Kinematic boundary conditions and surface elevation equation

At the surface and at the bottom, kinematic boundary conditions result from the requirement that the particles at the boundaries are moving along these boundaries:

$$w= \partial_t \zeta + u \partial_x\zeta + v \partial_y \zeta z=\zeta,$$ (6)

$$w= - u \partial_xH - v \partial_y H z=-H.$$ (7)