On the decrease of velocity with depth in irrotational periodic water waves
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On the decrease of velocity with depth in irrotational periodic water waves Luigi Roberti1 Received: 24 March 2020 / Accepted: 25 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We give an alternative proof for a classical result (due to Longuet-Higgins) that provides an estimate for the decay rate with depth of the velocity beneath two-dimensional, spatially periodic, irrotational water waves over a flat bed. Furthermore, an improvement to the same estimate is presented. Keywords Water waves · Irrotational flow · Spatial periodicity · Decay Mathematics Subject Classification 35Q35 · 76B15
1 Introduction The aim of this paper is to prove rigorous quantitative results regarding the attenuation of the motion that a water flow beneath a spatially periodic irrotational wave undergoes with increasing depth, which we do by adapting the methods developed in [1] for water waves with constant non-zero vorticity. This approach improves the classical result of Longuet-Higgins (see [10]). The model for the water wave problem that we will consider in the sequel relies upon the following basic assumptions: – We consider two-dimensional water waves, that is, the flow is supposed to be moving only in one horizontal direction, denoted by x, and one vertical direction, labelled by y and measured upwards. This assumption, although being a considerable mathematical simplification, does still embrace a vast range of observable phenomena.
Communicated by Adrian Constantin.
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Luigi Roberti [email protected] Faculty of Mathematics, University of Vienna, Oskar–Morgenstern–Platz 1, 1090 Vienna, Austria
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L. Roberti
– As mentioned above, we look for spatially periodic solutions, i.e. flows that are periodic in the x-direction with given period λ > 0 (also called the wavelength in this context). Note that we are not assuming periodicity in time, meaning that our considerations do not just apply to travelling waves, but also to wave profiles that may evolve with time in a non-periodic way, though maintaining periodicity in x. – The fluid domain, at each time assumed to be unbounded in the horizontal direction, is supposed to be delimited by the free surface S(t) = {(x, y) ∈ R2 : y = η(x, t)} for some (unknown) function η ∈ C 1 (R+ , C 1 (R)) which is a perturbation of the mean water level y = 0, i.e.
λ
η(x, t) dx = 0
0
for each time t > 0, and a rigid flat bed, situated at y = −d for the (given) mean depth d > 0. The latter assumption is not only a matter of convenience, because extensive patches of near-flat continental shelves can be found in all major sea and ocean basins, covering approximately 7.5% of the Earth’s ocean floor (see [8]). Thus the fluid domain D(t) at time t > 0 is given by D(t) = {(x, y) ∈ R2 : −d < y < η(x, t)}. – We will work in the setting of inviscid theory, which is a good approximation for water (see [6]), especially for gravity water waves where the dissipation effects caused by viscosity are negligible over long periods of time (cf. [3]). – Finally, as anti
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