in the bottom boundary layer, u must drop from u$\_$pb at the "top" to at y $=0.$ We can continue to neglect the quadratic terms in the Navier Stokes equations as long as wave amplitude us sufficiently small. Assuming that delta$/$wavelength $<<1,$ where delta is some measure of bottom boundary thickness, the slender flow assumptions can be employed. The vertical relation reduces in the usual form to $($d$/$dy$)*($p$\_$d$)=0,$ where p$\_$d$=$ p$-$p$\_$h$,$ p$\_$h$=$ rho$*$g$*($H$\_$o$-$y$).$ The dynamic pressure p$\_$d thus can be evaluated from the potential flow extrapolated to the bed from $($d$/$dt$)*$u$\_$p $=(-1/$rho$)*($d$/$dx$)*$p$.$ The streamwise equation of momentum balance reduces to : $($d$/$dt$)*$u $=($d$/$dt$)*$u$\_$pb $+$ v$*($d$^2/$dy$^2)*$u$.$ show all the steps in the derivation of $($d$/$dt$)*$u $=($d$/$dt$)*$u$\_$pb $+$ v$*($d$^2/$dy$^2)*$u