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diff --git a/source/know/concept/boussinesq-wave-theory/index.md b/source/know/concept/boussinesq-wave-theory/index.md index 31228ba..ad2fe4c 100644 --- a/source/know/concept/boussinesq-wave-theory/index.md +++ b/source/know/concept/boussinesq-wave-theory/index.md @@ -20,7 +20,7 @@ which were not predicted by the linear theories existing at the time. ## Boundary conditions Consider the [Euler equations](/know/concept/euler-equations/) -for an incompressible fluid with negligible viscosity: +for an incompressible fluid with negligible [viscosity](/know/concept/viscosity/): $$\begin{aligned} \va{g} - \frac{\nabla p}{\rho} @@ -441,6 +441,168 @@ which are both on the order of $$h / \lambda$$. +## Dimensionless form + +Let us non-dimensionalize the equation by introducing +dimensionless quantities $$\tilde{\eta}$$, $$\tilde{t}$$ and $$\tilde{x}$$: + +$$\begin{aligned} + \tilde{\eta}(\tilde{x}, \tilde{t}) + = \frac{\eta(x, t)}{\eta_c} + \qquad \qquad + \tilde{t} + = \frac{t}{t_c} + \qquad \qquad + \tilde{x} + = \frac{x}{x_c} +\end{aligned}$$ + +Where $$\eta_c$$, $$t_c$$ and $$x_c$$ are unspecified scale parameters. +We rewrite the Boussinesq equation with these quantities +by using the chain rule of differentiation, +and divide by $$\eta_c / t_c^2$$: + +$$\begin{aligned} + 0 + &= \tilde{\eta}_{\tilde{t} \tilde{t}} + - \frac{g h t_c^2}{x_c^2} \tilde{\eta}_{\tilde{x} \tilde{x}} + - \pdvn{2}{}{x} \bigg( \frac{3 g \eta_c t_c^2}{2 x_c^2} \tilde{\eta}^2 + \frac{g h^3 t_c^2}{3 x_c^4} \tilde{\eta}_{\tilde{x} \tilde{x}} \bigg) +\end{aligned}$$ + +Now we must choose values for $$\eta_c$$, $$t_c$$ and $$x_c$$ +such that the prefactors become simple constants. +Conventionally it is demanded that: + +$$\begin{aligned} + \frac{g h t_c^2}{x_c^2} + = 1 + \qquad \qquad + \frac{3 g \eta_c t_c^2}{2 x_c^2} + = 3 + \qquad \qquad + \frac{g h^3 t_c^2}{3 x_c^4} + = 1 +\end{aligned}$$ + +Solving this system of equations yields the following values for the scale parameters: + +$$\begin{aligned} + \eta_c + = 2 h + \qquad \qquad + t_c + = \sqrt{\frac{h}{3 g}} + \qquad \qquad + x_c + = \frac{h}{\sqrt{3}} +\end{aligned}$$ + +And the Boussinesq equation is reduced to its standard dimensionless form: + +$$\begin{aligned} + \boxed{ + \pdvn{2}{\tilde{\eta}}{\tilde{t}} - \pdvn{2}{\tilde{\eta}}{\tilde{x}} + - \pdvn{2}{}{x} \bigg( 3 \tilde{\eta}^2 + \pdvn{2}{\tilde{\eta}}{\tilde{x}} \bigg) + = 0 + } +\end{aligned}$$ + +Many authors flip the sign of $$\tilde{\eta}_{\tilde{x}\tilde{x}\tilde{x}\tilde{x}}$$ +to get the so-called "good" Boussinesq equation +(as opposed to the "bad" one above). +For fluid surface waves, this is unphysical, +but it makes the problem more well-behaved mathematically; +the details are beyond the scope of this article. + + + +## Soliton solution + +Let us make an ansatz for $$\tilde{\eta}$$ that describes a wave +with a fixed shape propagating in the positive $$\tilde{x}$$-direction +at dimensionless velocity $$v$$: + +$$\begin{aligned} + \tilde{\eta}(\tilde{x}, \tilde{t}) + = \phi(\xi) + \qquad + \xi + \equiv \tilde{x} - v \tilde{t} + \qquad \implies \qquad + \pdv{}{\tilde{t}} + = -v \pdv{}{\xi} + \qquad + \pdv{}{\tilde{x}} + = \pdv{}{\xi} +\end{aligned}$$ + +With this, the Boussinesq equation becomes a nonlinear ordinary differential equation: + +$$\begin{aligned} + 0 + &= (v^2 - 1) \phi_{\xi \xi} - \pdvn{2}{}{\xi} (3 \phi^2 + \phi_{\xi \xi}) +\end{aligned}$$ + +We abbreviate $$w \equiv v^2 \!-\! 1$$ and integrate twice, +introducing integration constants $$A$$ and $$B$$: + +$$\begin{aligned} + w \phi - 3 \phi^2 - \phi_{\xi \xi} + = A \xi + B +\end{aligned}$$ + +We restrict ourselves to localized solutions +by demanding that $$\phi \to 0$$ for $$\xi \to \pm \infty$$. +This implies that also $$\phi_\xi \to 0$$ and $$\phi_{\xi \xi} \to 0$$, +meaning that we must set $$A = B = 0$$ to satisfy the equation at infinity. +The remaining terms are multiplied by $$\phi_\xi$$ to give: + +$$\begin{aligned} + 0 + &= w \phi \phi_\xi - 3 \phi^2 \phi_\xi - \phi_{\xi \xi} \phi_\xi + = \frac{1}{2} \pdv{}{\xi} \Big( w \phi^2 - 2 \phi^3 - (\phi_\xi)^2 \Big) +\end{aligned}$$ + +Integrating (and dropping the integration constant due to localization) yields: + +$$\begin{aligned} + (\phi_\xi)^2 + = \phi^2 (w - 2 \phi) +\end{aligned}$$ + +Because $$\phi_\xi$$ is real, we need the right-hand side to be positive, +so $$w > 2 \phi$$; for $$\phi \to 0$$, this means that $$w > 0$$. +This equation is similar to the one encountered when solving +the [Korteweg-de Vries equation](/know/concept/korteweg-de-vries-equation/) +and is integrated in the same way; refer there for details. +The result is: + +$$\begin{aligned} + \boxed{ + \tilde{\eta}(\tilde{x}, \tilde{t}) + = \frac{w}{2} \sech^2\!\bigg( \frac{\sqrt{w}}{2} \big( \tilde{x} - \sqrt{1 \!+\! w} \: \tilde{t} - \tilde{x}_0 \big) \bigg) + } +\end{aligned}$$ + +This is known as a **soliton**. +Reintroducing units by replacing $$\tilde{\eta} = \eta / \eta_c$$ etc. +leads to: + +$$\begin{aligned} + \boxed{ + \eta(x, t) + = w h \sech^2\!\bigg( \frac{\sqrt{3 w}}{2 h} \Big( x - \sqrt{(1 \!+\! w) g h} \: t - x_0 \Big) \bigg) + } +\end{aligned}$$ + +Note that Boussinesq's original calculation had $$(1 \!+\! w/2)$$ instead of $$\sqrt{1 \!+\! w}$$; +the former is simply a first-order approximation of the latter. +Recall that $$\sqrt{g h}$$ is the phase velocity of Lagrange's linear theory: +this shows that nonlinear waves are faster, +and speed up with amplitude. + + + ## References 1. J. Boussinesq, [Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond](http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16416&Deb=63&Fin=116&E=PDF), @@ -451,3 +613,6 @@ which are both on the order of $$h / \lambda$$. 3. D. Dutykh, F. Dias, [Dissipative Boussinesq equations](https://doi.org/10.1016/j.crme.2007.08.003), 2007, Elsevier. +4. M.B. Almatrafi, A.R. Alharbi, C. Tunç, + [Constructions of the soliton solutions to the good Boussinesq equation](doi.org/10.1186/s13662-020-03089-8), + 2020, Springer. |