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Uchiyama Yusuke. 2019 Surface Gravity and Capillary Waves. In Cochran, J. Kirk; Bokuniewicz, J.
Henry; Yager, L. Patricia (Eds.) Encyclopedia of Ocean Sciences, 3rd Edition. vol. 3, pp. 672-681,
Elsevier. ISBN: 978-0-12-813081-0

dx.doi.org/10.1016/B978-0-12-409548-9.11465-4© 2019 Elsevier Ltd. All rights reserved.

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Surface Gravity and Capillary Waves☆
Yusuke Uchiyama, Kobe University, Kobe, Japan© 2019 Elsevier Ltd. All rights reserved.

Introduction
Basic Formulations
Linear Waves
The Group Velocity
Second Order Quantities
Waves on Currents: Action Conservation
Nonlinear Effects
Resonant Interactions
Parasitic Capillary Waves
Wave Breaking
Further Reading

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Introduction
Ocean surface waves are the most common oceanographic phenomena that are known to the casual observer. They can at once be
the source of inspiration and primal fear. It is remarkable that the complex, random wave field of a storm-lashed sea can be studied
and modeled using well-developed theoretical concepts. Many of these concepts are based on linear or weakly nonlinear
approximations to the full nonlinear dynamics of ocean waves. Early contributors to these theories included such luminaries as
Cauchy, Poisson, Stokes, Lagrange, Airy, Kelvin, and Rayleigh. Many of the current challenges in the study of ocean surface waves are
related to nonlinear processes that have not been well understood yet. These include dynamical coupling between the atmosphereand the ocean, wave–wave interactions, and wave breaking.
For the purposes of this article, surface waves are considered to extend from low frequency swell from distant storms at periods of
10 s or more and wavelengths of hundreds of meters, to capillary waves with wavelengths of millimeters and frequencies of O(10)Hz. In between are wind waves with lengths of O(1 100) m and periods of O(1–10) s. Fig. 1 shows a spectrum of surface waves
measured from the Research Platform FLIP off the coast of Oregon. The spectrum, F, shows the distribution of energy in the wave
field as a function of frequency. The wind wave peak at approximately 0.13 Hz is well separated from the swell peak at
approximately 0.06 Hz.
Ocean surface waves play an important role in air–sea interaction. Momentum from the wind goes into both surface waves andcurrents. Ultimately the waves are dissipated either by viscosity or breaking, giving up their momentum to currents. Surface waves
affect upper-ocean mixing through both wave breaking and their role in the generation of Langmuir circulations. This breaking andmixing influence the temperature of the ocean surface and thus the thermodynamics of air–sea interaction. Surface waves impose
significant structural loads on ships and other structures. Remote sensing of the ocean surface, from local to global scales, depends
on the surface wave field.

Basic Formulations
The dynamics and kinematics of surface waves are described by solutions of the Navier–Stokes equations for an incompressible
viscous fluid, with appropriate boundary and initial conditions. Surface waves of the scale described here are usually generated by
the wind, so the complete problem would include the dynamics of both the water and the air above. However, the density of the air
is approximately 800 times smaller than that of the water, so many aspects of surface wave kinematics and dynamics may beconsidered without invoking dynamical coupling with the air above.
The influence of viscosity is represented by the Reynolds number of the flow, Re ¼ UL/n, where U is a characteristic velocity, L acharacteristic length scale, and n ¼ m/r is the kinematic viscosity, where m is the viscosity and r the density of the fluid. The Reynolds
number is the ratio of inertial forces to viscous forces in the fluid and if Re  1, the effects of viscosity are often confined to thinboundary layers, with the interior of the fluid remaining essentially inviscid (n ¼ m ¼ 0) by assuming a homogeneous fluid. In
contrast, internal waves in stratified fluids are rotational since they introduce baroclinic generation of vorticity in the interior of thefluid. Denoting the fluid velocity by u¼(u, v, w), the vorticity of the flow is given by z ¼ r  u. If z ¼ 0, the flow is said tobe irrotational. From Kelvin’s circulation theorem, the irrotational flow of an incompressible (r u ¼ 0) inviscid fluid will☆
Change History: July 2018: Yusuke Uchiyama added abstract, updated the text, equations, and further reading list based on the article from the 2nd edition of
Encyclopedia of Ocean Sciences, Elsevier.

This is an update of W.K. Melville, Surface Gravity and Capillary Waves, Encyclopedia of Ocean Sciences (2nd Edn), edited by John H. Steele, Academic Press, 2001,
pp. 573–581.

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(A)

673

_

Φ (m 2 H z 1 )16
Swell peak
12

Wind sea peak
Pitch _ roll

8
4

Heave

0

0

0.02

0.04

0.06

0.08

0.1
f (Hz)

0.12

0.14

0.16

0.18

0.2

(B)

_

Φ ( m 2 H z 1)

10 0
_

10 2
_

10 4
_

10 6
_
10 2

_

10 1

100
f (Hz)

Fig. 1 (A) Surface displacement spectrum measured with an electromechanical wave gauge from the Research Platform FLIP in 8 m s1 winds off the coast ofOregon. Note the wind-wave peak at 0.13 Hz, the swell at 0.06 Hz and the heave and pitch and roll of FLIP at 0.04 and 0.02 Hz respectively. (B) An extension of(A) with logarithmic spectral scale, note that from the wind sea peak to approximately 1 Hz the spectrum has a slope like f4, common in wind-wave spectra.
Reproduced with permission from Felizardo, F. C. and Melville, W. K. (1995). Correlations between ambient noise and the ocean surface wave field. Journal of
Physical Oceanography 25, 513–532.

remain irrotational as the flow evolves. The essential features of surface waves may be considered in the context of incompressible
irrotational flows.
For an irrotational flow, a scalar velocity potential f can be introduced, so that u ¼r f. Then, by virtue of incompressibility, fsatisfies Laplace’s equation
r2 f ¼ 0(1)

from the incompressible continuity. We denote the surface by z ¼ (x, y, t), where (x, y) are the horizontal coordinates, z is thevertical coordinate, and t is time. The kinematic condition at the impermeable bottom at z ¼  h, is one of no flow through theboundary:@f¼ 0 at z ¼ h
@z

(2)

@
@
@
þu þv
¼w
@t
@x
@y

(3)

@f 1 2þ u þ g ¼ ðpa  pÞ=r
@t 2

(4)

There are two boundary conditions at z ¼  :The first is a kinematic condition, which is equivalent to imposing the condition that elements of fluid at the surface remain at
the surface. The second is a dynamical condition, a Bernoulli equation, which is equivalent to stating that the pressure p at z ¼ ,
an infinitesimal distance beneath the surface, is just a constant atmospheric pressure, pa, plus a contribution from surface tension.
The effect of gravity is to impose a restoring force tending to bring the surface back to z ¼ 0. The effect of surface tension is to reducethe curvature of the surface.
Although this formulation of surface waves is considerably simplified already, there are profound difficulties in predicting theevolution of surface waves based on these equations. Although Laplace’s equation is linear, the surface boundary conditions are
nonlinear and apply on a surface whose specification is a part of the solution. Our ability to accurately predict the evolution of

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nonlinear waves is limited and largely dependent on numerical techniques. The usual practice is to linearize the boundary
conditions about z ¼ 0.

Linear Waves
Simple harmonic surface waves are characterized by an amplitude a, half the distance between the crests and the troughs, and awavenumber vector k with | k | ¼ k ¼ 2p/l, where l is the wavelength. The surface displacement is ¼ aeiðk:xst Þ(5)where s ¼ 2p/T is the radian frequency, x ¼ (x, y), and T is the wave period. Then ak is a measure of the slope of the waves, and if
ak  1, the surface boundary conditions can be linearized about z ¼ 0. Unless otherwise stated, the real part of complex expressions
is taken to obtain solutions.
Following linearization, the boundary conditions become
@
¼w
@t

0
1
@f
G @@ 2  @ 2  A
at
þ g ¼þ@t
r @x2 @y2

(6)
z¼0

(7)

where the linearized Laplace pressure is
 2@  @2
pa  p ¼ G
þ
@x2 @y2(8)

and G is the surface tension coefficient.
Substituting for  and satisfying Laplace’s equation and the boundary conditions at z ¼ 0 and h gives
ig0 a cosh kðzþ; hÞ,s cosh kh(9)s2 ¼ g0 k tanh kh

(10)



g0 ¼ g 1 þ Gk2 =r

(11)

f¼
where

and

which is effective gravity modified by surface tension. Equations relating the frequency and wavenumber, s ¼ s(k), are known as
dispersion relations, and for linear waves provide a fundamental description of the wave kinematics. If surface tension is negligible,
G  1 and g0 !g, then dispersion relation (10) becomes s2 ¼ gk tanh kh. The phase speed,c ¼ s=k ¼

 0
1=2
g
tanh khk(12)

is the speed at which lines of constant phase (e.g., wave crests) move.
For waves propagating in the x-direction, the velocity field is
u¼

g0 ak cosh ðz þ hÞ iðkxst Þescosh khv¼0

w¼

ig0 ak sinh ðz þ hÞ iðkxst Þescosh kh

(13)
(14)
(15)

and the pressure is
p ¼ rg0 

cosh ðz þ hÞ
cosh kh(16)

where the hydrostatic pressure is subtracted. The velocity decays with depth away from the surface, and, to leading order, elementsof fluid execute elliptical orbits as the waves propagate.
For shallow water, kh  1,

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ðu; v; w; pÞ ¼

 0

gk
; 0; 0; rg0 
s

675

(17)

so that there is no vertical motion, just a uniform sloshing back and forth in the horizontal plane in phase with the surface
displacement . The phase speed c ¼ (g0 h)1/2, is independent of the wavenumber. Such waves are said to be nondispersive. Waves
propagating toward shore eventually attain this condition, and, as the depth tends to zero, nonlinear effects become important as ak
increases.
For very deep water, kh  1,
 0

gk
ig0 k
; 0; 
; rg0 ekz ðu; v; w; pÞ ¼
(18)ssso that the water particles execute circular motions that decay exponentially with depth. The horizontal motion is in phase with thesurface displacement, and the phase speed of the waves
c ¼ ðg0 =kÞ

1=2

¼

hg 
k

1 þ Gk2 =rg

i1=2

(19)

These deep-water waves are dispersive; that is, the phase speed is a function of the wavenumber as shown in Fig. 2. The influenceof surface tension relative to gravity is determined by the value of the dimensionless parameter S ¼ Gk2/rg. When S ¼ 1, thewavelength l ¼ 1.7 cm and the phase speed is a minimum at c ¼ 23 cm s1. When S  1, surface tension is the dominant restoring
force over gravity, the wavelength is less than 1.7 cm, and the phase speed increases as the wavelength decreases. When S  1,
gravity is the dominant restoring force over surface tension, the wavelength is greater than 1.7 cm, and the phase speed increases as
the wavelength increases.

The Group Velocity
Using the superposition principle over a continuum of wavenumbers a general disturbance (in two spatial dimensions) can be
represented byZ 1ðx; t Þ ¼
aðkÞeiðkxst Þ dk
(20)
1

where, as above, only the real part of the integral is taken. Assuming the disturbance is confined to wavenumbers in theneighborhood of ko, and expanding s(k) about ko gives
ds
jk¼k0 þ . . .
dk

(21)

aðkÞeiðkko Þðxcg t Þ dk þ . . .

(22)sðkÞ ¼ sðko Þ þ ðk  ko Þ
whence:ðx; t Þ¼eiðko xsðko Þt Þ

Z 1
1

where

_

c (m s 1)

101/2

c = (g/k + Γk /ρ)

1

0.1
0.001

0.01

0.1

1

10

100

λ (m)

Fig. 2 The phase speed of surface gravity-capillary waves as a function of wavelength l. A minimum phase speed of 23 cm s1 occurs for l ¼ 0.017 m. Shorter
waves approach pure capillary waves, whereas longer waves become pure gravity waves. Note that there are both capillary and gravity waves for a given phasespeed. This is the basis of the generation of parasitic capillary waves on the forward face of steep gravity waves.

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 


ds
c
2kh
1þ
cg ¼  ¼
dk k¼k0 2
sinh 2kh(23)

is the group velocity. Eq. (22) demonstrates that the modulation of the pure harmonic wave propagates at the group velocity. Thisimplies that an isolated packet of waves centered around the wavenumber ko will propagate at the speed cg, so that an observer
wishing to follow waves of the same length must travel at the group velocity. Since the energy density is proportional to a2 (seebelow), it is also the speed at which the energy propagates. These properties of the group velocity apply to linear waves, and more
subtle effects may become important at large slopes. From Eq. (23), cg 6¼ c in general.
For deep-water gravity waves, kh  1,
1
1 g1=2cg ¼ c ¼
2
2 k

(24)

so the wave group travels at half the phase speed, with waves appearing at the rear of a group propagating forward and disappearing
at the front of the group.
For deep-water capillary waves,3
s2 ¼ Gk3 =r, c ¼ ðGk=rÞ1=2 , cg ¼ c
2(25)

so waves appear at the front of the group and disappear at the rear of the group as it propagates.
For shallow water gravity waves, kh  1, cg ¼ c.

Second Order Quantities
The energy density (per horizontal surface area) of surface waves is1
E ¼ rg0 a2
2(26)

being the sum of the kinetic and potential energies. In the case of gravity waves, the potential energy results from the displacement of
the surface about its equilibrium horizontal position. For capillary waves, the potential energy arises from the stretching of thesurface against the restoring force of surface tension.
The mean momentum density M is given by1
k E k
M ¼ rsa2 coth kh  ¼ 
2
k c k

(27)

To leading order, linear gravity waves transfer energy without transporting mass; however, there is a second order mass transport
associated with surface waves. In a Lagrangian description of the flow it can be shown that for irrotational inviscid wave motion themean horizontal Lagrangian velocity (Stokes drift), ust, of a particle of fluid originally at z ¼ zo is
ust ¼ ska2cosh 2kðz þ hÞ k

2 sinh 2 khk(28)

which reduces to (ak)2ce2kzo k/k when kh  1. This second order velocity arises from the fact that the orbits of the particles of
fluid are not closed. Integrating Eq. (28) over the depth it can be shown that this mean Lagrangian velocity accounts for thewave momentum M in the Eulerian description. The Stokes drift is important for representing scalar transport near the ocean
surface, but this transport is likely to be significantly enhanced by the intermittent larger velocities associated with wave
breaking.
Longer waves, or swell, from distant storms can travel great distances. An extreme example is the propagation of swellalong great circle routes from storms in the Southern Ocean to the coast of California. For waves to travel so far, the effects ofdissipation must be small. In deep water, where the wave motions have decayed away to negligible levels at depth, thecontributions to the dissipation come from the thin surface boundary layer and the rate of strain of the irrotational motionsin the bulk of the fluid. It can be shown that the integral is dominated by the latter contributions, and the timescale for the
decay of the wave energy is just
1 dE 1  2 1
¼ 4vk
(29)
te ¼ 
E dt
or s/8pnk2 wave periods. This gives negligible dissipation for long-period swell in deep water over scales of the ocean basins. Morerealistic models of wave dissipation must take into account breaking and near surface turbulence, sometimes parameterized as an
eddy viscosity, which is several orders of magnitude greater than the molecular viscosity. When waves propagate into shallow water,
the dominant dissipation may occur in the bottom boundary layer.

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Eq. (27) shows that dissipation of wave energy is concomitant with a reduction in wave momentum, but since momentum isconserved, the reduction of wave momentum is accompanied by a transfer of momentum from waves to currents. That is, netdissipative processes in the wave field lead to the generation of currents.

Waves on Currents: Action Conservation
Waves propagating in varying currents may exchange energy with the current, thus modifying the waves. Perhaps the most dramatic
examples of this effect come when waves propagating against a current become larger and steeper. Examples occur off the east coastof South Africa as waves from the Southern Ocean meet the Aghulas Current; as North Atlantic storms meet the northward flowing
Gulf Stream, or at the mouths of estuaries as shoreward propagating waves meet the ebb tide.
For currents U ¼ (U, V) that only change slowly on the scale of the wavelength, and a surface displacement of the form ¼ aðx; y; t Þeiyðx;y;tÞ

(30)

where a is the slowly varying amplitude and y is the phase. The absolute local frequency o ¼ @y/@t, and the x- and y-componentsof the local wavenumber are given by k ¼ @y/@x, l ¼ @y/@y. The frequency seen by an observer moving with the current U is@y
þ U:ry(31)
@twhich is equal to the intrinsic frequency s. Thus we obtains ¼ o  U:k(32)

the Doppler relationship. This frequency Doppler shifting leads to modification of dispersion relation (10).
We also have conservation of wavenumber k rearranged in the following form:@kþ ro ¼ 0@t(33)

which can be interpreted as the conservation of wave crests, where k is the spatial density of crests and o the wave flux.
The velocity of a wave packet along rays isdxi@s¼ Ui þ
¼ Ui þ cgi
dt
@xi

(34)

which is simply the vector sum of the local current and the group velocity in a fluid at rest. Furthermore, refraction is governed by@Uj @s
dki
¼ kj

dt
@xi @xi(35)

where the first term on the right represents refraction due to the current and the second is due to gradients in the waveguide, such aschanges in the depth. It is this latter term that results in waves, propagating from deep water toward a beach, refracting so that they
propagate normal to shore.
For steady currents, the absolute frequency is constant along rays but the intrinsic frequency may vary, and the dynamics lead to aremarkable and quite general result for linear waves. If E is the energy density then the quantity A ¼ E=s, the wave action, isconserved:

@A
@ 
þ
Ui þ cgi A ¼ 0@t @xi(36)

In other words, the variations in the intrinsic frequency s and the energy density E, are such as to conserve the quotient.
This theory permits the prediction of the change of wave properties as they propagate into varying currents and water depths. For
example, in the case of waves approaching an increasing counter current, the waves will move to shorter wavelengths (higher k),larger amplitudes, and hence greater slopes, ak. As the speed of the adverse current approaches the group velocity, the waves will be“blocked” and be unable to propagate further. In this simplest theory, a singularity occurs with the wave slope becoming infinite,
but higher order effects lead to reflection of the waves and the same blocking effect. This theory also forms the basis of models of
long-wave–short-wave interaction that are important for wind-wave generation and the interpretation of remote sensing measurements of the ocean surface, including the remote sensing of long nonlinear internal waves.

Nonlinear Effects
The nonlinearity of surface waves is represented by the wave slope, ak. For typical gravity waves at the ocean surface the average slope
may be O(102  101); small, but not negligibly so. Nonlinear effects may be weak and can be described as a perturbation to theEncyclopedia of Ocean Sciences, (2019)

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linear wave theory, using the slope as an expansion parameter. This approach, pioneered by Stokes in the mid-19th century, showedthat for uniform approach deep-water gravity waves,
s2 ¼ gk 1 þ a2 k2 þ . . .
(37)
and
1
 ¼ a cos y þ a2 k cos 2y þ . . .
2(38)

Weakly nonlinear gravity waves have a phase speed greater than linear waves of the same wavelength. The effect of the higher
harmonics on the shape of the waves leads to a vertical asymmetry with sharper crests and flatter troughs.
The largest such uniform wave train has a slope of ak ¼ 0.446 a phase speed of 1.11c, and a discontinuity in slope at the crest
containing an included angle of 120 degrees. This limiting form has sometimes been used as the basis for the models of wave
breaking; however, uniform wave trains are unstable to side-band instabilities at significantly lower slopes, and it is unlikely that
this limiting form is ever achieved in the ocean.
With the assumption of both weak nonlinearity and weak dispersion (or small bandwidth, dk/ko  1), it may be shown that if,hi(39)ðx; y; t Þ ¼ ℜ Aðx; y; t Þeiðko xso t Þ
where so ¼ s(ko) and ℜ means that the real part is taken, then the complex wave envelope A(x, y, t) satisfies a nonlinear Schrödinger
equation or one of its variants. Solutions of the nonlinear Schrödinger equation for initial conditions that decay sufficiently rapidly
in space evolve into a series of envelope solitons (solitary waves) and a dispersive tail. Solitons propagate as waves of permanent
form and survive interactions with other solitons with just a change of phase. Attempts have been made to describe ocean surface
waves as fields of interacting envelope solitons; however, instabilities of the two-dimensional soliton solutions, and the effects ofhigher-order nonlinearities, random phase and amplitude fluctuations in real wave fields give pause to the applicability of these
idealized theoretical results.

Resonant Interactions
Modeling the generation, propagation, interaction, and dissipation of wind-generated surface waves is of great importance for avariety of scientific, commercial, and social reasons. A rigorous theoretical foundation for all components of this problem does not
yet exist, but there is a rational theory for weakly nonlinear wave–wave interactions.
For linear waves freely propagating away from a storm, the spectral content at any later time is explicitly defined by the initialstorm conditions. For a nonlinear wave field, wave–wave interactions can lead to the generation of wavenumbers different fromthose comprising the initial disturbance. For surface gravity waves, these nonlinear effects lead to the generation of waves of lowerand higher wavenumber with time. The timescale for this evolution in a random homogeneous wave field is of the order of (ak)4
times a characteristic wave period; slow, but significant over the life of a storm.
The foundation of weakly nonlinear interactions between surface waves is the resonant interaction between waves satisfying thelinear dispersion relationship. It is a simple consequence of quadratic nonlinearity that pairs of interacting waves lead to thegeneration of waves having sum and difference frequencies relative to the original waves. Thus
k3 ¼ k1  k2 , s3 ¼ s1  s2

(40)

If in addition, si(i ¼ 1, 2, 3) satisfies the dispersion relationship, then the interaction is resonant. In the case of surface waves, thenonlinearities arise from the surface boundary conditions, and resonant triads are possible for gravity capillary waves, and gravity
waves in water of intermediate depth.
For deep-water gravity waves, cubic nonlinearity is required before resonance occurs between a quartet of wave components:
k1  k2  k3  k4 ¼ 0
s1  s2  s3  s4 . . . ¼ 0,

si ¼ ðgki Þ1=2

(41)

These quartet interactions comprise the basis of nonlinear wave–wave interactions in operational models of surface gravity
waves. Exact resonance is not required, since even with detuning significant energy transfer can occur across the spectrum. The
formal basis of these theories may be cast as problems of multiple spatial and temporal scales, and higher-order interactions should
be considered as these scales increase, and the wave slope increases.

Parasitic Capillary Waves
The longer gravity waves are the dominant waves at the ocean surface, but recent developments in air–sea interaction and remotesensing, have placed increasing importance on the shorter gravity-capillary waves. Measurements of gravity-capillary waves at sea arevery difficult to make and much of the detailed knowledge is based on laboratory experiments and theoretical models.

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Laboratory measurements suggest that the initial generation of waves at the sea surface occurs in the gravity-capillary
wave range, initially at wavelengths of O(1) cm. As the waves grow and the fetch increases, the dominant waves,those at the peak of the spectrum, move into the gravity-wave range. A simple estimate of the effects of surface tension
based on the surface tension parameter S using the gravity wavenumber k would suggest that they are unimportant, but as
the wave slope increases and the curvature at the crest increases, the contribution of the Laplace pressure near the crest
increases. A consequence is that so-called parasitic capillary waves may be generated on the forward face of the gravity wave
(Fig. 3).
The source of these parasitic waves can be represented as a perturbation to the underlying gravity wave caused by the localized
Laplace pressure component at the crest. This is analogous to the “fish-line” problem of Rayleigh, who showed that due to thedifferences in the group velocities, capillary waves are found ahead of, and gravity waves behind, a localized source in a stream. In
this context the capillary waves are considered to be steady relative to the crest. The possibility of the direct resonant generation ofcapillary waves by perturbations moving at or near the phase speed of longer gravity waves is implied by the form of the dispersion
curve in Fig. 2. Free surfaces of large curvature, as in parasitic capillary waves, are not irrotational and so the effects of viscosity intransporting vorticity and dissipating energy must be accounted for. Theoretical and numerical studies show that the viscous
dissipation of the longer gravity waves is enhanced by one to two orders of magnitude by the presence of parasitic capillary waves.
These studies also show that the observed high wavenumber cut-off in the surface wave spectrum that has been observed atwavelengths of approximately O(103  102) m can be explained by the properties of the spectrum of parasitic capillary waves
bound to short steep gravity waves.

Wave Breaking
Although weak resonant and near-resonant interactions of weakly nonlinear waves occur over slow timescales, breaking is a fast
process, lasting for times comparable to the wave period. However, the turbulence and mixing due to breaking may last for aconsiderable time after the event. Breaking, which is a transient, two-phase, turbulent, free-surface flow, is the least understood of
the surface wave processes. The energy and momentum lost from the wave field in breaking are available to generate turbulence andsurface currents, respectively. The air entrained by breaking may, through the associated buoyancy force on the bubbles, bedynamically significant over times comparable to the wave period as the larger bubbles rise and escape through the surface. The

(A)

(B)

(C)

(D)

Fig. 3 (A–D) Evolution of a gravity wave toward breaking in the laboratory. Note the generation of parasitic capillary waves on the forward face of the crest.
Reproduced with permission from Duncan, J. H. et al. (1994). The formation of a spilling breaker. Physics of Fluids 6, S2.

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Fig. 4 Waves in a storm in the North Atlantic in December 1993 in which winds were gusting up to 50–60 knots and wave heights of 12–15 m were reported.
Breaking waves are (A) large, (B) intermediate, and (C) small scale. Photographs by Terrill, E. and Melville, W. K. reproduced with permission from Melville, W.K.
(1996). The role of wave breaking in air–sea interaction. Annual Review of Fluid Mechanics 28, 279–321.

sound generated with the breakup of the air into bubbles is perhaps the dominant source of high frequency sound in the ocean, andmay be used diagnostically to characterize certain aspects of air–sea interaction. Fig. 4 shows examples of breaking waves in a North
Atlantic storm.
If waves break in shallow areas such as sandy beaches due to shoaling bathymetry (viz., depth-induced breaking), the dissipatedwave energy and momentum are transferred to interior water to excite wave set-up and littoral currents. They are crucial to beacherosion and accretion associated with intensive sediment transport, increase of sea surface height during storm surges, andswimming safety through offshore-directed rip currents, and thus have been a central topic of nearshore oceanography and coastal
engineering (e.g., Uchiyama et al., 2010).
Since direct measurements of breaking in the field are so difficult, much of our understanding of breaking comes from laboratory
experiments and simple modeling. For example, laboratory experiments and similarity arguments suggest that the rate of energy loss
per unit length of the breaking crest of a wave of phase speed c is proportional to rg1c5, with a proportionality factor that depends
on the wave slope, and perhaps other parameters. Attempts are underway to combine such simple modeling along with fieldmeasurements of the statistics of breaking fronts to give an estimate of the distribution of dissipation across the wave spectrum.
Recent developments in the measurement and modeling of breaking using optical, acoustical microwave and numerical techniques
hold the promise of significant progress in the next decade.

Further Reading
Komen GJ, Cavaleri L, Donelan M, et al. (1994) Dynamics and modeling of ocean waves. Cambridge: Cambridge University Press.
Lamb H (1945) Hydrodynamics. New York: Dover Publications.
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