Airy wave theory

Description

Airy wave theory uses a potential flow (or velocity potential) 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 or flow separation 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 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 non-linear wave motion – known as Stokes' wave theory – correct up to third order in the wave steepness. Even before Airy's linear theory, Gerstner derived a nonlinear trochoidal wave theory in 1802, 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:

η

(

x

,

t

)

=

a

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 ux and uz in the horizontal (x) and vertical (z) directions by:

u

x

=

∂

Φ

∂

x

and

u

z

=

∂

Φ

∂

z

.

{\displaystyle 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:

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 zeroth-order solution) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero.

Solution for a progressive monochromatic wave

For a propagating wave of a single frequency – a monochromatic wave – the surface elevation is of the form:

η

=

a

cos

⁡

(

k

x

−

ω

t

)

.

{\displaystyle \eta =a\cos(kx-\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:

Φ

=

ω

k

a

cosh

⁡

k

(

z

+

h

)

sinh

⁡

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 Earth-fixed 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 σ. So in pure wave motion, with U = 0, both frequencies ω and σ are equal. The wave number k (and wavelength 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 ξx and ξz are the time integrals of the oscillatory flow velocities ux and uz respectively.

Water depth is classified into three regimes:

deep water – for a water depth larger than half the wavelength, h > ⁠1/2⁠λ, 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 5% of the wavelength, h < ⁠1/20⁠λ, the phase speed of the waves is only dependent on water depth, and no longer a function of period or wavelength; and

intermediate depth – all other cases, ⁠1/20⁠λ < h < ⁠1/2⁠λ, where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory.

Surface tension effects

Due to surface tension, the dispersion relation changes to:

Ω

2

(

k

)

=

(

g

+

γ

ρ

k

2

)

k

tanh

⁡

k

h

,

{\displaystyle \Omega ^{2}(k)=\left(g+{\frac {\gamma }{\rho }}k^{2}\right)k\,\tanh kh,}

with γ the surface tension in newtons per metre. All above equations for linear waves remain the same, if the gravitational acceleration g is replaced by

g

~

=

g

+

γ

ρ

k

2

Interfacial waves

Surface waves are a special case of interfacial waves, on the interface between two fluids of different density.

Two layers of infinite depth

Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation ω2 = Ω2(k) is given through

Ω

2

(

k

)

=

|

k

|

(

ρ

−

ρ

′

ρ

+

ρ

′

g

+

γ

ρ

+

ρ

′

k

2

)

,

{\displaystyle \Omega ^{2}(k)=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}g+{\frac {\gamma }{\rho +\rho '}}k^{2}\right),}

where ρ and ρ′ are the densities of the two fluids, below (ρ) and above (ρ′) the interface, respectively. Further γ is the surface tension on the interface.

Two layers between horizontal rigid planes

For two homogeneous layers of fluids, of mean thickness h below the interface and h′ above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship ω2 = Ω2(k) for gravity waves is provided by:

Ω

2

(

k

)

=

g

k

(

ρ

−

ρ

′

)

ρ

coth

⁡

k

h

+

ρ

′

coth

⁡

k

h

′

,

{\displaystyle \Omega ^{2}(k)={\frac {gk(\rho -\rho ')}{\rho \coth kh+\rho '\coth kh'}},}

where again ρ and ρ′ are the densities below and above the interface, while coth is the hyperbolic cotangent function. For the case ρ′ is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth h.

Two layers bounded above by a free surface

In this case the dispersion relation allows for two modes: a barotropic mode where the free surface amplitude is large compared with the amplitude of the interfacial wave, and a baroclinic mode where the opposite is the case – the interfacial wave is higher than and in antiphase with the free surface wave. The dispersion relation for this case is of a more complicated form.

Second-order wave properties

Several second-order wave properties, ones that are quadratic in the wave amplitude a, can be derived directly from Airy wave theory. They are of importance in many practical applications, such as forecasts of wave conditions. Using a WKBJ approximation, second-order wave properties also find their applications in describing waves in case of slowly varying bathymetry, 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.

Table of second-order wave properties

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

The last four equations describe the evolution of slowly varying wave trains over bathymetry in interaction with the mean flow, and can be derived from a variational principle: Whitham's averaged Lagrangian method. In the mean horizontal-momentum equation, d(x) is the still water depth, that is, the bed underneath the fluid layer is located at z = −d. Note that the mean-flow velocity in the mass and momentum equations is the mass transport velocity Ũ, including the splash-zone effects of the waves on horizontal mass transport, and not the mean Eulerian velocity (for example, 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 Epot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:

E

pot

=

∫

−

h

η

ρ

g

z

d

z

¯

−

∫

−

h

0

ρ

g

z

d

z

=

1

2

ρ

g

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 slowly varying wave trains, propagating in slowly varying bathymetry and mean-flow fields, a similar and conserved wave quantity, the wave action A = ⁠E/σ⁠:

∂

A

∂

t

+

∇

⋅

[

(

U

+

c

g

)

A

]

=

0

,

{\displaystyle {\frac {\partial {\mathcal {A}}}{\partial t}}+\nabla \cdot \left[\left(\mathbf {U} +\mathbf {c} _{g}\right){\mathcal {A}}\right]=0,}

with (U + cg) A the action flux and cg = cgek 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:

Wave mass flux and wave momentum

The mean horizontal momentum per unit area M induced by the wave motion – and also the wave-induced mass flux or mass transport – is:

M

=

∫

−

h

η

ρ

(

U

+

u

x

)

d

z

¯

−

∫

−

h

0

ρ

U

d

z

=

E

c

p

e

k

,

{\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\\[6px]&={\frac {E}{c_{p}}}\mathbf {e} _{k},\end{aligned}}}

Stokes drift

When following a single particle in pure wave motion (U = 0), according to linear Airy wave theory, a first approximation gives closed elliptical orbits for water particles. However, for nonlinear waves, particles exhibit a Stokes drift for which a second-order expression can be derived from the results of Airy wave theory (see the table above on second-order wave properties). The Stokes drift velocity ūS, which is the particle drift after one wave cycle divided by the period, can be estimated using the results of linear theory:

u

¯

S

=

1

2

σ

k

a

2

cosh

⁡

2

k

(

z

+

h

)

sinh

2

⁡

k

h

e

k

,

{\displaystyle {\bar {\mathbf {u} }}_{S}={\tfrac {1}{2}}\sigma ka^{2}{\frac {\cosh 2k(z+h)}{\sinh ^{2}kh}}\mathbf {e} _{k},}