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I am learning Fluid mechanics by reading Acheson's book entitled "Elementary Fluid Dynamics". Below is from problem 3.1.

Consider the Euler equation for an ideal fluid in the irrotational case. We are studying two-dimensional water waves, so the velocity vector is on the form: ${\bf{u}}=[u(x,y,t),\,v(x,y,t),\,0]$. Because of ''irrotationality'', there is a velocity potential $\phi$ such that ${\bf{u}}=\nabla \phi$. By the incompressibility condition, $\phi$ satisfies the Laplace equation. Let $y=\eta(x,t)$ be the equation of the free surface. Note that the bottom of the water is at $y=-h$, where $h$ is the depth. Then \begin{equation} \frac{\partial\eta}{\partial t}+u\frac{\partial\eta}{\partial x}=v,\;\;{\mbox{on}}\;\;y=\eta(x,t).\end{equation} Then the Euler equation on the free surface (once integrated) gives (note that both the pressure at the free surface and the density are considered constant and can be absorbed in the constant of integration): $$\frac{\partial\phi}{\partial t}+\frac{1}{2}(u^2+v^2)+g\eta=0,\;\;{\mbox{on}}\;\;y=\eta(x,t).$$ We now consider ${\bf{u}}$ and $\eta$ small (in a sense to be determined later) and linearized the equations above: $$\frac{\partial\eta}{\partial t}=v,\;\;{\mbox{on}}\;\;y=0.$$ $$\frac{\partial\phi}{\partial t}+g\eta=0,\;\;{\mbox{on}}\;\;y=0.$$ We look for sinusoidal traveling wave solution of the form $\eta=A\cos{\left(kx-\omega t\right)}$. With the condition that $v=\phi_y=0$ at $y=-h$ (the bottom), with the Laplace equation and the two linear equations above, we find that $$ \phi=\frac{A\omega}{k\sinh(kh)}\cosh{(k(y+h))}\sin{(kx-\omega t)}$$ and the dispersion relation $\omega^2={g}{k}\tanh(kh)$.

Now, to determine the sense in which ${\bf{u}}$ and $\eta$ are small, we need to compare $u^2+v^2$ to $g\eta$. The term $u^2+v^2$ is of order $A^2\omega^2=A^2{g}{k}\tanh(kh)$ and the term $g\eta$ is of order $gA$. Then one gets the condition $A\ll\lambda$ by asking that $u^2+v^2\ll g\eta$. One gets the same condition by comparing the other nonlinear terms to the linear ones in the two nonlinear equations above.

My question is this: I should be able to also deduct that the displacement of the free surface is small w.r.t. the depth, i.e. $A\ll h$. How is this condition obtained?

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  • $\begingroup$ I think that the condition $\eta \ll h$ might just be the regime in which the shallow water results are valid. If your initial conditions lead to a situation where this is no longer true, then you can no longer use the shallow water equations. I don't think the condition can be 'derived.' $\endgroup$ – kleingordon Aug 1 '14 at 0:52
  • $\begingroup$ Related: physics.stackexchange.com/q/92983 . $\endgroup$ – David Hammen Sep 12 '14 at 18:41
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As far as I can tell this condition is not apparent at this order via these BC considerations. But consider the conservation of energy (to the order you've considered) in a time independent system, ie

$$\frac{d(c_g E)}{dx} = 0,$$ here $E\sim a^2$ and $c_g \sim \sqrt{h}$ so that $a\sim h^{\frac{-1}{4}}$, which is a relationship between the two scales you're interested in.

This clearly breaks down for waves that have $ak$ get large. when that's the case, the KdV equation becomes appropriate, and in that equation there is an inherent scaling between $hk$ and $ak$.

Nick

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