On the existence theory for irrotational water waves

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm. Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and where g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.