A classical and central problem in the theory of water waves is to classify parameter regimes for which non-trivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number Fr (a non-dimensional wave speed) plays the central role. So far, the best analytical result Fr 2 was obtained by Starr (1947 J. Mar. Res. , vol. 6, pp. 175–193), while the numerical evidence of Longuet-Higgins & Fenton (1974 Proc. A, vol. 340, pp. 471–493) states Fr 1. 294. On the other hand, as shown recently by Kozlov (2023 On the first bifurcation of Stokes waves), the hypothetical upper bound Fr 1. 399 is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy and rigorously establish the improved upper bound Fr 1. 3451, which is the first rigorous improvement of Starr’s bound. In this process, we establish several new inequalities for the relative horizontal velocity, which are of separate interest and for which we delicately make use of the bound on the slope of the surface profile established by Amick (1987 Arch. Ration. Mech. Anal. , vol. 99, pp. 91–114). As an application we show that the velocity at the bottom below the crest of any solitary wave does not exceed 47\, \% of the propagation speed.
Lokharu et al. (2026) studied this question.