Airy wave theory

In

Airy wave theory uses a

• a is the wave amplitude in metres,
• cos is the
  cosine
  function,
• k is the
  angular wavenumber in radians per metre, related to the wavelength
  λ by k = ⁠2π/λ⁠,
• ω is the
  period T and frequency
  f by ω = ⁠2π/T⁠ = 2πf.

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

${\displaystyle H=2a\quad {\text{and}}\quad a={\tfrac {1}{2}}H,}$

valid in the present case of linear periodic waves.

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 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 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 free-surface value at a depth of half a wavelength.

In a similar fashion, there is also a pressure 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.

The waves propagate in the horizontal direction, with

components u_x and u_z in the horizontal (x) and vertical (z) directions by:

${\displaystyle u_{x}={\frac {\partial \Phi }{\partial x}}\quad {\text{and}}\quad u_{z}={\frac {\partial \Phi }{\partial z}}.}$

Then, due to the

Laplace equation

:

$${\displaystyle {\frac {\partial ^{2}\Phi }{\partial x^{2}}}+{\frac {\partial ^{2}\Phi }{\partial z^{2}}}=0.}$$

1

The bed being impermeable, leads to the kinematic bed boundary-condition:

$${\displaystyle {\frac {\partial \Phi }{\partial z}}=0\quad {\text{ at }}z=-h.}$$

2

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 free-surface boundary-condition:

$${\displaystyle {\frac {\partial \eta }{\partial t}}={\frac {\partial \Phi }{\partial z}}\quad {\text{ at }}z=\eta (x,t).}$$

3

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

$${\displaystyle {\frac {\partial \Phi }{\partial t}}+g\eta =0\quad {\text{ at }}z=\eta (x,t).}$$

4

Because this is a linear theory, in both free-surface 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.

For a propagating wave of a single frequency – a

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:^[8]

• deep water – for a water depth larger than half 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),^[9]
• 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;^[10] 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.

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[7]
quantity	symbol	units	deep water (h > ⁠1/2⁠λ)	shallow water (h < ⁠1/20⁠λ)	intermediate depth (all λ and h)
surface elevation	${\displaystyle \eta (\mathbf {x} ,t)}$	m	${\displaystyle a\cos \theta (\mathbf {x} ,t)}$
wave phase	${\displaystyle \theta (\mathbf {x} ,t)}$	rad	${\displaystyle \mathbf {k} \cdot \mathbf {x} -\omega t}$
observed angular frequency	${\displaystyle \omega }$	rad·s−1	${\displaystyle \left(\omega -\mathbf {k} \cdot \mathbf {U} \right)^{2}={\bigl (}\Omega (k){\bigr )}^{2}\quad {\text{ with }}\quad k=|\mathbf {k} |}$
intrinsic angular frequency	${\displaystyle \sigma }$	rad·s−1	${\displaystyle \quad \sigma ^{2}={\bigl (}\Omega (k){\bigr )}^{2}\quad {\text{ with }}\quad \sigma =\omega -\mathbf {k} \cdot \mathbf {U} }$
unit vector in the wave propagation direction	${\displaystyle \mathbf {e} _{k}}$	–	${\displaystyle {\frac {\mathbf {k} }{k}}}$
dispersion relation	${\displaystyle \Omega (k)}$	rad·s−1	${\displaystyle \Omega (k)={\sqrt {gk}}}$	${\displaystyle \Omega (k)=k{\sqrt {gh}}}$	${\displaystyle \Omega (k)={\sqrt {gk\tanh kh}}}$
phase speed	${\displaystyle c_{p}={\frac {\Omega (k)}{k}}}$	m·s−1	${\displaystyle {\sqrt {\frac {g}{k}}}={\frac {g}{\sigma }}}$	${\displaystyle {\sqrt {gh}}}$	${\displaystyle {\sqrt {{\frac {g}{k}}\tanh kh}}}$
group speed	${\displaystyle c_{g}={\frac {\partial \Omega }{\partial k}}}$	m·s−1	${\displaystyle {\tfrac {1}{2}}{\sqrt {\frac {g}{k}}}={\tfrac {1}{2}}{\frac {g}{\sigma }}}$	${\displaystyle {\sqrt {gh}}}$	${\displaystyle {\tfrac {1}{2}}c_{p}\left(1+kh{\frac {1-\tanh ^{2}kh}{\tanh kh}}\right)}$
ratio	${\displaystyle {\frac {c_{g}}{c_{p}}}}$	–	${\displaystyle {\tfrac {1}{2}}}$	${\displaystyle 1}$	${\displaystyle {\tfrac {1}{2}}\left(1+kh{\frac {1-\tanh ^{2}kh}{\tanh kh}}\right)}$
horizontal velocity	${\displaystyle \mathbf {u} _{x}(\mathbf {x} ,z,t)}$	m·s−1	${\displaystyle \mathbf {e} _{k}\sigma a\,e^{kz}\cos \theta }$	${\displaystyle \mathbf {e} _{k}{\sqrt {\frac {g}{h}}}a\cos \theta }$	${\displaystyle \mathbf {e} _{k}\sigma a{\frac {\cosh k(z+h)}{\sinh kh}}\cos \theta }$
vertical velocity	${\displaystyle u_{z}(\mathbf {x} ,z,t)}$	m·s−1	${\displaystyle \sigma a\,e^{kz}\sin \theta }$	${\displaystyle \sigma a{\frac {z+h}{h}}\sin \theta }$	${\displaystyle \sigma a{\frac {\sinh k(z+h)}{\sinh kh}}\sin \theta }$
horizontal particle excursion	${\displaystyle {\boldsymbol {\xi }}_{x}(\mathbf {x} ,z,t)}$	m	${\displaystyle -\mathbf {e} _{k}a\,e^{kz}\sin \theta }$	${\displaystyle -\mathbf {e} _{k}{\frac {1}{kh}}a\sin \theta }$	${\displaystyle -\mathbf {e} _{k}a{\frac {\cosh k(z+h)}{\sinh kh}}\sin \theta }$
vertical particle excursion	${\displaystyle \xi _{z}(\mathbf {x} ,z,t)}$	m	${\displaystyle a\,e^{kz}\cos \theta }$	${\displaystyle a{\frac {z+h}{h}}\cos \theta }$	${\displaystyle a{\frac {\sinh k(z+h)}{\sinh kh}}\cos \theta }$
pressure oscillation	${\displaystyle p(\mathbf {x} ,z,t)}$	N·m−2	${\displaystyle \rho ga\,e^{kz}\cos \theta }$	${\displaystyle \rho ga\cos \theta }$	${\displaystyle \rho ga{\frac {\cosh k(z+h)}{\cosh kh}}\cos \theta }$

Due to surface tension, the dispersion relation changes to:^[11]

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

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^[12]

${\displaystyle {\tilde {g}}=g+{\frac {\gamma }{\rho }}k^{2}.}$

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

The group velocity ⁠∂Ω/∂k⁠ of capillary waves – dominated by surface tension effects – is greater than the phase velocity ⁠Ω/k⁠. 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.^[13]

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

Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation ω² = Ω²(k) is given through[11]^[14][15]

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

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.

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.

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:^[16]

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

where again ρ and ρ′ are the densities below and above the interface, while coth is the

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

, 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.

Second-order quantities and their dynamics, using results of Airy wave theory
quantity	symbol	units	formula
mean wave-energy density per unit horizontal area	${\displaystyle E}$	J·m−2	${\displaystyle E={\tfrac {1}{2}}\rho ga^{2}}$
radiation stress or excess horizontal momentum flux due to the wave motion	${\displaystyle S_{xx}}$	N·m−1	${\displaystyle S_{xx}=\left(2{\frac {c_{g}}{c_{p}}}-{\tfrac {1}{2}}\right)E}$
wave action	${\displaystyle {\mathcal {A}}}$	J·s·m−2	${\displaystyle {\mathcal {A}}={\frac {E}{\sigma }}={\frac {E}{\omega -kU}}}$
mean mass-flux due to the wave motion or the wave pseudo-momentum	${\displaystyle M}$	kg·m−1·s−1	${\displaystyle M={\frac {E}{c_{p}}}=k{\frac {E}{\sigma }}}$
mean horizontal mass-transport velocity	${\displaystyle {\tilde {U}}}$	m·s−1	${\displaystyle {\tilde {U}}=U+{\frac {M}{\rho h}}=U+{\frac {E}{\rho hc_{p}}}}$
Stokes drift	${\displaystyle {\bar {u}}_{S}}$	m·s−1	${\displaystyle {\bar {u}}_{S}={\tfrac {1}{2}}\sigma ka^{2}{\frac {\cosh 2k(z+h)}{\sinh ^{2}kh}}}$
wave-energy propagation		J·m−2·s−1	${\displaystyle {\frac {\partial E}{\partial t}}+{\frac {\partial }{\partial x}}{\bigl (}(U+c_{g})E{\bigr )}+S_{xx}{\frac {\partial U}{\partial x}}=0}$
wave action conservation		J·m−2	${\displaystyle {\frac {\partial {\mathcal {A}}}{\partial t}}+{\frac {\partial }{\partial x}}{\bigl (}(U+c_{g}){\mathcal {A}}{\bigr )}=0}$
wave- crest conservation		rad·m−1·s−1	${\displaystyle {\frac {\partial k}{\partial t}}+{\frac {\partial \omega }{\partial x}}=0\quad {\text{with}}\quad \omega =\Omega (k)+kU}$
mean mass conservation		kg·m−2·s−1	${\displaystyle {\frac {\partial }{\partial t}}(\rho h)+{\frac {\partial }{\partial x}}\left(\rho h{\tilde {U}}\right)=0}$
mean horizontal-momentum evolution		N·m−2	${\displaystyle {\frac {\partial }{\partial t}}\left(\rho h{\tilde {U}}\right)+{\frac {\partial }{\partial x}}\left(\rho h{\tilde {U}}^{2}+{\tfrac {1}{2}}\rho gh^{2}+S_{xx}\right)=\rho gh{\frac {\partial d}{\partial x}}}$

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

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.[20] 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:^[21]

${\displaystyle {\begin{aligned}E_{\text{pot}}&={\overline {\int _{-h}^{\eta }\rho gz\,\mathrm {d} z}}-\int _{-h}^{0}\rho gz\,\mathrm {d} z\\[6px]&={\overline {{\tfrac {1}{2}}\rho g\eta ^{2}}}={\tfrac {1}{4}}\rho ga^{2}.\end{aligned}}}$

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).

The mean kinetic energy density per unit horizontal area E_kin of the wave motion is similarly found to be:^[21]

${\displaystyle {\begin{aligned}E_{\text{kin}}&={\overline {\int _{-h}^{0}{\tfrac {1}{2}}\rho \left[\left|\mathbf {U} +\mathbf {u} _{x}\right|^{2}+u_{z}^{2}\right]\,\mathrm {d} z}}-\int _{-h}^{0}{\tfrac {1}{2}}\rho \left|\mathbf {U} \right|^{2}\,\mathrm {d} z\\[6px]&={\tfrac {1}{4}}\rho {\frac {\sigma ^{2}}{k\tanh kh}}a^{2},\end{aligned}}}$

with σ the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for surface gravity waves is:

${\displaystyle E_{\text{kin}}={\tfrac {1}{4}}\rho ga^{2}.}$

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.^[22][23] Adding potential and kinetic contributions, E_pot and E_kin, the mean energy density per unit horizontal area E of the wave motion is:

${\displaystyle E=E_{\text{pot}}+E_{\text{kin}}={\tfrac {1}{2}}\rho ga^{2}.}$

In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving^[22]

${\displaystyle E_{\text{pot}}=E_{\text{kin}}={\tfrac {1}{4}}\left(\rho g+\gamma k^{2}\right)a^{2},}$

so

${\displaystyle E=E_{\text{pot}}+E_{\text{kin}}={\tfrac {1}{2}}\left(\rho g+\gamma k^{2}\right)a^{2},}$

with γ the surface tension.

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:^[19][24][25]

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

with (U + c_g) A the action flux and c_g = c_ge_k the group velocity vector. Action conservation forms the basis for many wind wave models and wave turbulence models.^[26] It is also the basis of coastal engineering models for the computation of wave shoaling.^[27] Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:^[28]

${\displaystyle {\frac {\partial E}{\partial t}}+\nabla \cdot \left[\left(\mathbf {U} +\mathbf {c} _{g}\right)E\right]+{\boldsymbol {S}}:\left(\nabla \mathbf {U} \right)=0,}$

with:

• (U + c_g)E is the mean wave energy density flux,
• S is the radiation stress tensor and
• ∇U is the mean-velocity shear rate tensor.

In this equation in non-conservation form, the Frobenius inner product S : (∇U) 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, ∇U = 0, the mean wave energy density E is conserved. The two tensors S and ∇U are in a Cartesian coordinate system of the form:^[29]

${\displaystyle {\begin{aligned}{\boldsymbol {S}}&={\begin{pmatrix}S_{xx}&S_{xy}\\S_{yx}&S_{yy}\end{pmatrix}}={\boldsymbol {I}}\left({\frac {c_{g}}{c_{p}}}-{\frac {1}{2}}\right)E+{\frac {1}{k^{2}}}{\begin{pmatrix}k_{x}k_{x}&k_{x}k_{y}\\[2ex]k_{y}k_{x}&k_{y}k_{y}\end{pmatrix}}{\frac {c_{g}}{c_{p}}}E,\\[6px]{\boldsymbol {I}}&={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\\[6px]\nabla \mathbf {U} &={\begin{pmatrix}\displaystyle {\frac {\partial U_{x}}{\partial x}}&\displaystyle {\frac {\partial U_{y}}{\partial x}}\\[2ex]\displaystyle {\frac {\partial U_{x}}{\partial y}}&\displaystyle {\frac {\partial U_{y}}{\partial y}}\end{pmatrix}},\end{aligned}}}$

with k_x and k_y the components of the wavenumber vector k and similarly U_x and U_y the components in of the mean velocity vector U.

The mean horizontal