Boussinesq approximation (water waves)

non-linearity

.

In

The Boussinesq approximation for

harbours

.

While the Boussinesq approximation is applicable to fairly long waves – that is, when the

Stokes expansion

is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).

Boussinesq approximation

gravitational acceleration

is 9.81 m/s².

The essential idea in the Boussinesq approximation is the elimination of the vertical

water waves

. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation.

This elimination of the vertical coordinate was first done by

wave of translation

). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.

The steps in the Boussinesq approximation are:

• a
  Taylor expansion is made of the horizontal and vertical flow velocity (or velocity potential) around a certain elevation
  ,
• this
  Taylor expansion is truncated to a finite
  number of terms,
• the conservation of mass (see
  partial derivatives
  .

Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the resulting

).

As an example, consider potential flow over a horizontal bed in the ${\displaystyle (x,z)}$ plane, with ${\displaystyle x}$ the horizontal and ${\displaystyle z}$ the vertical

. The bed is located at ${\displaystyle z=-h}$, where ${\displaystyle h}$ is the ${\displaystyle \varphi (x,z,t)}$ around the bed level ${\displaystyle z=-h}$:[2]

${\displaystyle {\begin{aligned}\varphi \,=\,&\varphi _{b}\,+\,(z+h)\,\left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-h}\,+\,{\frac {1}{2}}\,(z+h)^{2}\,\left[{\frac {\partial ^{2}\varphi }{\partial z^{2}}}\right]_{z=-h}\,\\&+\,{\frac {1}{6}}\,(z+h)^{3}\,\left[{\frac {\partial ^{3}\varphi }{\partial z^{3}}}\right]_{z=-h}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,\left[{\frac {\partial ^{4}\varphi }{\partial z^{4}}}\right]_{z=-h}\,+\,\cdots ,\end{aligned}}}$

where ${\displaystyle \varphi _{b}(x,t)}$ is the velocity potential at the bed. Invoking Laplace's equation for ${\displaystyle \varphi }$, as valid for incompressible flow, gives:

${\displaystyle {\begin{aligned}\varphi \,=\,&\left\{\,\varphi _{b}\,-\,{\frac {1}{2}}\,(z+h)^{2}\,{\frac {\partial ^{2}\varphi _{b}}{\partial x^{2}}}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,{\frac {\partial ^{4}\varphi _{b}}{\partial x^{4}}}\,+\,\cdots \,\right\}\,\\&+\,\left\{\,(z+h)\,\left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-h}\,-\,{\frac {1}{6}}\,(z+h)^{3}\,{\frac {\partial ^{2}}{\partial x^{2}}}\left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-h}\,+\,\cdots \,\right\}\\=\,&\left\{\,\varphi _{b}\,-\,{\frac {1}{2}}\,(z+h)^{2}\,{\frac {\partial ^{2}\varphi _{b}}{\partial x^{2}}}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,{\frac {\partial ^{4}\varphi _{b}}{\partial x^{4}}}\,+\,\cdots \,\right\},\end{aligned}}}$

since the vertical velocity ${\displaystyle \partial \varphi /\partial z}$ is zero at the – impermeable – horizontal bed ${\displaystyle z=-h}$. This series may subsequently be truncated to a finite number of terms.

Original Boussinesq equations

Derivation

For

in the ${\displaystyle (x,z)}$ plane, the elevation ${\displaystyle z=\eta (x,t)}$ are:[3]

${\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}\,&+\,u\,{\frac {\partial \eta }{\partial x}}\,-\,w\,=\,0\\{\frac {\partial \varphi }{\partial t}}\,&+\,{\frac {1}{2}}\,\left(u^{2}+w^{2}\right)\,+\,g\,\eta \,=\,0,\end{aligned}}}$

where:

• ${\displaystyle u}$ is the horizontal flow velocity component: ${\displaystyle u=\partial \varphi /\partial x}$,
• ${\displaystyle w}$ is the vertical flow velocity component: ${\displaystyle w=\partial \varphi /\partial z}$,
• ${\displaystyle g}$ is the acceleration by gravity.

Now the Boussinesq approximation for the velocity potential ${\displaystyle \varphi }$, as given above, is applied in these

terms with respect to ${\displaystyle \eta }$ and ${\displaystyle u_{b}}$ are retained (with ${\displaystyle u_{b}=\partial \varphi _{b}/\partial x}$ the horizontal velocity at the bed ${\displaystyle z=-h}$). The

partial differential equations

are obtained:

set A – Boussinesq (1872), equation (25)

${\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}\,&+\,{\frac {\partial }{\partial x}}\,\left[\left(h+\eta \right)\,u_{b}\right]\,=\,{\frac {1}{6}}\,h^{3}\,{\frac {\partial ^{3}u_{b}}{\partial x^{3}}},\\{\frac {\partial u_{b}}{\partial t}}\,&+\,u_{b}\,{\frac {\partial u_{b}}{\partial x}}\,+\,g\,{\frac {\partial \eta }{\partial x}}\,=\,{\frac {1}{2}}\,h^{2}\,{\frac {\partial ^{3}u_{b}}{\partial t\,\partial x^{2}}}.\end{aligned}}}$

This set of equations has been derived for a flat horizontal bed, i.e. the mean depth ${\displaystyle h}$ is a constant independent of position ${\displaystyle x}$. When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations.

Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation ${\displaystyle \eta }$:

set B – Boussinesq (1872), equation (26)

${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}\,-\,gh\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}\,-\,gh\,{\frac {\partial ^{2}}{\partial x^{2}}}\left({\frac {3}{2}}\,{\frac {\eta ^{2}}{h}}\,+\,{\frac {1}{3}}\,h^{2}\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}\right)\,=\,0.}$

From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number. In

, using the water depth ${\displaystyle h}$ and gravitational acceleration ${\displaystyle g}$ for non-dimensionalization, this equation reads, after

${\displaystyle {\frac {\partial ^{2}\psi }{\partial \tau ^{2}}}\,-\,{\frac {\partial ^{2}\psi }{\partial \xi ^{2}}}\,-\,{\frac {\partial ^{2}}{\partial \xi ^{2}}}\left(\,3\,\psi ^{2}\,+\,{\frac {\partial ^{2}\psi }{\partial \xi ^{2}}}\,\right)\,=\,0,}$

with:

${\displaystyle \psi \,=\,{\frac {1}{2}}\,{\frac {\eta }{h}}}$	: the dimensionless surface elevation,
${\displaystyle \tau \,=\,{\sqrt {3}}\,t\,{\sqrt {\frac {g}{h}}}}$	: the dimensionless time, and
${\displaystyle \xi \,=\,{\sqrt {3}}\,{\frac {x}{h}}}$	: the dimensionless horizontal position.

Linear frequency dispersion

.

The linear frequency dispersion characteristics for the above set A of equations are:^[5]

${\displaystyle c^{2}\,=\;gh\,{\frac {1\,+\,{\frac {1}{6}}\,k^{2}h^{2}}{1\,+\,{\frac {1}{2}}\,k^{2}h^{2}}},}$

with:

• ${\displaystyle c}$ the
  phase speed
  ,
• ${\displaystyle k}$ the
  wave number
  (${\displaystyle k=2\pi /\lambda }$, with ${\displaystyle \lambda }$ the
  wave length
  ).

The

in the phase speed ${\displaystyle c}$ for set A, as compared with linear theory for water waves, is less than 4% for a relative wave number ${\displaystyle kh<\pi /2}$. So, in engineering applications, set A is valid for wavelengths ${\displaystyle \lambda }$ larger than 4 times the water depth ${\displaystyle h}$.

The linear frequency dispersion characteristics of equation B are:^[5]

${\displaystyle c^{2}\,=\,gh\,\left(1\,-\,{\frac {1}{3}}\,k^{2}h^{2}\right).}$

The relative error in the phase speed for equation B is less than 4% for ${\displaystyle kh<2\pi /7}$, equivalent to wave lengths ${\displaystyle \lambda }$ longer than 7 times the water depth ${\displaystyle h}$, called fairly long waves.^[6]

For short waves with ${\displaystyle k^{2}h^{2}>3}$ equation B become physically meaningless, because there are no longer

. The original set of two (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.

The shallow water equations have a relative error in the phase speed less than 4% for wave lengths ${\displaystyle \lambda }$ in excess of 13 times the water depth ${\displaystyle h}$.

Boussinesq-type equations and extensions

There are an overwhelming number of

which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as the Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them Boussinesq-type equations. Strictly speaking, the Boussinesq equations is the above-mentioned set B, since it is used in the analysis in the remainder of his 1872 paper.

Some directions, into which the Boussinesq equations have been extended, are:

• varying bathymetry,
• improved frequency dispersion,
• improved
  non-linear
  behavior,
• making a
  elevations
  ,
• dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
• inclusion of
  wave breaking
  ,
• inclusion of surface tension,
• extension to
  mass density
  ,
• derivation from a variational principle.

Further approximations for one-way wave propagation

While the Boussinesq equations allow for waves traveling simultaneously in opposing directions, it is often advantageous to only consider waves traveling in one direction. Under small additional assumptions, the Boussinesq equations reduce to:

• the
  wave propagation in one horizontal dimension
  ,
• the
  wave propagation in two horizontal dimensions
  ,
• the
  modulated
  waves).

Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called cnoidal waves. These are approximate solutions of the Boussinesq equation.

Numerical models

For the simulation of wave motion near coasts and harbours, numerical models – both commercial and academic – employing Boussinesq-type equations exist. Some commercial examples are the Boussinesq-type wave modules in MIKE 21 and SMS. Some of the free Boussinesq models are Celeris,^[7] COULWAVE,^[8] and FUNWAVE.^[9] Most numerical models employ finite-difference, finite-volume or finite element techniques for the discretization of the model equations. Scientific reviews and intercomparisons of several Boussinesq-type equations, their numerical approximation and performance are e.g. Kirby (2003), Dingemans (1997, Part 2, Chapter 5) and Hamm, Madsen & Peregrine (1993).

Notes

References

• Boussinesq, J. (1871). "Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire"
  . Comptes Rendus de l'Académie des Sciences. 72: 755–759.
• Boussinesq, J. (1872). "Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond". Journal de Mathématiques Pures et Appliquées
  . Deuxième Série. 17: 55–108.
• Dingemans, M.W. (1997). Wave propagation over uneven bottoms. Advanced Series on Ocean Engineering 13. World Scientific, Singapore.
  ISBN 978-981-02-0427-3. Archived from the original
  on 2012-02-08. Retrieved 2008-01-21. See Part 2, Chapter 5.
• Hamm, L.; Madsen, P.A.;
  doi:10.1016/0378-3839(93)90044-9
  .
• Johnson, R.S. (1997). A modern introduction to the mathematical theory of water waves. Cambridge Texts in Applied Mathematics. Vol. 19. Cambridge University Press.
  ISBN 0-521-59832-X
  .
• Kirby, J.T. (2003). "Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents". In Lakhan, V.C. (ed.). Advances in Coastal Modeling. Elsevier Oceanography Series. Vol. 67. Elsevier. pp. 1–41.
  ISBN 0-444-51149-0
  .
• S2CID 119385147
  .
• ISBN 0-12-493250-9
  .