# 5.4.2: Dynamic Pressure

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Pressure gradients in coastal waters are mainly due to mean water level variations (**hydrostatic pressure**) and fluctuations of pressure due to waves. The hydrostatic pressure due to mean water level variation is \(p_0 = -\rho g z\) and hence linearly increases from zero at the water surface \(z = -\) to \(p_0 = \rho gh\) at the bottom \(z = -h\). In the case of waves, the total pressure is the sum of this hydrostatic pressure \(p_0\) (from \(z = -h\) to \(z = \eta\)) plus the wave-induced or dynamic pressure \(p_0\) wave from (from \(z = -h\) to \(z = \eta\)). Wave-induced pressure oscillations are different under wave crest and wave trough and – in intermediate and deep water – reduce with depth below the free surface (Fig. 5.22a).

According to linear theory, the wave-induced pressure varies harmonically (in phase with the surface elevation \(\eta\)) with amplitude:

\[\hat{p} = \dfrac{\rho g H}{2} \dfrac{\cosh k(h + z)}{\cosh kh} \label{eq5.4.2.1}\]

which reduces in shallow water to:

\[\hat{p} = \dfrac{\rho g H}{2}\]

Hence, in shallow water the hydrostatic dynamic pressure varies linearly with the free surface elevation \(p_{\text{wave}} = \tilde{p} = \rho g \eta\) (Fig. 5.22b). The tilde indicates the purely oscillatory character.

To derive Eq. \(\ref{eq5.4.2.1}\) the amplitude was assumed to be very small in order to linearise the free surface boundary condition. It is therefore not valid in the region between the trough and the crest elevation. We can however assume that the dynamic pressure is hydrostatic between wave trough and wave crest: \(\tilde{p} = \rho g \eta\).