Which other type of condition we can prescribe to specify the potential function $\phi(x,y,z)$ at the water surface ?

At the boundaries of the fluid domain we can also specify stress (force per unit area) or force. For the case of ideal fluid, the stress is due only to the pressure. At interface of water and air, we impose that there is no pressure jump. This is a Dynamic Boundary Condition, since we prescribed force or stress (pressure) at the boundary.