A semi-infinite channel of finite depth is occupied by an ideal fluid layer initially at rest . the vertical finite end of the channel is fixed and only a part of the horizontal bottom , with finite support , is set in a bounded motion .

Find the resulting free surface elevation at any subsequent instant of time .

My question is what does it mean that horizontal bottom with finite support ?

Is that means that we have a wave maker on the horizontal bottom ?

We know that : in the fluid mass $$\phi_{xx} + \phi_{yy} = 0$$ And at $x=0$ we have $$\frac{\partial \phi}{\partial x} = 0$$ And on $y=0$ we have $$\frac{\partial^2 \phi}{\partial t^2} + g \frac{\partial \phi}{\partial y} = 0$$ And we have initial condition $\phi(x,y,0)=0$ How should I write the condition on the bottom ? Is it correct if i wrote $\frac{\partial \phi}{\partial y} = f(x,t)$ at y = $-h$

Crossposted at MSE: https://math.stackexchange.com/questions/3067996/question-on-free-surface-elevation-of-water-wave