Shoaling of Sixth-order Stokes Waves on a Current
Jonsson and Arneborg (1995) combined energy flux and set-down to make
shoaling predictions for fourth-order Stokes waves with and without a net
volume flux. With basis in their expressions, sixth-order expressions
are derived and combined to make shoaling predictions correct to
sixth order with an arbitrary net volume flux. The new sixth-order results
are compared with the fourth-order results and the practically exact
results obtained by Sobey and Bando (1991) with a Fourier 18 model. The
effects of introducing sixth-order theory rather than the fourth-order
theory in shoaling calculations are in general found to be small. As
expected the deviations increase with increasing wave-steepness,
decreasing depth and opposing currents. Also as expected, the results
obtained with the sixth-order expressions improve the results obtained
with the fourth-order expressions when compared to the results of Sobey
and Bando. As novelties, some considerations regarding the consistency of
odd- and even-order shoaling calculations, and the magnitude of the
bottom slope, are presented. Furthermore a comparison between the
wave-induced current and the total current is given.