Estimation of wave heights from pressure data at the bed in the presence of uniform underlying currents

We provide some bounds on the estimated wave heights (valid for all ranges of amplitudes) from pressure measurements at the flat bed for steady water waves flowing over underlying uniform currents. The derived upper bound for the wave height for the case when the speed of the underlying current is greater than the wave speed is different from the one when there is no current. The results in this paper also confirm that waves of nontrivial amplitudes cannot exist where the speed of the underlying current is same as the wave speed, a result that was previously proved by Basu (2016) in a different context following another approach.