In
fluid dynamics,
Airy wave
theory (often referred to as
linear wave
theory) gives a
linearised
description of the
propagation of
gravity waves on the surface of a
homogeneous
fluid layer. The theory assumes
that the fluid layer has a uniform mean depth, and that the
fluid flow is
inviscid,
incompressible and
irrotational. This theory was first published,
in correct form, by
George Biddell
Airy in the 19
^{th} century.
Airy wave theory is often applied in
ocean engineering and
coastal engineering for the modelling of
random sea states —
giving a description of the wave
kinematics and
dynamics
of highenough accuracy for many purposes.
Further, several secondorder nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects.
Description
Airy wave theory uses a
potential
flow 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
viscosity,
vorticity,
turbulence
and/or
flow separation into account.
This is due to the fact that for the oscillatory part of the fluid
motion, waveinduced vorticity is restricted to some thin
oscillatory
Stokes boundary
layers at the boundaries of the fluid domain.
Airy wave theory is often used in
ocean engineering and
coastal engineering. Especially for
random waves, sometimes called
wave turbulence, the evolution of the wave
statistics — including the wave
spectrum —
is predicted well over not too long distances (in terms of
wavelengths) and in not too shallow water.
Diffraction is one of the wave effects which can
be described with Airy wave theory. Further, by using the
WKBJ approximation,
wave shoaling and
refraction can be predicted..
Earlier attempts to describe surface gravity waves using potential
flow were made by, among others,
Laplace,
Poisson,
Cauchy and
Kelland. But
Airy was the first to publish the
correct derivation and formulation in 1841. Soon after, in 1847,
the linear theory of Airy was extended by
Stokes for
nonlinear wave motion, correct up to
third order in the
wave steepness. Even before Airy's linear theory,
Gerstner derived a nonlinear
trochoidal wave theory in 1804, which
however is not
irrotational.
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
η(
x,
t) of one
wave component is
sinusoidal, as a
function of horizontal position
x and time
t:
 \eta(x,t)\, =\, a\, \cos\, \left( kx\, \, \omega t\right)
where
 k\,=\,\frac{2\pi}{\lambda},\,
 \omega\,=\,\frac{2\pi}{T}\,=\,2\pi\,f.\,
The waves propagate along the water surface with the
phase speed c_{p}:
 c_p\, =\, \frac{\omega}{k}\, =\, \frac{\lambda}{T}.
The angular wavenumber
k and frequency
ω are not
independent parameters (and thus also wavelength
λ and
period
T are not independent), but are coupled. Surface
gravity waves on a fluid are
dispersive waves — exhibiting
frequency dispersion — meaning that each wavenumber has its own
frequency and phase speed.
Note that in engineering the
wave height
H — the difference in elevation between
crest and
trough — is often used:
 H\, =\, 2\, a \qquad \text{and} \qquad a\, =\, \frac12\,
H,
valid in the present case of linear periodic waves.
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.
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 finite depth—with the ellipses becoming flatter near
the bottom of the fluid layer. So while the wave propagates, the
fluid particles just orbit (oscillate) around their
average 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 freesurface value at
a depth of half a wavelength.
In a similar fashion, there is also a pressure oscillation
underneath the free surface, with waveinduced pressure
oscillations reducing with depth below the free surface — in the
same way as for the orbital motion of fluid parcels.
Mathematical formulation of the wave motion
Flow problem formulation
The waves propagate in the horizontal direction, with
coordinate x, and a fluid
domain bound above by a free surface at
z =
η(
x,
t), with
z the vertical
coordinate (positive in the upward direction) and
t being
time. The level
z = 0
corresponds with the mean surface elevation. The
impermeable bed underneath the fluid layer is
at
z = 
h. Further, the flow is assumed
to be
incompressible and
irrotational — a good
approximation of the flow in the fluid interior for waves on a
liquid surface — and
potential
theory can be used to describe the flow. The velocity potential
Φ(
x,
z,
t) is related to the
flow velocity components
u_{x} and
u_{z}
in the horizontal (
x) and vertical (
z) directions
by:
u_x\, =\, \frac{\partial\Phi}{\partial x}
\quad \text{and} \quad
u_z\, =\, \frac{\partial\Phi}{\partial z}.
Then, due to the
continuity equation for
an incompressible flow, the potential
Φ has to satisfy the
Laplace equation:
(1) \qquad
\frac{\partial^2\Phi}{\partial x^2}\, +\,
\frac{\partial^2\Phi}{\partial z^2}\, =\, 0.
Boundary conditions 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
zerothorder solution) of the flow is.
Here, we assume the base state is rest, implying the mean flow
velocities are zero.
The bed being impermeable, leads to the
kinematic bed boundarycondition:
 (2) \qquad \frac{\partial\Phi}{\partial z}\, =\, 0 \quad \text{
at } z\, =\, h.
In case of deep water — by which is meant
infinite water depth, from a mathematical point of
view — the flow velocities have to go to zero in the
limit as the vertical coordinate goes to
minus infinity:
z → ∞.
At the free surface, for
infinitesimal
waves, the vertical motion of the flow has to be equal to the
vertical velocity of the free surface. This leads to the kinematic
freesurface boundarycondition:
 (3) \qquad \frac{\partial\eta}{\partial t}\, =\,
\frac{\partial\Phi}{\partial z} \quad \text{ at } z\, =\,
\eta(x,t).
If the free surface elevation
η(
x,
t) 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
Bernoulli's equation 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
dynamic
freesurface boundary condition:
 (4) \qquad \frac{\partial\Phi}{\partial t}\, +\, g\, \eta\, =\,
0 \quad \text{ at } z\, =\, \eta(x,t).
Because this is a linear theory, in both freesurface boundary
conditions — the kinematic and the dynamic one, equations (3) and
(4) — the value of
Φ and ∂
Φ/∂
z at the
fixed mean level
z = 0 is used.
Solution for a progressive monochromatic wave
For a propagating wave of a single frequency — a
monochromatic wave — the surface elevation is
of the form:
 \eta\, =\, a\, \cos\, ( k x\, \, \omega t ).
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:
 \Phi\, =\, \frac{\omega}{k}\, a\, \frac{\cosh\, \bigl( k\,
(z+h) \bigr)}{\sinh\, (k\, h)}\, \sin\, ( k x\, \, \omega t),
with sinh and cosh the
hyperbolic
sine and
hyperbolic cosine
function, respectively.But
η and
Φ also have to
satisfy the dynamic boundary condition, which results in
nontrivial (nonzero) values for the wave amplitude
a
only if the linear
dispersion
relation is satisfied:
 \omega^2\, =\, g\, k\, \tanh\, ( k h ),
with tanh the
hyperbolic tangent.
So angular frequency
ω and wavenumber
k — or
equivalently period
T and wavelength
λ — cannot
be chosen independently, but are related. This means that wave
propagation at a fluid surface is an
eigenproblem. When
ω and
k
satisfy the dispersion relation, the wave amplitude
a can
be chosen freely (but small enough for Airy wave theory to be a
valid approximation).
Table of wave quantities
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
x = (
x,
y)
plane. The
wavenumber vector is
k, and is perpendicular to the cams of
the
wave crests. Secondly, allowance
is made for a mean flow velocity
U, in
the horizontal direction and uniform over (independent of) depth
z. This introduces a
Doppler
shift in the dispersion relations. At an Earthfixed location,
the
observed angular frequency (or
absolute angular
frequency) is
ω. On the other hand, in a
frame of reference moving with the mean
velocity
U (so the mean velocity as
observed from this reference frame is zero), the angular frequency
is different. It is called the
intrinsic angular frequency
(or
relative angular frequency), denoted as
σ. So
in pure wave motion, with
U=
0, both
frequencies
ω and
σ are equal. The wave number
k (and wave length
λ) are independent of the
frame of reference, and have no
Doppler shift (for monochromatic waves).
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
ξ_{x} and
ξ_{z} are the time
integrals of the oscillatory flow velocities
u_{x} and
u_{z} respectively.
Water depth is classified into three regimes:
 deep water — for a water depth larger than
half the wavelength, h > ½
λ, the phase speed of the waves
is hardly influenced by depth (this is the case for most wind waves
on the sea and ocean surface),
 shallow water — for a water depth smaller than
the wavelength divided by 20, h λ, the phase
speed of the waves is only dependent on water depth, and no longer
a function of period or wavelength;The error
in the phase speed is less than 2% if wavelength effects are
neglected for h λ. and
 intermediate depth — all other cases,
λ h ½ λ, where both water depth and
period (or wavelength) have a significant influence on the solution
of Airy wave theory.
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.
Properties of gravity waves on the surface of deep
water, shallow water and at intermediate depth, according to Airy
wave theory 
quantity 
symbol 
units 
deep water( h > ½ λ ) 
shallow water( h 0.05 λ ) 
intermediate depth( all λ and h
) 
surface elevation 
\eta(\boldsymbol{x},t)\, 
m 
a\, \cos\, \theta(\boldsymbol{x},t)\, 
wave phase 
\theta(\boldsymbol{x},t)\, 
rad 
\boldsymbol{k}\cdot\boldsymbol{x}\, \, \omega\,
t\, 
observed angular
frequency 
\omega\, 
rad / s 
\left( \omega\, \,
\boldsymbol{k}\cdot\boldsymbol{U} \right)^2\, =\, \bigl( \Omega(k)
\bigr)^2 \quad \text{ with } \quad k\,=\, \boldsymbol{k} \, 
intrinsic angular frequency 
\sigma\, 
rad / s 
\quad \sigma^2\, =\, \bigl( \Omega(k) \bigr)^2
\quad \text{ with } \quad \sigma\, =\, \omega\, \,
\boldsymbol{k}\cdot\boldsymbol{U}\, 
unit vector in the wave propagation direction 
\boldsymbol{e}_k\, 
– 
\frac{\boldsymbol{k}}{k}\, 
dispersion
relation 
\Omega(k)\, 
rad / s 
\Omega(k)\, =\, \sqrt{g\, k} 
\Omega(k)\, =\, k\, \sqrt{g\, h}\, 
\Omega(k)\, =\, \sqrt{g\, k\, \tanh\, (k\, h)}\, 
phase speed 
c_p=\frac{\Omega(k)}{k}\, 
m / s 
\sqrt{\frac{g}{k}}\, =\, \frac{g}{\sigma}\, 
\sqrt{g h} 
\sqrt{\frac{g}{k}\, \tanh\, (k\, h)\,} 
group speed 
c_g = \frac{\partial\Omega}{\partial k} 
m / s 
\frac{1}{2}\, \sqrt{\frac{g}{k}}\, =\, \frac{1}{2}\,
\frac{g}{\sigma}\, 
\sqrt{g h}\, 
\frac{1}{2}\, c_p\, \left( 1\, +\, k\, h\, \frac{1\, \,
\tanh^2\, (k\, h)}{\tanh\, (k\, h)} \right) 
ratio 
\frac{c_g}{c_p}\, 
– 
\frac{1}{2}\, 
1\, 
\frac{1}{2}\, \left( 1\, +\, k\, h\, \frac{1\, \, \tanh^2\,
(k\, h)}{\tanh\, (k\, h)} \right) 
horizontal velocity 
\boldsymbol{u}_x(\boldsymbol{x},z,t)\, 
m / s 
\boldsymbol{e}_k\, \sigma\, a\; \text{e}^{\displaystyle k\,
z}\, \cos\, \theta\, 
\boldsymbol{e}_k\, \sqrt{\frac{g}{h}}\, a\, \cos\,
\theta\, 
\boldsymbol{e}_k\, \sigma\, a\, \frac{\cosh\, \bigl( k\, (z+h)
\bigr)}{\sinh\, (k\, h)}\, \cos\, \theta\, 
vertical velocity 
u_z(\boldsymbol{x},z,t)\, 
m / s 
\sigma\, a\; \text{e}^{\displaystyle k\, z}\, \sin\,
\theta\, 
\sigma\, a\, \frac{z\, +\, h}{h}\, \sin\, \theta\, 
\sigma\, a\, \frac{\sinh\, \bigl( k\, (z+h) \bigr)}{\sinh\,
(k\, h)}\, \sin\, \theta\, 
horizontal particle excursion 
\boldsymbol{\xi}_x(\boldsymbol{x},z,t)\, 
m 
\boldsymbol{e}_k\, a\; \text{e}^{\displaystyle k\, z}\, \sin\,
\theta\, 
\boldsymbol{e}_k\, \frac{1}{k\, h}\, a\, \sin\, \theta\, 
\boldsymbol{e}_k\, a\, \frac{\cosh\, \bigl( k\, (z+h)
\bigr)}{\sinh\, (k\, h)}\, \sin\, \theta\, 
vertical particle excursion 
\xi_z(\boldsymbol{x},z,t)\, 
m 
a\; \text{e}^{\displaystyle k\, z}\, \cos\, \theta\, 
a\, \frac{z\, +\, h}{h}\, \cos\, \theta\, 
a\, \frac{\sinh\, \bigl( k\, (z+h) \bigr)}{\sinh\, (k\, h)}\,
\cos\, \theta\, 
pressure oscillation 
p(\boldsymbol{x},z,t)\, 
N / m^{2} 
\rho\, g\, a\; \text{e}^{\displaystyle k\, z}\, \cos\,
\theta\, 
\rho\, g\, a\, \cos\, \theta\, 
\rho\, g\, a\, \frac{\cosh\, \bigl( k\, (z+h) \bigr)}{\cosh\,
(k\, h)}\, \cos\, \theta\, 
Surface tension effects
Due to
surface tension, the
dispersion relation changes to:Phillips (1977), p. 37.
 \Omega^2(k)\, =\, \left( g\, +\, \frac{\gamma}{\rho}\, k^2
\right)\, k\; \tanh\, ( k\, h ),
with
γ the surface tension, with
SI
units in N/m
^{2}. All above equations for linear waves
remain the same, if the gravitational acceleration
g is
replaced by
 \tilde{g}\, =\, g\, +\, \frac{\gamma}{\rho}\, k^2.
As a result of surface tension, the waves propagate faster. Surface
tension only has influence for short waves, with wavelengths less
than a few
decimeters in case of a
water–air interface. For very short wavelengths — two
millimeter in case of the interface between air
and water – gravity effects are negligible.
Interfacial waves
Surface gravity waves are a special case of interfacial waves, on
the
interface between two
fluids of different
density. Consider two
fluids separated by an interface, and without further boundaries.
Then their dispersion relation becomes:
\Omega^2(k)\, =\, k\, \left( \frac{\rho\rho'}{\rho+\rho'} g\,
+\, \frac{\sigma}{\rho+\rho'}\, k^2 \right),
where
ρ and
ρ‘ are the densities of the two
fluids, below (
ρ) and above (
ρ‘) the interface,
respectively. For interfacial waves to exist, the lower layer has
to be heavier than the upper one,
ρ > ρ‘. Otherwise,
the interface is unstable and a
Rayleigh–Taylor
instability develops.
Secondorder wave properties
Several
secondorder wave
properties,
i.e. quadratic in the
wave amplitude
a, can be derived directly from Airy wave
theory. They are of importance in many practical applications,
e.g forecasts of wave conditions.
Using a
WKBJ approximation,
secondorder wave properties also find their applications in
describing waves in case of slowlyvarying
bathymetry, and meanflow variations of currents
and surface elevation. As well as in the description of the wave
and meanflow interactions due to time and spacevariations in
amplitude, frequency, wavelength and direction of the wave field
itself.
Table of secondorder wave properties
In the table below, several secondorder wave properties — as well
as the dynamical equations they satisfy in case of slowlyvarying
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. Further on in this section, more
detailed descriptions and results are given for the general case of
propagation in twodimensional horizontal space.
Secondorder quantities and their dynamics, using
results of Airy wave theory 
quantity 
symbol 
units 
formula 
mean waveenergy density per unit horizontal area 
E\, 
J / m^{2} 
E\, =\, \frac12\, \rho\, g\, a^2\, 
radiation stress or excess horizontal momentum flux due to the wave
motion 
S_{xx}\, 
N / m 
S_{xx}\, =\, \left( 2\, \frac{c_g}{c_p}\, \, \frac12 \right)\,
E\, 
wave action 
\mathcal{A}\, 
J·s / m^{2} 
\mathcal{A}\, =\, \frac{E}{\sigma}\, =\, \frac{E}{\omega\, \,
k\, U}\, 
mean massflux due to the wave motion or the wave
pseudomomentum 
M\, 
kg / (m·s) 
M\, =\, \frac{E}{c_p}\, =\, k\, \frac{E}{\sigma}\, 
mean horizontal masstransport velocity 
\tilde{U}\, 
m / s 
\tilde{U}\, =\, U\, +\, \frac{M}{\rho\, h}\, =\, U\, +\,
\frac{E}{\rho\, h\, c_p}\, 
Stokes drift 
\bar{u}_S\, 
m / s 
\bar{u}_S\, =\, \frac12\, \sigma\, k\, a^2\, \frac{\cosh\,
2\,k\,(z+h)}{\sinh^2\, (k\,h)}\, 
waveenergy propagation 

J / (m^{2}·s) 
\frac{\partial E}{\partial t}\, +\, \frac{\partial}{\partial x}
\Bigl( (U\, +\, c_g)\, E \Bigr)\, +\, S_{xx}\, \frac{\partial
U}{\partial x}\, =\,0\, 
wave action conservation 

J / m^{2} 
\frac{\partial \mathcal{A}}{\partial t}\, +\,
\frac{\partial}{\partial x} \Bigl( (U\, +\, c_g)\, \mathcal{A}
\Bigr)\, =\, 0\, 
wavecrest conservation 

rad / (m·s) 
\frac{\partial k}{\partial t}\, +\, \frac{\partial
\omega}{\partial x}\, =\, 0\, with \omega\, =\,
\Omega(k)\, +\, k\, U\, 
mean mass conservation 

kg / (m^{2}·s) 
\frac{\partial}{\partial t}\Bigl( \rho\, h \Bigr)\, +\,
\frac{\partial}{\partial x} \Bigl( \rho\, h\, \tilde{U} \Bigr)\,
=\, 0\, 
mean horizontalmomentum evolution 

N / m^{2} 
\frac{\partial}{\partial t}\Bigl( \rho\, h\, \tilde{U} \Bigr)\,
+\, \frac{\partial}{\partial x} \left( \rho\, h\, \tilde{U}^2\, +\,
\frac12\, \rho\, g\, h^2\, +\, S_{xx} \right)\, =\, \rho\, g\, h\,
\frac{\partial d}{\partial x}\, 
The last four equations describe the evolution of slowlyvarying
wave trains over
bathymetry in
interaction with the mean flow, and can be derived from a
variational principle: Whitham's average
Lagrangian method. , p. 559. In the mean
horizontalmomentum equation,
d(
x) is the still
water depth,
i.e. the bed underneath the fluid layer is
located at
z = –
d. Note that the
meanflow velocity in the mass and momentum equations is the
mass transport velocity \tilde{U}, including the
splashzone effects of the waves on horizontal mass transport, and
not the mean
Eulerian velocity (e.g.
as measured with a fixed flow meter).
Wave energy density
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 waves, the most used
energy measure is the mean wave energy density per unit horizontal
area. It is the sum of the
kinetic
and
potential energy 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
E_{pot} of the surface
gravity waves, which is the deviation of the potential energy due
to the presence of the waves:Phillips (1977), p. 39.
 E_\text{pot}\, =\, \overline{\int_{h}^{\eta}
\rho\,g\,z\;\text{d}z}\, \, \int_{h}^0 \rho\,g\,z\;
\text{d}z\,
=\, \overline{\frac12\,\rho\,g\,\eta^2}\,
=\, \frac14\, \rho\,g\,a^2,
with an overbar denoting 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).
The mean kinetic energy density per unit horizontal area
E_{kin} of the wave motion is similarly found to
be:
E_\text{kin}\, =\, \overline{\int_{h}^0 \frac12\, \rho\, \left[\, \left \boldsymbol{U}\, +\, \boldsymbol{u}_x \right^2\, +\, u_z^2\, \right]\; \text{d}z}\,
\, \int_{h}^0 \frac12\, \rho\, \left \boldsymbol{U} \right^2\; \text{d}z\,
=\, \frac14\, \rho\, \frac{\sigma^2}{k\, \tanh\, (k\, h)}\,a^2,
with
σ the intrinsic frequency, see the
table of wave quantities. Using
the dispersion relation, the result for surface gravity waves
is:
 E_\text{kin}\, =\, \frac14\, \rho\, g\, a^2.
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
conservative system.. Adding potential
and kinetic contributions,
E_{pot} and
E_{kin}, the mean energy density per unit
horizontal area
E of the wave motion is:
 E\, =\, E_\text{pot}\, +\, E_\text{kin}\, =\, \frac12\, \rho\,
g\, a^2.
In case of surface tension effects not being negligible, their
contribution also adds to the potential and kinetic energy
densities, givingPhillips (1977), p. 38.
E_\text{pot}\, =\, E_\text{kin}\, =\, \frac14\, \left( \rho\, g\, +\, \gamma\, k^2 \right)\, a^2,
\qquad \text{so} \qquad
E\, =\, E_\text{pot}\, +\, E_\text{kin}\, =\, \frac12\, \left( \rho\, g\, +\, \gamma\, k^2 \right)\, a^2,
with
γ the
surface
tension.
Wave action, wave energy flux and radiation stress
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
dissipative effects), 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
slowlyvarying wave trains, propagating in slowlyvarying
bathymetry and meanflow fields, a similar and
conserved wave quantity, the wave action \mathcal{A}=E/\sigma:
Phillips (1977), p. 26.
 \frac{\partial \mathcal{A}}{\partial t}\, +\, \nabla\cdot\left[
\left(\boldsymbol{U}+\boldsymbol{c}_g\right)\, \mathcal{A}\right]\,
=\, 0,
with \left(\boldsymbol{U}+\boldsymbol{c}_g\right)\, \mathcal{A} the
action
flux and
\boldsymbol{c}_g=c_g\,\boldsymbol{e}_k the
group velocity vector. Action conservation
forms the basis for many
wind wave
models and
wave turbulence
models. It is also the basis of
coastal engineering models for the
computation of
wave shoaling.
Expanding the above wave action conservation equation leads to the
following evolution equation for the wave energy density:Phillips
(1977), p. 66.
 \frac{\partial E}{\partial t}\, +\, \nabla\cdot\left[\left(
\boldsymbol{U}+\boldsymbol{c}_g\right)\, E \right]\, +\,
\mathbb{S}:\left(\nabla\boldsymbol{U}\right)\, =\, 0,
with:
 \left( \boldsymbol{U}+\boldsymbol{c}_g\right)\, E is the mean
wave energy density flux,
 \mathbb{S} is the radiation stress tensor
and
 \nabla\boldsymbol{U} is the meanvelocity shearrate tensor.
In this equation in nonconservation form, the
Frobenius inner product
\mathbb{S}:(\nabla\boldsymbol{U}) is the source term describing the
energy exchange of the wave motion with the mean flow. Only in case
the mean shearrate is zero, \nabla\boldsymbol{U}=\mathsf{0}, the
mean wave energy density E is conserved. The two tensors \mathbb{S}
and \nabla\boldsymbol{U} are in a
Cartesian coordinate system of
the form:
\begin{align}
\mathbb{S}\, &=\, \begin{pmatrix} S_{xx} & S_{xy} \\ S_{yx} & S_{yy} \end{pmatrix}\,
=\, \mathbb{I}\, \left( \frac{c_g}{c_p}  \frac12 \right)\, E\,
+\, \frac{1}{k^2}\, \begin{pmatrix} k_x\, k_x & k_x\, k_y \\[2ex] k_y\, k_x & k_y\, k_y \end{pmatrix}\, \frac{c_g}{c_p}\, E,
\\
\mathbb{I}\, &=\, \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\quad \text{and}
\\
\nabla \boldsymbol{U}\, &=\,
\begin{pmatrix}
\displaystyle \frac{\partial U_x}{\partial x} & \displaystyle \frac{\partial U_y}{\partial x}
\\[2ex]
\displaystyle \frac{\partial U_x}{\partial y} & \displaystyle \frac{\partial U_y}{\partial y}
\end{pmatrix},
\end{align}
with k_x and k_y the components of the wavenumber vector
\boldsymbol{k} and similarly U_x and U_y the components in of the
mean velocity vector \boldsymbol{U}.
Wave mass flux and wave momentum
The mean horizontal
momentum per unit area
\boldsymbol{M} induced by the wave motion — and also the
waveinduced
mass flux or mass
transport — is:Phillips (1977), pp.
39–40 & 61.
\boldsymbol{M}\, =\,
\overline{\int_{h}^\eta \rho\, \left( \boldsymbol{U}+\boldsymbol{u}_x\right)\; \text{d}z}\,
\, \int_{h}^0 \rho\, \boldsymbol{U}\; \text{d}z\,
=\, \frac{E}{c_p}\, \boldsymbol{e}_k,
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.
Stokes
already identified two possible definitions of
phase velocity for periodic nonlinear waves:
 Stokes first definition of wave celerity (S1) — with the mean Eulerian flow velocity
equal to zero for all elevations z below the wave troughs, and
 Stokes second definition of wave celerity (S2) — with
the mean mass transport equal to zero.
The above relation between wave momentum
M and wave energy density
E is
valid within the framework of Stokes' first definition.
However, for waves perpendicular to a coast line or in closed
laboratory wave channel, 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 waveinduced mass flux in the
wave propagation direction, which has to be balanced by a mean flow
U in the opposite direction — called the
undertow.
So in general, there are quite some subtleties involved. Therefore
also the term pseudomomentum of the waves is used instead of wave
momentum.
Mass and momentum evolution equations
For slowlyvarying
bathymetry, wave and
meanflow fields, the evolution of the mean flow can de described
in terms of the mean masstransport velocity \tilde{\boldsymbol{U}}
defined as:Phillips (1977), pp. 61–63.
 \tilde{\boldsymbol{U}}\, =\, \boldsymbol{U}\, +\,
\frac{\boldsymbol{M}}{\rho\,h}.
Note that for deep water, when the mean depth
h goes to
infinity, the mean Eulerian velocity \boldsymbol{U} and mean
transport velocity \tilde{\boldsymbol{U}} become equal.
The equation for mass conservation is:
\frac{\partial}{\partial t}\left( \rho\, h\, \right)\,
+\, \nabla \cdot \left( \rho\, h\,\tilde{\boldsymbol{U}} \right)\,
=\, 0,
where
h(
x,
t) is the
mean waterdepth, slowly varying in space and time.Similarly, the
mean horizontal momentum evolves as:
\frac{\partial}{\partial t}\left( \rho\, h\, \tilde{\boldsymbol{U}}\right)\,
+\, \nabla \cdot \left( \rho\, h\, \tilde{\boldsymbol{U}} \otimes \tilde{\boldsymbol{U}}\, +\, \frac12\,\rho\,g\,h^2\,\mathbb{I}\, +\, \mathbb{S} \right)\,
=\, \rho\, g\, h\, \nabla d,
with
d the stillwater depth (the sea bed is at
z=–
d), \mathbb{S} is the wave radiationstress
tensor, \mathbb{I} is the
identity matrix and \otimes is the
dyadic product:
\tilde{\boldsymbol{U}} \otimes \tilde{\boldsymbol{U}}\, =\,
\begin{pmatrix}
\tilde{U}_x\, \tilde{U}_x & \tilde{U}_x\, \tilde{U}_y
\\[2ex]
\tilde{U}_y\, \tilde{U}_x & \tilde{U}_y\, \tilde{U}_y
\end{pmatrix}.
Note that mean horizontal
momentum is only
conserved if the sea bed is horizontal (
i.e the
stillwater depth
d is a constant), in agreement with
Noether's theorem.
The system of equations is closed through the description of the
waves. Wave energy propagation is described through the waveaction
conservation equation (without dissipation and nonlinear wave
interactions):Phillips (1977), p. 66.
\frac{\partial}{\partial t} \left( \frac{E}{\sigma}\, \right)
+\, \nabla \cdot \left[ \left( \boldsymbol{U} +\boldsymbol{c}_g \right)\, \frac{E}{\sigma} \right]\,
=\, 0.
The wave kinematics are described through the wavecrest
conservation equation:
 \frac{\partial \boldsymbol{k}}{\partial t}\, +\, \nabla
\omega\, =\, \boldsymbol{0},
with the angular frequency
ω a function of the (angular)
wavenumber k,
related through the
dispersion
relation. For this to be possible, the wave field must be
coherent. By taking the
curl of the wavecrest conservation, it can be seen
that an initially
irrotational
wavenumber field stays irrotational.
Stokes drift
When following a single particle in pure wave motion
(\boldsymbol{U}=\boldsymbol{0}), according to linear Airy wave
theory the particles are in closed elliptical orbit. However, in
nonlinear waves this is no longer the case and the particles
exhibit a
Stokes drift. The Stokes
drift velocity \bar{\boldsymbol{u}}_S, which is the Stokes drift
after one wave cycle divided by the
period,
can be estimated using the results of linear theory:Phillips
(1977), p. 44.
 \bar{\boldsymbol{u}}_S\, =\, \frac12\, \sigma\, k\, a^2\,
\frac{\cosh\, 2\,k\,(z+h)}{\sinh^2\, (k\,h)}\,
\boldsymbol{e}_k,
so it varies as a function of elevaton. The given formula is for
Stokes first definition of wave celerity. When
\rho\,\bar{\boldsymbol{u}}_S is
integrated
over depth, the expression for the mean wave momentum
\boldsymbol{M} is recovered.
See also
Notes
 Craik (2004).
 Dean & Dalrymple (1991).
 Phillips (1977), §3.2, pp. 37–43 and §3.6, pp. 60–69.
 Stokes (1847).
 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.
 Dean & Dalrymple (1991) pp. 64–65
 The error in the phase speed is less than 0.2% if depth
h is taken to be infinite, for h > ½
λ.
 Lighthill (1978), p. 223.
 Lamb, H.
(1994), §267, page 458–460.
 Dingemans (1997), Section 2.1.1, p. 45.
 See for example: the High seas forecasts of NOAA's National Weather service.
 Phillips (1977), p. 23–25.
 Reprinted as Appendix in: Theory of Sound
1, MacMillan, 2nd revised edition, 1894.
 Phillips (1977), pp. 179–183.
 Phillips (1977), pp. 70–74.
 Phillips (1977), p. 68.
 Phillips (1977), p. 40.
 Phillips (1977), p. 70.
 Phillips (1977), p. 23.
References
Historical
 . Also: "Trigonometry, On the Figure of the Earth, Tides and
Waves", 396 pp.
 Reprinted in:
Further reading
 Two parts, 967 pages.
 Originally published in 1879, the 6^{th} extended
edition appeared first in 1932.
 504 pp.
External links