In the context of a school project, I'm trying to redo the derivation of the Korteweg-de Vries equation. At the point where the normal free surface condition is combined with the effect of surface tension, I'm lost.
(Notation: y = function describing the heigth of the water wave at a point x and a time t, p = atmospheric pressure, T= surface tension, p' = pressure just below water surface)
Korteweg and de Vries state in their paper that $p' = p - T\frac{\partial^2 y}{\partial x^2}$. I was surprised by this statement, as I recall from some class that the Young-Laplace equation states that $\Delta p = p'-p = T(\frac{1}{R_x})$, $R_x$ being the radius of curvature in the $x$-direction (I'm doing 1D so there is no other direction). I thought that $1/R_x$ was then given by $\frac{|y''(x)|}{(1+(y'(x))^2)^{3/2}}$, but I don't see why that would be equal to $\frac{\partial^2 y}{\partial x^2}$ in this case.
I'd really appreciate it if someone could shine a light on this matter.