For this case the dynamic condition is simply one of continuity of pressure across the surface
where 
is the air presure and P the water pressure.
For simplicity suppose is assumed dependent upon z
only.
A Taylor's series expansion of (3.6) then gives
A linearization for small 
may be justified if
and 
are
approximated by their hydrostatic values, which gives
where 
for practical purposes can be replaced
by 
since 
.