Undertow (water waves)

An exact relation for the mass flux of a nonlinear periodic wave on an inviscid fluid layer was established by Levi-Civita in 1924.[9] In a frame of reference according to Stokes' first definition of wave celerity, the mass flux ${\displaystyle M_{w}}$ of the wave is related to the wave's kinetic energy density ${\displaystyle E_{k}}$ (integrated over depth and thereafter averaged over wavelength) and phase speed ${\displaystyle c}$ through:

${\displaystyle M_{w}={\frac {2E_{k}}{c}}.}$

Similarly, Longuet Higgins showed in 1975 that – for the common situation of zero mass flux towards the shore (i.e. Stokes' second definition of wave celerity) – normal-incident periodic waves produce a depth- and time-averaged undertow velocity:[10]

${\displaystyle {\bar {u}}=-{\frac {2E_{k}}{\rho ch}},}$

with ${\displaystyle h}$ the mean water depth and ${\displaystyle \rho }$ the fluid density. The positive flow direction of ${\displaystyle {\bar {u}}}$ is in the wave propagation direction.

For small-amplitude waves, there is equipartition of kinetic (${\displaystyle E_{k}}$) and potential energy (${\displaystyle E_{p}}$):

${\displaystyle E_{w}=E_{k}+E_{p}\approx 2E_{k}\approx 2E_{p},}$

with ${\displaystyle E_{w}}$ the total energy density of the wave, integrated over depth and averaged over horizontal space. Since in general the potential energy ${\displaystyle E_{p}}$ is much easier to measure than the kinetic energy, the wave energy is approximately ${\displaystyle {E_{w}\approx {\tfrac {1}{8}}\rho gH^{2}}}$ (with ${\displaystyle H}$ the wave height). So

${\displaystyle {\bar {u}}\approx -{\frac {1}{8}}{\frac {gH^{2}}{ch}}.}$

For irregular waves the required wave height is the root-mean-square wave height ${\displaystyle H_{\text{rms}}\approx {\sqrt {8}}\;\sigma ,}$ with ${\displaystyle \sigma }$ the standard deviation of the free-surface elevation.[11] The potential energy is ${\displaystyle E_{p}={\tfrac {1}{2}}\rho g\sigma ^{2}}$ and ${\displaystyle E_{w}\approx \rho g\sigma ^{2}.}$

The distribution of the undertow velocity over the water depth is a topic of ongoing research.[5][6][7]