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| {{About|the Boussinesq approximation for waves on a free-moving fluid surface||Boussinesq approximation (disambiguation){{!}}Boussinesq approximation}}
| | by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Should you loved this short article and you wish to receive details with regards to [http://www.ifollosports.com/users/terriebeanewyjxs wordpress backup plugin] assure visit our own webpage. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. This plugin allows a blogger get more Facebook fans on the related fan page. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>But before choosing any one of these, let's compare between the two. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. The company gains commission from the customers' payment. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products and services for your customers. In fact portfolio Word - Press themes is a smooth and attractive but considerably flawed Word - Press theme in creating simpler to the photographers or designers to develop a specific internet site showcasing their most current perform since it appear modern-day and has fantastic typography and large photographs which would develop an attractive wanting portfolio internet site. While deciding couple should consider the expertise of the doctor,clinics success rate,the costs of fertility treatment,including fertility tests and IVF costs and overall ones own financial budget. with posts or testimonials updated as they are uploaded to a particular section of the website. Press CTRL and the numbers one to six to choose your option. |
| [[File:Bous2d berk03 z.gif|thumb|right|300px|Simulation of periodic waves over an underwater [[shoal]] with a Boussinesq-type model. The waves propagate over an elliptic-shaped underwater shoal on a plane beach. This example combines several effects of [[waves and shallow water]], including [[refraction]], [[diffraction]], shoaling and weak [[non-linearity]].]]
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| In [[fluid dynamics]], the '''Boussinesq approximation''' for [[water waves]] is an [[approximation]] valid for weakly [[non-linear]] and [[#Linear frequency dispersion|fairly long waves]]. The approximation is named after [[Joseph Boussinesq]], who first derived them in response to the observation by [[John Scott Russell]] of the [[wave of translation]] (also known as [[solitary wave (water waves)|solitary wave]] or [[soliton]]). The 1872 paper of Boussinesq introduces the equations now known as the '''Boussinesq equations'''.<ref>This paper (Boussinesq, 1872) starts with: ''"Tous les ingénieurs connaissent les belles expériences de J. Scott Russell et M. Basin sur la production et la propagation des ondes solitaires"'' (''"All engineers know the beautiful experiments of J. Scott Russell and M. Basin on the generation and propagation of solitary waves"'').</ref>
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| The Boussinesq approximation for [[water waves]] takes into account the vertical structure of the horizontal and vertical [[flow velocity]]. This results in [[non-linear]] [[partial differential equations]], called Boussinesq-type equations, which incorporate [[dispersion (water waves)|frequency dispersion]] (as opposite to the [[shallow water equations]], which are not frequency-dispersive). In [[coastal engineering]], Boussinesq-type equations are frequently used in [[computer models]] for the [[simulation]] of [[water waves]] in [[waves and shallow water|shallow]] [[sea]]s and [[harbours]].
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| While the Boussinesq approximation is applicable to fairly long waves – that is, when the [[wavelength]] is large compared to the water depth – the [[Stokes expansion]] is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).
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| ==Boussinesq approximation==
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| [[File:Boussinesq_T06H18d5.svg|thumb|right|300px|Periodic waves in the Boussinesq approximation, shown in a vertical [[cross section (geometry)|cross section]] in the [[wave propagation]] direction. Notice the flat [[trough (physics)|trough]]s and sharp [[crest (physics)|crest]]s, due to the wave nonlinearity. This case (drawn on [[scale (ratio)|scale]]) shows a wave with the [[wavelength]] equal to 39.1 [[metre|m]], the wave height is 1.8 m (''i.e.'' the difference between crest and trough elevation), and the mean water depth is 5 m, while the [[Earth's gravity|gravitational acceleration]] is 9.81 m/s<sup>2</sup>.]]
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| The essential idea in the Boussinesq approximation is the elimination of the vertical [[coordinate]] from the flow equations, while retaining some of the influences of the vertical structure of the flow under [[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.
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| This elimination of the vertical coordinate was first done by [[Joseph Boussinesq]] in 1871, to construct an approximate solution for the solitary wave (or [[wave of translation]]). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.
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| The steps in the Boussinesq approximation are:
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| *a [[Taylor expansion]] is made of the horizontal and vertical [[flow velocity]] (or [[velocity potential]]) around a certain [[elevation]],
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| *this [[Taylor expansion]] is truncated to a [[Wikt:finite|finite]] number of terms,
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| *the conservation of mass (see [[continuity equation]]) for an [[incompressible flow]] and the zero-[[Curl (mathematics)|curl]] condition for an [[irrotational flow]] are used, to replace vertical [[partial derivatives]] of quantities in the [[Taylor expansion]] with horizontal [[partial derivatives]].
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| Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate.
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| As a result, the resulting [[partial differential equations]] are in terms of [[Function (mathematics)|functions]] of the horizontal [[coordinates]] (and [[time]]).
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| As an example, consider [[potential flow]] over a horizontal bed in the (''x,z'') plane, with ''x'' the horizontal and ''z'' the vertical [[coordinate]]. The bed is located at {{nowrap|''z'' {{=}} −''h''}}, where ''h'' is the [[mean]] water depth. A [[Taylor expansion]] is made of the [[velocity potential]] ''φ(x,z,t)'' around the bed level {{nowrap|''z'' {{=}} −''h''}}:<ref>Dingemans (1997), p. 477.</ref>
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| {{clear}}
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| :<math>
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| \begin{align}
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| \varphi\, =\, &
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| \varphi_b\,
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| +\, (z+h)\, \left[ \frac{\partial \varphi}{\partial z } \right]_{z=-h}\,
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| +\, \frac{1}{2}\, (z+h)^2\, \left[ \frac{\partial^2 \varphi}{\partial z^2} \right]_{z=-h}\,
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| \\ &
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| +\, \frac{1}{6}\, (z+h)^3\, \left[ \frac{\partial^3 \varphi}{\partial z^3} \right]_{z=-h}\,
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| +\, \frac{1}{24}\, (z+h)^4\, \left[ \frac{\partial^4 \varphi}{\partial z^4} \right]_{z=-h}\,
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| +\, \cdots,
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| \end{align}
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| </math>
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|
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| where ''φ<sub>b</sub>(x,t)'' is the velocity potential at the bed. Invoking [[Laplace's equation]] for ''φ'', as valid for [[incompressible flow]], gives:
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| :<math>
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| \begin{align}
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| \varphi\, =\, &
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| \left\{\,
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| \varphi_b\,
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| -\, \frac{1}{2}\, (z+h)^2\, \frac{\partial^2 \varphi_b}{\partial x^2}\,
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| +\, \frac{1}{24}\, (z+h)^4\, \frac{\partial^4 \varphi_b}{\partial x^4}\,
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| +\, \cdots\,
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| \right\}\,
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| \\
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| & +\,
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| \left\{\,
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| (z+h)\, \left[ \frac{\partial \varphi}{\partial z} \right]_{z=-h}\,
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| -\, \frac16\, (z+h)^3\, \frac{\partial^2}{\partial x^2} \left[ \frac{\partial \varphi}{\partial z} \right]_{z=-h}\,
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| +\, \cdots\,
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| \right\}
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| \\
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| =\, &
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| \left\{\,
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| \varphi_b\,
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| -\, \frac{1}{2}\, (z+h)^2\, \frac{\partial^2 \varphi_b}{\partial x^2}\,
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| +\, \frac{1}{24}\, (z+h)^4\, \frac{\partial^4 \varphi_b}{\partial x^4}\,
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| +\, \cdots\,
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| \right\},
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| \end{align}
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| </math>
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| since the vertical velocity {{nowrap|∂''φ'' / ∂''z''}} is zero at the – impermeable – horizontal bed {{nowrap|''z'' {{=}} −''h''}}. This series may subsequently be truncated to a finite number of terms.
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| ==Original Boussinesq equations==
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| ===Derivation===
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| For [[water waves]] on an [[incompressible fluid]] and [[irrotational flow]] in the (''x'',''z'') plane, the [[boundary conditions]] at the [[free surface]] elevation {{nowrap|''z'' {{=}} ''η''(''x'',''t'')}} are:<ref>Dingemans (1997), p. 475.</ref>
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| :<math>
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| \begin{align}
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| \frac{\partial \eta}{\partial t}\, &+\, u\, \frac{\partial \eta}{\partial x}\, -\, w\, =\, 0
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| \\
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| \frac{\partial \varphi}{\partial t}\, &+\, \frac{1}{2}\, \left( u^2 + w^2 \right)\, +\, g\, \eta\, =\, 0,
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| \end{align}
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| </math>
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| where:
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| :''u'' is the horizontal [[flow velocity]] component: {{nowrap|''u'' {{=}} ∂''φ'' / ∂''x''}},
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| :''w'' is the vertical [[flow velocity]] component: {{nowrap|''w'' {{=}} ∂''φ'' / ∂''z''}},
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| :''g'' is the [[acceleration]] by [[gravity]].
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| Now the Boussinesq approximation for the [[velocity potential]] ''φ'', as given above, is applied in these [[boundary conditions]]. Further, in the resulting equations only the [[linear]] and [[quadratic equation|quadratic]] terms with respect to ''η'' and ''u<sub>b</sub>'' are retained (with {{nowrap|''u''<sub>b</sub> {{=}} ∂''φ''<sub>b</sub> / ∂''x''}} the horizontal velocity at the bed {{nowrap|''z'' {{=}} −''h''}}). The [[cubic equation|cubic]] and higher order terms are assumed to be [[negligible]]. Then, the following [[partial differential equations]] are obtained:
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| ;set A – Boussinesq (1872), equation (25)
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| :<math>
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| \begin{align}
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| \frac{\partial \eta}{\partial t}\,
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| & +\, \frac{\partial}{\partial x}\, \left[ \left( h + \eta \right)\, u_b \right]\,
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| =\, \frac{1}{6}\, h^3\, \frac{\partial^3 u_b}{\partial x^3},
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| \\
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| \frac{\partial u_b}{\partial t}\,
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| & +\, u_b\, \frac{\partial u_b}{\partial x}\,
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| +\, g\, \frac{\partial \eta}{\partial x}\,
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| =\, \frac{1}{2}\, h^2\, \frac{\partial^3 u_b}{\partial t\, \partial x^2}.
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| \end{align}
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| </math>
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| This set of equations has been derived for a flat horizontal bed, ''i.e.'' the mean depth ''h'' is a constant independent of position ''x''. When the right-hand sides of the above equations are set to zero, they reduce to the [[shallow water equations]].
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| 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 ''η'':
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| ;set B – Boussinesq (1872), equation (26)
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| :<math>
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| \frac{\partial^2 \eta}{\partial t^2}\,
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| -\, g h\, \frac{\partial^2 \eta}{\partial x^2}\,
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| -\, g h\, \frac{\partial^2}{\partial x^2}
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| \left(
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| \frac{3}{2}\, \frac{\eta^2}{h}\,
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| +\, \frac{1}{3}\, h^2\, \frac{\partial^2 \eta}{\partial x^2}
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| \right)\, =\, 0.
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| </math>
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| From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the [[Ursell number]].
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| In [[dimensionless quantities]], using the water depth ''h'' and gravitational acceleration ''g'' for non-dimensionalization, this equation reads, after [[Normalisable wave function|normalization]]:
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|
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| :<math>
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| \frac{\partial^2 \psi}{\partial \tau^2}\,
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| -\, \frac{\partial^2 \psi}{\partial \xi^2}\,
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| -\, \frac{\partial^2}{\partial \xi^2}
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| \left(\,
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| \frac{1}{2}\, \psi^2\,
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| +\, \frac{\partial^2 \psi}{\partial \xi^2}\,
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| \right)\, =\, 0,
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| </math> | |
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| with: | |
| {|
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| |-
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| |<math>\psi\, =\, 3\, \frac{\eta}{h}</math> ||: the dimensionless surface elevation,
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| |-
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| |<math>\tau\, =\, \sqrt{3}\, t\, \sqrt{\frac{g}{h}}</math> ||: the dimensionless time, and
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| |-
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| |<math>\xi\, =\, \sqrt{3}\, \frac{x}{h}</math> ||: the dimensionless horizontal position.
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| |}
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| [[File:disp_bouss.svg|thumb|350px|right|Linear phase speed squared ''c''<sup>2</sup>/(''gh'') as a function of relative wave number ''kh''.<br>
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| '''A''' = Boussinesq (1872), equation (25),<br>
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| '''B''' = Boussinesq (1872), equation (26),<br>
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| '''C''' = full linear wave theory, see [[dispersion (water waves)]]
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| ]]
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| ===Linear frequency dispersion===
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| [[Water waves]] of different [[wave length]]s travel with different [[phase speed]]s, a phenomenon known as [[dispersion (water waves)|frequency dispersion]]. For the case of [[infinitesimal]] wave [[amplitude]], the terminology is ''linear frequency dispersion''. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid [[approximation]].
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| The linear [[dispersion (water waves)|frequency dispersion]] characteristics for the above set '''A''' of equations are:<ref name=Dingemans521>Dingemans (1997), p. 521.</ref>
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| :<math> c^2\, =\; g h\, \frac{ 1\, +\, \frac{1}{6}\, k^2 h^2 }{ 1\, +\, \frac{1}{2}\, k^2 h^2 }, </math>
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| with:
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| *''c'' the [[phase speed]],
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| *''k'' the [[wave number]] ({{nowrap|''k'' {{=}} 2π / ''λ''}}, with ''λ'' the [[wave length]]).
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| The [[relative error]] in the phase speed ''c'' for set '''A''', as compared with [[Airy wave theory|linear theory for water waves]], is less than 4% for a relative wave number {{nowrap|''kh'' < ½ π}}. So, in [[engineering]] applications, set '''A''' is valid for wavelengths ''λ'' larger than 4 times the water depth ''h''. | |
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| The linear [[dispersion (water waves)|frequency dispersion]] characteristics of equation '''B''' are:<ref name=Dingemans521/>
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| :<math> c^2\, =\, g h\, \left( 1\, -\, \frac{1}{3}\, k^2 h^2 \right). </math>
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| The relative error in the phase speed for equation '''B''' is less than 4% for {{nowrap|''kh'' < 2π/7}}, equivalent to wave lengths ''λ'' longer than 7 times the water depth ''h'', called '''fairly long waves'''.<ref>Dingemans (1997), p. 473 & 516.</ref>
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|
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| For short waves with {{nowrap|''k''<sup>2</sup> ''h''<sup>2</sup> > 3}} equation '''B''' become physically meaningless, because there are no longer [[real-valued]] [[solutions]] of the [[phase speed]].
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| The original set of two [[partial differential equations]] (Boussinesq, 1872, equation 25, see set '''A''' above) does not have this shortcoming.
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| The [[shallow water equations]] have a relative error in the phase speed less than 4% for wave lengths ''λ'' in excess of 13 times the water depth ''h''.
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| ==Extensions==
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| There are an overwhelming number of [[mathematical models]] 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.
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| Some directions, into which the Boussinesq equations have been extended, are:
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| *varying [[bathymetry]],
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| *improved [[dispersion (water waves)|frequency dispersion]],
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| *improved [[non-linear]] behavior,
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| *making a [[Taylor expansion]] around different vertical [[elevations]],
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| *dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
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| *inclusion of [[wave breaking]],
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| *inclusion of [[surface tension]],
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| *extension to [[internal waves]] on an [[Interface (chemistry)|interface]] between fluid domains of different [[mass density]],
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| *derivation from a [[variational principle]].
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| ==Further approximations for one-way wave propagation==
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| 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 [[Korteweg–de Vries equation]] for [[wave propagation]] in one horizontal [[dimension]],
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| *the [[Kadomtsev–Petviashvili equation]] for [[wave propagation]] in two horizontal [[dimension]]s,
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| *the [[nonlinear Schrödinger equation]] (NLS equation) for the [[complex number|complex valued]] [[amplitude]] of [[narrowband]] waves (slowly [[modulated]] waves).
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| Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called [[cnoidal wave]]s. These are approximate solutions of the Boussinesq equation.
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| ==References==
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| *{{cite journal | author= J. Boussinesq | authorlink=Joseph Boussinesq | year= 1871 | title= Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire |journal= Comptes Rendus de l'Academie des Sciences | volume= 72 | pages= 755–759 | url= http://gallica.bnf.fr/ark:/12148/bpt6k3029x/f759 }}
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| *{{cite journal | author= J. Boussinesq | authorlink=Joseph Boussinesq | year= 1872 | title= 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= [[Journal de Mathématiques Pures et Appliquées]]|series= Deuxième Série | volume= 17 | pages= 55–108 | url= http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16416&Deb=63&Fin=116&E=PDF }}
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| *{{cite book | author= M.W. Dingemans | year= 1997 | title=Wave propagation over uneven bottoms | series= Advanced Series on Ocean Engineering '''13''' | publisher= World Scientific, Singapore | url= http://www.worldscibooks.com/engineering/1241.html | isbn= 981-02-0427-2 }} ''See Part 2, Chapter 5''.
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| *{{cite journal | author= D.H. Peregrine | authorlink=Howell Peregrine | year= 1967 | title= Long waves on a beach | journal= Journal of Fluid Mechanics | volume= 27 | issue= 4 | pages= 815–827 |url= http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=381506# | doi= 10.1017/S0022112067002605 |bibcode = 1967JFM....27..815P }}
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| *{{cite conference | author= D.H. Peregrine | authorlink=Howell Peregrine | year=1972 | title= Equations for water waves and the approximations behind them | booktitle= Waves on Beaches and Resulting Sediment Transport | editor= Ed. R.E. Meyer | publisher= Academic Press | pages= 95–122 | id= ISBN 0-12-493250-9 }}
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| == Notes ==
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| {{reflist}}
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| {{physical oceanography}}
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| [[Category:Fluid dynamics]]
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| [[Category:Water waves]]
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| [[Category:Equations of fluid dynamics]]
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