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 ?
The support of a function is the part of the domain where the function value is nonzero. What this is saying is that a finite section of the bottom of the channel is set in motion. If $x$ is along the channel, maybe the section from $x=0$ to $x=1$ is alternately lifted and dropped, so the bottom there has elevation say $y=A\sin \omega t$. This would be produced by having a plate of that size on a piston, moving up and down with amplitude $A$ and frequency $\frac \omega{2\pi}$. The rest of the bottom of the channel would be fixed at $y=0$