Shoaling of sixth-order Stokes waves on a current

Jonsson and Arneborg (Jonsson, I.G., Arneborg, L., 1995. Energy properties and shoaling of higher-order Stokes waves on a current. Ocean Engng 22, 819-857.) 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 (Sobey R.J., Bando K., 1991. Variations on higher order shoaling. J. Waterway, Port, Coastal Ocean Engng ASCE 117, 348-368) 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. (C) 2000 Published by Elsevier Science Ltd.