Remarks on Bernoulli constants, gauge conditions and phase velocities in the context of water waves

This short note is about the gauge condition for the velocity potential, the definitions of the Bernoulli constant and of the velocity speeds in the context of water waves. These definitions are often implicit and thus the source of confusion in the literature. This note aims at addressing this issue. The discussion is related to water waves because the confusion are frequent in this field, but it is relevant for more general problems in fluid mechanics.