Finding Free surface elevation in semi-infinite channel

Question

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

Answer 1

What I seem to understand is that the bottom is at $y=-h+v(x-ct)$ where $v$ has compact support. That corresponds to an obstacle moving at a constant speed towards the infinite end. But it might mean something else, such as $y=-h+v(x)\sin(\omega t)$. In any case the condition at the (moving) bottom has to be $\frac{\partial\phi}{\partial n}=0$ where $n$ is the instantaneous normal to the bottom. Clearly the fluid motion is incompressible with velocity $\nabla\phi$.