803: 2790: 2788: 2785: 2784: 2767: 2750: 2734: 2720: 2717: 2716: 2696: 2688: 2685: 2682: 2679: 2667: 2652: 2636: 2631: 2627: 2616: 2612: 2598: 2591: 2588: 2587: 2557: 2554: 2542: 2527: 2511: 2506: 2501: 2492: 2484: 2481: 2477: 2466: 2462: 2448: 2441: 2439: 2435: 2434: 2419: 2416: 2413: 2405: 2391: 2388: 2371: 2368: 2357: 2350: 2347: 2346: 2328: 2325: 2314: 2311: 2308: 2305: 2295: 2278: 2275: 2264: 2257: 2254: 2253: 2239: 2235: 2216: 2210:shallow water 2209: 2205: 2186: 2179: 2176: 2173: 2170: 2169: 2164: 2161: 2154: 2135: 2131: 2111: 2108: 2105: 2100: 2081: 2076: 2075:shallow water 2073: 2070: 2065: 2046: 2041: 2037: 2034: 2033: 2028: 2024: 2021: 2017: 2013: 2006: 2002: 1997: 1994:are the time 1991: 1987: 1980: 1976: 1970: 1968: 1947: 1943: 1934: 1930: 1925: 1920: 1912: 1908: 1904: 1903:Doppler shift 1895: 1890: 1885: 1880: 1874: 1870: 1866: 1854: 1840: 1820: 1797: 1794: 1791: 1788: 1785: 1782: 1779: 1776: 1771: 1767: 1759: 1758: 1757: 1755: 1739: 1735: 1708: 1702: 1699: 1696: 1693: 1690: 1684: 1681: 1675: 1672: 1669: 1666: 1658: 1655: 1652: 1646: 1643: 1640: 1634: 1629: 1626: 1621: 1611: 1610: 1609: 1592: 1586: 1583: 1580: 1577: 1574: 1568: 1565: 1562: 1559: 1556: 1549: 1548: 1547: 1545: 1544:monochromatic 1539: 1529: 1525: 1515: 1495: 1494: 1489: 1488: 1478: 1471: 1469: 1455: 1449: 1446: 1443: 1437: 1434: 1431: 1422: 1419: 1416: 1413: 1410: 1404: 1383: 1382: 1379: 1377: 1372: 1366: 1362: 1358: 1348: 1341: 1339: 1325: 1319: 1316: 1313: 1307: 1304: 1301: 1289: 1272: 1266: 1258: 1245: 1244: 1241: 1239: 1238:infinitesimal 1234: 1230: 1225: 1221: 1212: 1205: 1203: 1189: 1186: 1183: 1180: 1177: 1168: 1165: 1159: 1138: 1137: 1134: 1132: 1127: 1125: 1121: 1113: 1106: 1104: 1090: 1087: 1079: 1075: 1061: 1050: 1042: 1038: 1024: 1009: 1008: 1005: 1003: 995: 976: 970: 953: 948: 944: 930: 913: 908: 904: 896: 895: 894: 870: 869:flow velocity 864: 860: 856: 851: 847: 843: 839: 834: 830: 825: 819: 804: 800: 796: 792: 784: 769: 767: 762: 759: 749: 741: 737: 720: 717: 711: 708: 702: 699: 689: 686: 683: 680: 673: 672: 671: 669: 665: 658: 653: 651: 616: 611: 608: 603: 598: 595: 590: 585: 581: 573: 572: 571: 563: 554: 546: 538: 529: 521: 514: 510: 503: 495: 487: 478: 470: 466: 462: 455: 452: 445: 442: 435: 434: 433: 415: 411: 408: 405: 402: 399: 395: 391: 388: 385: 382: 376: 373: 370: 364: 357: 356: 355: 345: 339: 335: 331: 325: 323: 319: 315: 311: 307: 303: 299: 295: 291: 287: 283: 279: 274: 272: 268: 267:wave shoaling 264: 260: 256: 252: 248: 244: 240: 235: 233: 229: 225: 221: 217: 213: 209: 198: 190: 186: 182: 173: 165: 153: 145: 141: 137: 133: 125: 116: 114: 113:shallow water 110: 105: 103: 99: 96: 92: 88: 84: 81: 77: 73: 68: 66: 62: 58: 54: 50: 46: 42: 41:gravity waves 38: 34: 30: 26: 22: 13025:Water column 12973:Oceanography 12948:Observations 12943:Explorations 12913:Marginal sea 12906: 12864:OSTM/Jason-2 12696:Volcanic arc 12671:Slab suction 12388:Head of tide 12278:Loop Current 12218:Ekman spiral 12004:Stokes drift 11914:Gravity wave 11889:Cnoidal wave 11858: 11780:, retrieved 11776:the original 11763: 11759: 11717: 11686: 11656: 11621: 11590: 11563: 11538: 11534: 11513: 11503: 11499: 11485: 11432: 11426: 11407: 11401: 11392: 11345: 11341: 11332: 11323: 11314: 11305: 11296: 11287: 11278: 11269: 11236: 11232: 11223: 11203: 11200: 11186: 11182: 11169: 11136: 11108: 11077: 11071: 11053: 11047: 11029: 11023: 11014: 11002: 10993: 10984: 10963: 10944: 10938: 10929: 10910: 10900: 10891: 10830: 10826: 10817: 10808: 10799: 10772: 10746: 10742: 10690:Cnoidal wave 10657: 10646: 10642: 10639: 10636: 10484: 10480: 10472:Stokes drift 10466: 10462: 10459: 10456:Stokes drift 10450:Stokes drift 10444:irrotational 10426: 10416: 10351: 10229: 10214: 9963: 9953: 9947: 9943: 9936: 9747: 9740: 9736: 9732: 9729: 9627: 9621: 9615: 9608: 9541:defined as: 9537: 9530: 9522: 9512: 9508:wave channel 9505: 9495: 9492: 9486: 9465: 9452: 9249: 9242: 9231: 9211: 9194: 8639: 8632: 8622: 8618: 8610: 8605: 8604: 8598: 8587: 8571: 8563: 8558: 8554: 8550: 8543: 8386: 8382: 8378: 8374: 8369: 8365: 8350: 8346: 8342: 8338: 8219: 8211: 8189: 8173: 8061: 7955: 7859: 7850: 7843: 7768: 7464: 7461: 7458: 7239: 7227: 7210: 7205: 7201: 7194: 7190: 7171: 6312:Stokes drift 5753: 5724:second-order 5721: 5697: 5681: 5666: 5660:where again 5659: 5530: 5526: 5519: 5512: 5501: 5492: 5480: 5455: 5447: 5438: 5420:and density 5411: 5409:and density 5404: 5377: 5373: 5370: 5365: 5359: 5348: 5341: 5191: 5187: 5184: 5168: 5153: 5130: 5113: 5101: 5026: 4922: 4909: 4877: 4873: 4863: 4845: 4841: 4605:oscillation 2233: 2214: 2203: 2184: 2159: 2152: 2133: 2129: 2109: 2098: 2079: 2074: 2063: 2044: 2035: 2022: 2015: 2011: 2004: 2000: 1989: 1985: 1978: 1974: 1971: 1945: 1941: 1932: 1928: 1923: 1910: 1906: 1893: 1883: 1872: 1868: 1864: 1860: 1839:eigenproblem 1812: 1723: 1607: 1541: 1523: 1513: 1491: 1485: 1483: 1472: 1364: 1360: 1356: 1353: 1342: 1235: 1228: 1217: 1206: 1128: 1118: 1107: 991: 862: 858: 854: 842:irrotational 832: 828: 817: 802: 798: 794: 790: 780: 763: 754: 735: 654: 631: 559: 552: 544: 536: 527: 493: 485: 476: 439:is the wave 431: 337: 333: 329: 326: 322:irrotational 275: 236: 205: 196: 188: 184: 180: 171: 163: 151: 106: 95:second-order 69: 61:irrotational 28: 24: 18: 13091:Water waves 13015:Thermocline 12732:Mesopelagic 12705:Ocean zones 12676:Slab window 12541:Hydrography 12481:Abyssal fan 12448:Tidal range 12438:Tidal power 12433:Tidal force 12318:Rip current 12253:Gulf Stream 12213:Ekman layer 12203:Downwelling 12178:Baroclinity 12165:Circulation 12061:Wave height 12051:Wave action 12034:megatsunami 12014:Stokes wave 11974:Rossby wave 11939:Kelvin wave 11919:Green's law 11802:Water waves 11477:Airy, G. B. 11348:: 331–347. 10833:: 667–681. 10711:Stokes wave 8359:the action 8200:wave action 6003:wave action 5469:phase speed 5109:surfactants 3151:group speed 2974:phase speed 2348:wave phase 2069:phase speed 1935:), denoted 1889:wave crests 1877:plane. The 871:components 824:impermeable 657:wave height 644:and period 562:phase speed 310:third order 259:Diffraction 146:divided by 119:Description 109:wave height 37:propagation 13085:Categories 12953:Reanalysis 12852:Satellites 12833:Sofar bomb 12681:Subduction 12656:Ridge push 12551:Ocean bank 12531:Contourite 12458:Tide gauge 12443:Tidal race 12428:Tidal bore 12418:Slack tide 12383:Earth tide 12303:Ocean gyre 12123:Wind setup 12118:Wind fetch 12081:Wave setup 12076:Wave radar 12071:Wave power 11969:Rogue wave 11899:Dispersion 11782:2013-05-05 11748:Flügge, S. 11506:: 441–455. 11470:Historical 11465:References 10717:Wave power 10653:integrated 10423:wavenumber 9533:bathymetry 8609: : (∇ 8593:shear rate 8196:bathymetry 7174:bathymetry 5744:bathymetry 5708:baroclinic 5700:barotropic 5386:develops. 5104:decimeters 2180:deep water 2040:wavelength 2036:deep water 1881:vector is 1879:wavenumber 1536:See also: 783:coordinate 650:dispersive 469:wavelength 443:in metres, 344:sinusoidal 302:non-linear 271:refraction 224:turbulence 136:Dispersion 87:kinematics 83:sea states 49:fluid flow 33:linearised 31:) gives a 12815:Acoustics 12767:Sea level 12666:Slab pull 12603:tectonics 12511:Cold seep 12473:Landforms 12350:Whirlpool 12345:Upwelling 12128:Wind wave 12056:Wave base 11984:Sea state 11904:Edge wave 11894:Cross sea 11772:612422741 11261:202575349 11189:: 21–26. 11131:, p. 559. 10855:113694460 10598:⁡ 10562:⁡ 10540:σ 10513:¯ 10391:ω 10388:∇ 10376:∂ 10366:∂ 10324:σ 10281:⋅ 10278:∇ 10266:σ 10248:∂ 10244:∂ 10180:~ 10161:~ 10140:~ 10121:~ 10098:~ 10079:~ 10058:~ 10039:~ 10015:~ 10004:⊗ 9998:~ 9916:∇ 9907:ρ 9870:ρ 9849:~ 9838:⊗ 9832:~ 9818:ρ 9810:⋅ 9807:∇ 9793:~ 9779:ρ 9765:∂ 9761:∂ 9698:~ 9684:ρ 9676:⋅ 9673:∇ 9659:ρ 9645:∂ 9641:∂ 9585:ρ 9560:~ 9378:ρ 9365:− 9361:∫ 9357:− 9352:¯ 9309:ρ 9304:η 9296:− 9292:∫ 9259:transport 9255:mass flux 9158:∂ 9143:∂ 9127:∂ 9112:∂ 9094:∂ 9079:∂ 9063:∂ 9048:∂ 9021:∇ 8787:− 8643:are in a 8510:∇ 8448:⋅ 8445:∇ 8433:∂ 8425:∂ 8268:⋅ 8265:∇ 8253:∂ 8241:∂ 8131:γ 8122:ρ 8019:γ 8010:ρ 7925:ρ 7813:ρ 7728:⁡ 7712:σ 7706:ρ 7652:ρ 7627:− 7623:∫ 7619:− 7614:¯ 7536:ρ 7511:− 7507:∫ 7424:ρ 7404:¯ 7394:η 7387:ρ 7344:ρ 7331:− 7327:∫ 7323:− 7318:¯ 7299:ρ 7294:η 7286:− 7282:∫ 7178:mean flow 7149:∂ 7141:∂ 7129:ρ 7089:ρ 7062:~ 7049:ρ 7035:∂ 7031:∂ 7015:~ 7003:ρ 6989:∂ 6985:∂ 6938:~ 6926:ρ 6912:∂ 6908:∂ 6894:ρ 6882:∂ 6878:∂ 6827:Ω 6821:ω 6799:∂ 6794:ω 6791:∂ 6776:∂ 6768:∂ 6675:∂ 6671:∂ 6657:∂ 6645:∂ 6599:∂ 6591:∂ 6524:∂ 6520:∂ 6506:∂ 6498:∂ 6455:⁡ 6419:⁡ 6397:σ 6370:¯ 6330:¯ 6279:ρ 6255:ρ 6234:~ 6201:~ 6167:σ 6074:− 6071:ω 6057:σ 5966:− 5835:ρ 5766:quantity 5736:forecasts 5728:quadratic 5712:antiphase 5704:amplitude 5628:⁡ 5618:ρ 5605:⁡ 5599:ρ 5587:ρ 5583:− 5580:ρ 5547:Ω 5302:ρ 5295:ρ 5291:γ 5273:ρ 5266:ρ 5257:ρ 5253:− 5250:ρ 5208:Ω 5171:interface 5072:ρ 5069:γ 5052:~ 5003:⁡ 4976:ρ 4973:γ 4939:Ω 4810:θ 4807:⁡ 4792:⁡ 4766:⁡ 4751:ρ 4730:θ 4727:⁡ 4715:ρ 4694:θ 4691:⁡ 4665:ρ 4586:θ 4583:⁡ 4568:⁡ 4542:⁡ 4512:θ 4509:⁡ 4464:θ 4461:⁡ 4388:ξ 4361:θ 4358:⁡ 4343:⁡ 4317:⁡ 4293:− 4272:θ 4269:⁡ 4233:− 4212:θ 4209:⁡ 4174:− 4120:ξ 4092:θ 4089:⁡ 4074:⁡ 4048:⁡ 4036:σ 4015:θ 4012:⁡ 3985:σ 3964:θ 3961:⁡ 3938:σ 3858:θ 3855:⁡ 3840:⁡ 3814:⁡ 3802:σ 3769:θ 3766:⁡ 3715:θ 3712:⁡ 3689:σ 3581:⁡ 3567:⁡ 3554:− 3383:⁡ 3369:⁡ 3356:− 3260:σ 3186:∂ 3181:Ω 3178:∂ 3126:⁡ 3060:σ 3001:Ω 2949:⁡ 2923:Ω 2877:Ω 2834:Ω 2798:Ω 2697:⋅ 2689:− 2686:ω 2680:σ 2647:Ω 2628:σ 2599:σ 2522:Ω 2493:⋅ 2485:− 2482:ω 2449:ω 2436:observed 2417:ω 2414:− 2406:⋅ 2358:θ 2315:θ 2312:⁡ 2265:η 2171:quantity 1996:integrals 1789:⁡ 1768:ω 1700:ω 1697:− 1685:⁡ 1670:⁡ 1644:⁡ 1627:ω 1619:Φ 1584:ω 1581:− 1569:⁡ 1557:η 1528:is used. 1438:η 1417:η 1402:∂ 1397:Φ 1394:∂ 1308:η 1287:∂ 1282:Φ 1279:∂ 1264:∂ 1259:η 1256:∂ 1184:− 1157:∂ 1152:Φ 1149:∂ 1131:kinematic 1072:∂ 1067:Φ 1058:∂ 1035:∂ 1030:Φ 1021:∂ 968:∂ 963:Φ 960:∂ 928:∂ 923:Φ 920:∂ 609:λ 596:ω 520:frequency 453:function, 441:amplitude 409:ω 406:− 392:⁡ 365:η 350:and time 220:vorticity 216:viscosity 98:nonlinear 13048:Category 13000:Seawater 12727:Littoral 12722:Deep sea 12581:Seamount 12463:Tideline 12408:Rip tide 12338:shutdown 12308:Overflow 12041:Undertow 11884:Clapotis 11754:(eds.), 11716:(1977). 11685:(1978). 11675:15017127 11655:(1986). 11640:30070401 11620:(1994). 11618:Lamb, H. 11609:36126836 11582:22907242 11541:: 1–28. 11370:18232994 11177:(1877). 11107:(1974). 11007:Lamb, H. 10791:45200228 10749:: 1–28. 10665:See also 10436:coherent 10217:momentum 9518:undertow 9468:celerity 9257:or mass 9245:momentum 7218:Eulerian 6755:rad·m·s 5869:momentum 5775:formula 5722:Several 5638:′ 5621:′ 5590:′ 5467:and the 5400:inviscid 5305:′ 5276:′ 5260:′ 4603:pressure 1220:infinite 766:pressure 314:Gerstner 255:spectrum 91:dynamics 53:inviscid 13058:Commons 12928:Mooring 12878:Related 12869:Jason-3 12859:Jason-1 12742:Pelagic 12737:Oceanic 12712:Benthic 12029:Tsunami 11999:Soliton 11736:7319931 11709:504 pp. 11705:2966533 11543:Bibcode 11350:Bibcode 11241:Bibcode 10961:⁠ 10949:⁠ 10927:⁠ 10915:⁠ 10835:Bibcode 9975:is the 9967:is the 9481:troughs 8595:tensor. 8575:is the 8224:⁠ 8208:⁠ 7231:kinetic 7182:Whitham 6866:kg·m·s 6123:kg·m·s 5769:symbol 5675:is the 5506:⁠ 5488:⁠ 5460:⁠ 5443:⁠ 5175:density 5158:⁠ 5144:⁠ 5135:⁠ 5120:⁠ 4923:Due to 4871:√ 4868:⁠ 4854:⁠ 4839:√ 2231:⁠ 2219:⁠ 2201:⁠ 2189:⁠ 2174:symbol 2150:⁠ 2138:⁠ 2127:⁠ 2115:⁠ 2096:⁠ 2084:⁠ 2061:⁠ 2049:⁠ 1841:. When 1518:⁠ 1503:⁠ 1490:) and ( 1376:dynamic 807:, with 758:average 549:⁠ 532:⁠ 507:is the 498:⁠ 481:⁠ 465:radians 459:is the 449:is the 290:Kelland 282:Poisson 278:Laplace 194:√ 176:⁠ 160:⁠ 149:√ 102:tsunami 12747:Photic 12576:Seabed 11989:Seiche 11770: 11750:& 11734: 11724: 11703: 11693: 11673: 11663: 11638: 11628: 11607: 11597: 11580: 11570: 11439: 11414: 11368: 11259: 11127:815118 11125: 11115: 11060: 11036: 10853: 10789: 10779: 10491:period 9959:tensor 9730:where 9455:Stokes 9261:– is: 8580:tensor 8544:with: 6485:J·m·s 6032:J·s·m 5772:units 5478:(with 5342:where 4907:Note: 3413:ratio 2177:units 2104:period 2067:, the 668:trough 513:period 451:cosine 432:where 298:Stokes 292:. But 286:Cauchy 247:random 80:random 12938:Ocean 12907:Alvin 12757:Swash 12601:Plate 12546:Knoll 12536:Guyot 12491:Atoll 12370:Tides 12133:model 12019:Swell 11851:Waves 11521:–229. 11366:S2CID 11257:S2CID 10947:< 10913:> 10851:S2CID 10724:Notes 9937:with 9483:, and 9195:with 8339:with 8174:with 7769:with 6748:crest 6746:wave- 5376:> 5027:with 2242:(all 2217:< 2187:> 2136:< 2132:< 2082:< 2047:> 1913:) is 1813:with 1724:with 1224:limit 664:crest 140:Phase 45:fluid 12905:DSV 12890:Argo 12752:Surf 12208:Eddy 11768:OCLC 11732:OCLC 11722:ISBN 11701:OCLC 11691:ISBN 11671:OCLC 11661:ISBN 11636:OCLC 11626:ISBN 11605:OCLC 11595:ISBN 11578:OCLC 11568:ISBN 11437:ISBN 11412:ISBN 11123:OCLC 11113:ISBN 11083:NOAA 11058:ISBN 11034:ISBN 10787:OCLC 10777:ISBN 10589:sinh 10559:cosh 10440:curl 9971:and 9222:and 9202:and 8636:and 8399:and 8391:the 8363:and 8361:flux 8178:the 7857:and 7725:tanh 7233:and 6973:N·m 6816:with 6632:J·m 6446:sinh 6416:cosh 6353:m·s 6218:m·s 5908:N·m 5872:flux 5673:coth 5664:and 5625:coth 5602:coth 5529:= Ω( 5346:and 5190:= Ω( 5114:The 5000:tanh 4789:cosh 4763:cosh 4565:sinh 4539:sinh 4340:sinh 4314:cosh 4071:sinh 4045:sinh 3928:m·s 3837:sinh 3811:cosh 3667:m·s 3578:tanh 3558:tanh 3380:tanh 3360:tanh 3206:m·s 3123:tanh 3030:m·s 2946:tanh 2821:rad· 2613:rad· 2463:rad· 2246:and 2009:and 1983:and 1953:and 1931:(or 1909:(or 1845:and 1817:the 1815:tanh 1786:tanh 1744:and 1736:and 1732:the 1730:cosh 1728:and 1726:sinh 1667:sinh 1641:cosh 1500:and 1231:→ −∞ 878:and 840:and 666:and 518:and 300:for 294:Airy 288:and 269:and 241:and 210:(or 142:and 89:and 74:and 59:and 11806:MIT 11804:at 11551:doi 11519:197 11358:doi 11346:106 11249:doi 11237:302 11191:doi 11081:of 10843:doi 10831:173 10751:doi 10651:is 9951:), 9946:= – 8582:and 8097:kin 8084:pot 8062:so 7985:kin 7972:pot 7905:kin 7892:pot 7863:kin 7854:pot 7793:kin 7491:kin 7468:kin 7266:pot 7243:pot 7204:= − 7184:'s 5807:·m 5471:by 5436:by 4804:cos 4724:cos 4688:cos 4655:·m 4580:cos 4506:cos 4458:cos 4355:sin 4266:sin 4206:sin 4086:sin 4009:sin 3958:sin 3852:cos 3763:cos 3709:cos 2390:rad 2309:cos 1867:= ( 1682:sin 1566:cos 1526:= 0 939:and 831:= − 820:= 0 695:and 551:= 2 525:by 474:by 463:in 447:cos 389:cos 226:or 51:is 39:of 19:In 13087:: 11762:, 11758:, 11730:. 11699:. 11669:. 11651:; 11634:. 11603:. 11576:. 11549:. 11539:36 11537:. 11502:. 11451:^ 11378:^ 11364:. 11356:. 11344:. 11255:. 11247:. 11235:. 11211:^ 11185:. 11181:. 11157:^ 11145:^ 11121:. 11090:^ 10972:^ 10958:20 10875:^ 10863:^ 10849:. 10841:. 10829:. 10785:. 10763:^ 10747:36 10745:. 10731:^ 10465:= 10337:0. 10227:. 9979:: 9961:, 9520:. 9235:. 8621:= 8553:+ 8373:= 8354:) 8345:+ 8227:: 8206:= 8182:. 5690:. 5509:). 5491:Ω( 5486:= 5441:= 5177:. 5123:∂Ω 5111:. 4878:ρg 4876:/ 4844:/ 4842:gσ 4431:m 4164:m 3458:– 2751:– 2296:m 2250:) 2237:) 2228:20 2207:) 2124:20 2093:20 2042:, 1944:= 1506:∂Φ 1233:. 1091:0. 1004:: 861:, 857:, 853:Φ( 793:= 570:: 553:πf 530:= 479:= 354:: 324:. 284:, 280:, 265:, 222:, 218:, 197:gh 179:. 152:gh 55:, 23:, 11835:e 11828:t 11821:v 11808:. 11764:9 11738:. 11707:. 11677:. 11642:. 11611:. 11584:. 11557:. 11553:: 11545:: 11504:8 11445:. 11420:. 11372:. 11360:: 11352:: 11263:. 11251:: 11243:: 11204:1 11197:. 11193:: 11187:9 11129:. 10967:. 10964:λ 10955:/ 10952:1 10945:h 10933:. 10930:λ 10924:2 10921:/ 10918:1 10911:h 10906:h 10857:. 10845:: 10837:: 10793:. 10757:. 10753:: 10677:. 10658:M 10647:S 10643:ū 10640:ρ 10622:, 10617:k 10612:e 10604:h 10601:k 10593:2 10583:) 10580:h 10577:+ 10574:z 10571:( 10568:k 10565:2 10551:2 10547:a 10543:k 10534:2 10531:1 10525:= 10520:S 10509:u 10485:S 10481:ū 10467:0 10463:U 10427:k 10419:ω 10402:, 10398:0 10394:= 10385:+ 10379:t 10370:k 10334:= 10330:] 10321:E 10315:) 10309:g 10304:c 10299:+ 10295:U 10290:( 10285:[ 10275:+ 10271:) 10263:E 10258:( 10251:t 10221:d 10200:. 10195:) 10187:y 10177:U 10168:y 10158:U 10147:x 10137:U 10128:y 10118:U 10105:y 10095:U 10086:x 10076:U 10065:x 10055:U 10046:x 10036:U 10026:( 10021:= 10011:U 9994:U 9973:⊗ 9964:I 9954:S 9948:d 9944:z 9939:d 9922:, 9919:d 9913:h 9910:g 9904:= 9900:) 9895:S 9891:+ 9887:I 9881:2 9877:h 9873:g 9864:2 9861:1 9855:+ 9845:U 9828:U 9821:h 9814:( 9804:+ 9800:) 9789:U 9782:h 9775:( 9768:t 9743:) 9741:t 9739:, 9737:x 9735:( 9733:h 9715:, 9712:0 9709:= 9705:) 9694:U 9687:h 9680:( 9670:+ 9666:) 9662:h 9655:( 9648:t 9622:Ũ 9616:U 9611:h 9594:. 9588:h 9580:M 9574:+ 9570:U 9566:= 9556:U 9538:Ũ 9513:U 9501:E 9496:M 9477:z 9434:, 9429:k 9424:e 9415:p 9411:c 9407:E 9402:= 9392:z 9388:d 9382:U 9373:0 9368:h 9348:z 9344:d 9338:) 9332:x 9327:u 9322:+ 9318:U 9313:( 9299:h 9285:= 9277:M 9250:M 9232:U 9226:y 9224:U 9219:x 9217:U 9212:k 9206:y 9204:k 9199:x 9197:k 9176:, 9171:) 9161:y 9151:y 9147:U 9130:y 9120:x 9116:U 9097:x 9087:y 9083:U 9066:x 9056:x 9052:U 9038:( 9033:= 9025:U 9014:, 9009:) 9003:1 8998:0 8991:0 8986:1 8980:( 8975:= 8967:I 8959:, 8956:E 8949:p 8945:c 8939:g 8935:c 8927:) 8919:y 8915:k 8909:y 8905:k 8897:x 8893:k 8887:y 8883:k 8873:y 8869:k 8863:x 8859:k 8851:x 8847:k 8841:x 8837:k 8830:( 8821:2 8817:k 8813:1 8808:+ 8805:E 8801:) 8795:2 8792:1 8780:p 8776:c 8770:g 8766:c 8759:( 8754:I 8750:= 8745:) 8737:y 8734:y 8730:S 8722:x 8719:y 8715:S 8705:y 8702:x 8698:S 8690:x 8687:x 8683:S 8676:( 8671:= 8663:S 8640:U 8638:∇ 8633:S 8628:E 8623:0 8619:U 8617:∇ 8613:) 8611:U 8606:S 8588:U 8586:∇ 8572:S 8564:E 8562:) 8559:g 8555:c 8551:U 8549:( 8529:, 8526:0 8523:= 8519:) 8514:U 8506:( 8502:: 8498:S 8494:+ 8490:] 8486:E 8482:) 8476:g 8471:c 8466:+ 8462:U 8457:( 8452:[ 8442:+ 8436:t 8428:E 8387:k 8383:e 8379:g 8375:c 8370:g 8366:c 8356:A 8351:g 8347:c 8343:U 8341:( 8324:, 8321:0 8318:= 8314:] 8308:A 8302:) 8296:g 8291:c 8286:+ 8282:U 8277:( 8272:[ 8262:+ 8256:t 8246:A 8220:σ 8216:/ 8212:E 8204:A 8176:γ 8159:, 8154:2 8150:a 8145:) 8139:2 8135:k 8128:+ 8125:g 8118:( 8111:2 8108:1 8102:= 8093:E 8089:+ 8080:E 8076:= 8073:E 8047:, 8042:2 8038:a 8033:) 8027:2 8023:k 8016:+ 8013:g 8006:( 7999:4 7996:1 7990:= 7981:E 7977:= 7968:E 7941:. 7936:2 7932:a 7928:g 7919:2 7916:1 7910:= 7901:E 7897:+ 7888:E 7884:= 7881:E 7868:E 7860:E 7851:E 7829:. 7824:2 7820:a 7816:g 7807:4 7804:1 7798:= 7789:E 7771:σ 7750:, 7745:2 7741:a 7734:h 7731:k 7722:k 7716:2 7700:4 7697:1 7691:= 7681:z 7677:d 7670:2 7665:| 7661:U 7657:| 7646:2 7643:1 7635:0 7630:h 7610:z 7606:d 7600:] 7594:2 7589:z 7585:u 7581:+ 7576:2 7571:| 7565:x 7560:u 7555:+ 7551:U 7546:| 7540:[ 7530:2 7527:1 7519:0 7514:h 7500:= 7487:E 7465:E 7440:. 7435:2 7431:a 7427:g 7418:4 7415:1 7409:= 7398:2 7390:g 7381:2 7378:1 7369:= 7359:z 7355:d 7350:z 7347:g 7339:0 7334:h 7314:z 7310:d 7305:z 7302:g 7289:h 7275:= 7262:E 7240:E 7214:Ũ 7206:d 7202:z 7197:) 7195:x 7193:( 7191:d 7152:x 7144:d 7135:h 7132:g 7126:= 7122:) 7116:x 7113:x 7109:S 7105:+ 7100:2 7096:h 7092:g 7083:2 7080:1 7074:+ 7069:2 7059:U 7052:h 7045:( 7038:x 7026:+ 7022:) 7012:U 7006:h 6999:( 6992:t 6952:0 6949:= 6945:) 6935:U 6929:h 6922:( 6915:x 6903:+ 6900:) 6897:h 6891:( 6885:t 6845:U 6842:k 6839:+ 6836:) 6833:k 6830:( 6824:= 6811:0 6808:= 6802:x 6785:+ 6779:t 6771:k 6730:0 6727:= 6722:) 6715:A 6710:) 6705:g 6701:c 6697:+ 6694:U 6691:( 6686:( 6678:x 6666:+ 6660:t 6650:A 6611:0 6608:= 6602:x 6594:U 6583:x 6580:x 6576:S 6572:+ 6567:) 6562:E 6559:) 6554:g 6550:c 6546:+ 6543:U 6540:( 6535:( 6527:x 6515:+ 6509:t 6501:E 6461:h 6458:k 6450:2 6440:) 6437:h 6434:+ 6431:z 6428:( 6425:k 6422:2 6408:2 6404:a 6400:k 6391:2 6388:1 6382:= 6377:S 6367:u 6337:S 6327:u 6290:p 6286:c 6282:h 6275:E 6270:+ 6267:U 6264:= 6258:h 6251:M 6246:+ 6243:U 6240:= 6231:U 6198:U 6164:E 6159:k 6156:= 6149:p 6145:c 6141:E 6136:= 6133:M 6109:M 6080:U 6077:k 6067:E 6062:= 6054:E 6049:= 6044:A 6016:A 5986:E 5982:) 5975:2 5972:1 5959:p 5955:c 5949:g 5945:c 5939:2 5935:( 5931:= 5926:x 5923:x 5919:S 5892:x 5889:x 5885:S 5846:2 5842:a 5838:g 5829:2 5826:1 5820:= 5817:E 5805:J 5790:E 5732:a 5688:h 5684:′ 5682:ρ 5669:′ 5667:ρ 5662:ρ 5645:, 5635:h 5631:k 5614:+ 5611:h 5608:k 5594:) 5577:( 5574:k 5571:g 5565:= 5562:) 5559:k 5556:( 5551:2 5533:) 5531:k 5527:ω 5522:′ 5520:h 5515:h 5502:k 5498:/ 5495:) 5493:k 5483:p 5481:c 5475:p 5473:c 5465:g 5456:λ 5452:/ 5448:π 5446:2 5439:k 5434:k 5430:λ 5426:a 5422:ρ 5418:h 5414:′ 5412:ρ 5407:′ 5405:h 5380:′ 5378:ρ 5374:ρ 5366:γ 5362:′ 5360:ρ 5355:ρ 5351:′ 5349:ρ 5344:ρ 5327:, 5323:) 5317:2 5313:k 5298:+ 5286:+ 5283:g 5269:+ 5243:( 5238:| 5234:k 5230:| 5226:= 5223:) 5220:k 5217:( 5212:2 5194:) 5192:k 5188:ω 5154:k 5150:/ 5147:Ω 5131:k 5129:∂ 5126:/ 5087:. 5082:2 5078:k 5064:+ 5061:g 5058:= 5049:g 5033:g 5029:γ 5012:, 5009:h 5006:k 4996:k 4992:) 4986:2 4982:k 4968:+ 4965:g 4961:( 4957:= 4954:) 4951:k 4948:( 4943:2 4910:σ 4899:. 4896:g 4894:c 4889:p 4887:c 4883:. 4874:σ 4864:λ 4860:/ 4857:1 4846:ρ 4798:h 4795:k 4784:) 4781:h 4778:+ 4775:z 4772:( 4769:k 4757:a 4754:g 4721:a 4718:g 4683:z 4680:k 4676:e 4671:a 4668:g 4653:N 4638:) 4635:t 4632:, 4629:z 4626:, 4622:x 4618:( 4615:p 4574:h 4571:k 4560:) 4557:h 4554:+ 4551:z 4548:( 4545:k 4533:a 4501:h 4497:h 4494:+ 4491:z 4485:a 4453:z 4450:k 4446:e 4441:a 4417:) 4414:t 4411:, 4408:z 4405:, 4401:x 4397:( 4392:z 4349:h 4346:k 4335:) 4332:h 4329:+ 4326:z 4323:( 4320:k 4308:a 4303:k 4298:e 4263:a 4257:h 4254:k 4250:1 4243:k 4238:e 4201:z 4198:k 4194:e 4189:a 4184:k 4179:e 4150:) 4147:t 4144:, 4141:z 4138:, 4134:x 4130:( 4125:x 4080:h 4077:k 4066:) 4063:h 4060:+ 4057:z 4054:( 4051:k 4039:a 4004:h 4000:h 3997:+ 3994:z 3988:a 3953:z 3950:k 3946:e 3941:a 3914:) 3911:t 3908:, 3905:z 3902:, 3898:x 3894:( 3889:z 3885:u 3846:h 3843:k 3832:) 3829:h 3826:+ 3823:z 3820:( 3817:k 3805:a 3797:k 3792:e 3760:a 3754:h 3751:g 3743:k 3738:e 3704:z 3701:k 3697:e 3692:a 3684:k 3679:e 3653:) 3650:t 3647:, 3644:z 3641:, 3637:x 3633:( 3628:x 3623:u 3594:) 3587:h 3584:k 3573:h 3570:k 3562:2 3551:1 3545:h 3542:k 3539:+ 3536:1 3532:( 3525:2 3522:1 3498:1 3474:2 3471:1 3440:p 3436:c 3430:g 3426:c 3396:) 3389:h 3386:k 3375:h 3372:k 3364:2 3353:1 3347:h 3344:k 3341:+ 3338:1 3334:( 3328:p 3324:c 3317:2 3314:1 3288:h 3285:g 3257:g 3249:2 3246:1 3240:= 3234:k 3231:g 3222:2 3219:1 3189:k 3172:= 3167:g 3163:c 3132:h 3129:k 3118:k 3115:g 3088:h 3085:g 3057:g 3052:= 3046:k 3043:g 3014:k 3010:) 3007:k 3004:( 2995:= 2990:p 2986:c 2955:h 2952:k 2943:k 2940:g 2935:= 2932:) 2929:k 2926:( 2900:h 2897:g 2892:k 2889:= 2886:) 2883:k 2880:( 2854:k 2851:g 2846:= 2843:) 2840:k 2837:( 2823:s 2807:) 2804:k 2801:( 2768:k 2764:k 2735:k 2730:e 2701:U 2693:k 2683:= 2668:2 2662:) 2656:) 2653:k 2650:( 2642:( 2637:= 2632:2 2615:s 2572:| 2567:k 2562:| 2558:= 2555:k 2543:2 2537:) 2531:) 2528:k 2525:( 2517:( 2512:= 2507:2 2502:) 2497:U 2489:k 2478:( 2465:s 2420:t 2410:x 2402:k 2375:) 2372:t 2369:, 2365:x 2361:( 2332:) 2329:t 2326:, 2322:x 2318:( 2306:a 2282:) 2279:t 2276:, 2272:x 2268:( 2248:h 2244:λ 2234:λ 2225:/ 2222:1 2215:h 2212:( 2204:λ 2198:2 2195:/ 2192:1 2185:h 2182:( 2153:λ 2147:2 2144:/ 2141:1 2134:h 2130:λ 2121:/ 2118:1 2099:λ 2090:/ 2087:1 2080:h 2064:λ 2058:2 2055:/ 2052:1 2045:h 2016:z 2012:u 2005:x 2001:u 1990:z 1986:ξ 1979:x 1975:ξ 1963:λ 1959:k 1955:σ 1951:ω 1946:0 1942:U 1937:σ 1924:U 1915:ω 1899:z 1894:U 1884:k 1875:) 1873:y 1871:, 1869:x 1865:x 1851:a 1847:k 1843:ω 1835:λ 1831:T 1827:k 1823:ω 1798:, 1795:h 1792:k 1783:k 1780:g 1777:= 1772:2 1750:a 1746:Φ 1742:η 1709:, 1706:) 1703:t 1694:x 1691:k 1688:( 1676:h 1673:k 1662:) 1659:h 1656:+ 1653:z 1650:( 1647:k 1635:a 1630:k 1622:= 1593:. 1590:) 1587:t 1578:x 1575:k 1572:( 1563:a 1560:= 1524:z 1514:z 1512:∂ 1509:/ 1498:Φ 1493:4 1487:3 1477:) 1475:4 1473:( 1456:. 1453:) 1450:t 1447:, 1444:x 1441:( 1435:= 1432:z 1423:0 1420:= 1414:g 1411:+ 1405:t 1367:) 1365:t 1363:, 1361:x 1359:( 1357:η 1347:) 1345:3 1343:( 1326:. 1323:) 1320:t 1317:, 1314:x 1311:( 1305:= 1302:z 1290:z 1273:= 1267:t 1229:z 1211:) 1209:2 1207:( 1190:. 1187:h 1181:= 1178:z 1169:0 1166:= 1160:z 1112:) 1110:1 1108:( 1088:= 1080:2 1076:z 1062:2 1051:+ 1043:2 1039:x 1025:2 998:Φ 977:. 971:z 954:= 949:z 945:u 931:x 914:= 909:x 905:u 891:z 887:x 882:z 880:u 875:x 873:u 865:) 863:t 859:z 855:x 833:h 829:z 818:z 813:t 809:z 805:) 803:t 801:, 799:x 797:( 795:η 791:z 786:x 721:, 718:H 712:2 709:1 703:= 700:a 690:a 687:2 684:= 681:H 660:H 646:T 642:λ 638:ω 634:k 617:. 612:T 604:= 599:k 591:= 586:p 582:c 567:p 565:c 556:. 545:T 541:/ 537:π 535:2 528:ω 523:f 516:T 505:ω 501:, 494:λ 490:/ 486:π 484:2 477:k 472:λ 457:k 437:a 416:) 412:t 403:x 400:k 396:( 386:a 383:= 380:) 377:t 374:, 371:x 368:( 352:t 348:x 340:) 338:t 336:, 334:x 332:( 330:η 189:C 185:B 181:A 172:λ 168:/ 164:h

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Index