Ig. Jonsson et H. Kofoedhansen, PRESSURE AND SET DOWN FOR HIGHER-ORDER STOKES WAVES ON CURRENT, Journal of waterway, port, coastal, and ocean engineering, 119(5), 1993, pp. 496-504
The Bernoulli equation for higher-order Stokes waves on a steady curre
nt over a horizontal, impermeable bed was analyzed, and an ambiguity w
as detected in the determination of the wave-induced pressure at even-
order theories. This leads to the introduction of a new (reduced) Bern
oulli constant. A simple relation between this quantity and the curren
t-wave set down is presented, and the set down correct to the fourth o
rder in wave steepness is calculated. The finding of the fourth-order
current-wave set down is a necessary step to allow calculation of the
wave height for fourth-order shoaling in the presence of a current. Fi
nally, the difference between a wave quantity belonging to nth-order t
heory and being correct to the nth order is discussed, and guidelines
are offered for the determination of the former and the corresponding
dispersion relation.