 Solution

Condition (1b) is the kinematic free-surface boundary condition: fluid particles on the surface remain there, since the material derivative of $y-\eta (x,t)$ vanishes. Condition (1c) is the dynamic free-surface boundary condition, obtained from the unsteady Bernoulli equation with constant atmospheric pressure and a choice of potential gauge. Condition (1d) is impermeability: no normal flow through the horizontal bed.