text: The horizontal flow velocity in the -direction follows directly from substituting the solution for the surface elevation  from Eq. () into the expression for  in Eq. ():
  and  an additional constant discharge.
  Note that –  when the width  and depth  are not constants – the term proportional to  implies an  (small) phase difference between elevation  and velocity .
  For sinusoidal waves with velocity amplitude  the flow velocities shoal to leading order as
  This could have been anticipated since for a horizontal bed  with  the wave amplitude.