# Talk:Shallow-water wave theory

## Comment on pressure in 3.4 and 6.1 by Niels Mejlhede Jensen (Denmark 2022.12.12)

3.4 Pressure variation induced by wave motion

The proposed formula

$p = −\rho g \, z + \rho g \, \eta \, K p(z), \; z=0, \quad K p(z) = \Large\frac{\cosh(k(h+z))}{\cosh kh}\normalsize.$

implies that at the mean water level (MWL, $z=0$) the pressure (by this classical Airy formula) is the hydrostatic pressure $p=\rho g\eta$.

However, the pressure is not hydrostatic, because the water has a vertical negative acceleration, and the acceleration is highest here near the surface, especially by the crest.

So, a better fully correct first order solution is

$p=−\rho g \, z+\rho g \, \eta \, Kp(z), \; z=0, \quad Kp(z) = \Large\frac{\cosh(k(h+z))}{\cosh(k(h+\eta))}\normalsize .$

This gives an acceleration reduced pressure reduction of important numerical relevance e.g. for the overturning stability moment of the caisson breakwater described in 6.1 Vertical Walls. This pressure formula (and other formulas) is proven to be correct on: http://lavigne.dk/waves/simplewavesformulas.pdf page 9).

The Airy potential first order wave theory for small waves does not include $\eta$ in $(h+\eta)$ (because $\eta$ is called here a second order term) but for the design wave height in engineering practice it is not a negligible term, and it is easy to include $\eta$ to obtain a more reliable pressure expression. On: http://lavigne.dk/waves/makingthesimplewaterwavetheory.pdf a simple non-potential wave theory is described to get this pressure formula. This is a more technically relevant first order wave theory where we need not assume irrotational flow and do not use the Laplace equation with velocity potentials, but instead use the conservation of mass equation and Newton’s second law directly on the physical water particles demanding the pressure at the surface to be precisely $p=0$ and the resulting water flow precisely $q=0$.

This easy first order wave theory gives, with $z=-h$ at the bottom, the following formulas for the standing wave:

$\eta = ½ H \, cos(\omega t) \cos(kx) ,$

$p = \rho g \, \Big[ - z + \eta \, \Large\frac{ \cosh(k(z+h))}{\cosh(k(h+\eta))}\normalsize \Big]$ is the water pressure ($p+ = \rho g \, \eta \, \Large\frac{\cosh(k(z+h))}{\cosh(k(h+\eta)}\normalsize$ is the wave pressure below MWL and trough),

$w = \Large\frac{\partial \eta}{\partial t} \, \frac{\sinh(k(z+h))}{\sinh(k(h+\eta))} \normalsize$ is the vertical water particle velocity,

$q = \Large\frac{HL}{2T}\normalsize \, \sin(\omega t) \, \sin(kx)$ is the horizontal water flow through a vertical,

$u = q \, k \, \Large\frac{\cosh(k(z+h))}{ \sinh(k(h+\eta))}\normalsize$ is the horizontal water particle velocity.

For the progressive wave:

$c^2 = (L/T)^2 = (g/k) \, \tanh (kh)$ is the celerity squared,

$q = c \eta$ ,

$u = c \, \eta \, k \, \Large\frac{\cosh(k(z+h))}{\sinh(k(h+\eta))}\normalsize$ .

Mean water level, MWL, $z=h$: pressure $p+ = \rho g \, \eta \, \Large\frac{\cosh(kh)}{\cosh(k(h+ \eta))}\normalsize$; (NOT Airy: $p+ = \rho g \, \eta$).

Surface of the trough: $p = 0$, so $p+ = \rho g \, \eta$ (negative); (NOT Airy: $p+ = \rho g \, \eta \, \Large\frac{\cosh(k(z+h))}{\cosh(kh)}\normalsize$).

Those are the improved formulas for a regular wave, and they have been proved to be mathematically and theoretically correct to first order (see the 2 internet references).

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