Boussinesq equation

The Boussinesq equations are nonlinear approximate equations for water waves in shallow water and partial differential equations that Joseph Boussinesq established. They are equations for momentum and mass conservation integrated over the water depth.

formulation

There are different versions of the Boussinesq equation, one also speaks of Boussinesq-like equations.

Originally, Boussinesq introduced the equation in 1871/72:

${\displaystyle u_{tt}-u_{xx}+{u^{2}}_{xx}-u_{xxxx}=0\qquad }$ (Equation 1)

a (also sometimes with a positive sign in front of or other coefficients). The equation is exactly integrable and has soliton solutions. The solitons of the equation behave unusually (for example, they can decay or form a singularity (collapse of the solitons) in a finite time). ${\displaystyle u_{xxxx}}$

Other versions in the literature are the modified Boussinesq equation:

${\displaystyle 3u_{tt}-6u_{t}u_{xx}-6{u_{x}}^{2}u_{xx}+u_{xxxx}=0}$

or the system of coupled equations:

${\displaystyle u_{t}-(uv)_{x}+v_{xxx}=0}$

${\displaystyle v_{t}+u_{x}+vv_{x}=0}$

There is also a linear Boussinesq equation:

${\displaystyle u_{tt}-\alpha ^{2}u_{xx}=\beta ^{2}u_{xxtt}}$

application

The Boussinesq equation is used to model water waves in shallow water (e.g. coast, harbors, the wavelength is large compared to the water depth). The shallow water equations (see below) are also used to simulate shallow water waves . They are also equations for momentum and mass balance integrated over the water depth, but the Boussinesq approximation also takes into account the dispersion of the waves (i.e. different speeds at different wavelengths). In comparison, the shallow water equations are simpler, restricted models, as they are based on a hydrostatic pressure distribution over the depth and do not take into account any vertical speed components, which is no longer permissible in the case of very high waves and water jumps, for example. If you want to model the behavior even more precisely locally, you have to resort to a simulation of the Navier-Stokes equations.

Derivation of the Boussinesq equation

What is meant here is equation (1). The flow is said to be eddy-free (so that a velocity potential can be given) and incompressible (it follows that the velocity potential is the solution of Laplace's equation). With velocity components in the horizontal direction x and in the vertical direction z, the deflection of the free liquid surface (rest position is at ) and the liquid depth (bottom at ), the incompressibility results in the kinematic boundary condition: ${\displaystyle \varphi (x,z)}$ ${\displaystyle u=\varphi _{x}}$ ${\displaystyle v=\varphi _{z}}$ ${\displaystyle \eta (x)}$ ${\displaystyle \eta =0}$ ${\displaystyle z=0}$ ${\displaystyle d}$ ${\displaystyle z=-d}$

${\displaystyle \eta _{t}+u\,\eta _{x}=v\qquad }$ (Equation 2)

The dynamic boundary condition results from the momentum balance (a form of the Bernoulli equation):

${\displaystyle \varphi _{t}+{\frac {u^{2}+v^{2}}{2}}+g\eta =0\qquad }$ (Equation 3)

These are the usual equations for free-edge water waves.

In addition, a boundary condition on the ground is considered . ${\displaystyle v=0}$

Boussinesq developed the velocity potential on the ground (index b) in a Taylor series:

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

If one uses the Laplace equation ( ) and the boundary condition follows: ${\displaystyle {\frac {\partial ^{2}\varphi }{\partial z^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x^{2}}}=0}$ ${\displaystyle \left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-d}=v_{z=-d}=v_{b}=0}$

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

If you insert the Taylor series up to the 4th order into equations (2), (3) for water waves and only keep terms that are linear or quadratic in and , you get: ${\displaystyle u_{b}={\frac {\varphi _{b}}{\partial x}}\,}$ ${\displaystyle \eta }$

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

An approximately constant water depth was assumed in the derivation. If you neglect the right side, you get the shallow water equations.

With a few additional assumptions, Boussinesq derived the equation from this

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

from. The equation can be formulated in dimensionless parameters after appropriate normalization:

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

With:

${\displaystyle \psi \,=\,{\frac {1}{2}}\,{\frac {\eta }{d}}}$	: the dimensionless wave height
${\displaystyle \tau \,=\,{\sqrt {3}}\,t\,{\sqrt {\frac {g}{d}}}}$	: the dimensionless time
${\displaystyle \xi \,=\,{\sqrt {3}}\,{\frac {x}{d}}}$	: the dimensionless horizontal position

Note: A linear approximation of the above water wave equations (2), (3) for small amplitudes is

${\displaystyle \eta _{t}=v}$

${\displaystyle \varphi _{t}+g\eta =0}$

at the free edge and with in the depth , which results in the linear theory of water waves (Airy theory). The Boussinesq approximation also allows the treatment of non-linear effects, which affects, for example, the dispersion of the water waves. ${\displaystyle v=0}$ ${\displaystyle d}$

literature

• Ablowitz, Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press 1991, p. 163ff (chapter 4)

Original works:

• Boussinesq: Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, Comptes Rendus de l'Academie des Sciences, Volume 72, 1871, pp. 755-759.
• Boussinesq: 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, Volume 17, 1872, pp. 55-108.

Web links

• Boussinesq equation, Mathworld

Individual evidence

1. ^ Ablowitz, Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press 1991, p. 52
2. ^ Based on Whitham, Linear and Nonlinear Waves 1974, z. B. prefactor 3 before ${\displaystyle {u^{2}}_{xx}}$
3. ↑ LV Bogdanov, VE Zakharov, The Boussinesq equation revisited, Physica D, Volume 165, 2002, pp. 137-162, pdf
4. ↑ Ablowitz, Clarkson, loc. cit.
5. ↑ Wolfram, Mathworld, article Boussinesq equation, see web links
6. ↑ Markus Witting, Heinz-Dieter Niemyer, Mathematical Modeling of Wave Run-Up and Overflow, The Coast, Volume 71, 2006, pp. 93-123, pdf
7. ↑ The u defined in this section has nothing to do with the u in the section above, where u simply gives the solution of the dimensionless Boussinesq equation derived below.
8. ↑ z. B. Lokenath Debnath, Nonlinear Water Waves, Academic Press 1994, p. 12 or RS Johnson, A modern introduction to the mathematical theory of water waves, Cambridge UP, 1997, p. 15f.