Under what conditions is it useful to assume the flow is irrotational? What kind of flows would this be a bad assumption for?
If we push a rod through the water we will have to exert a force, despite the fact that irrotational flow theory says we won’t. What two effects cause this resistance?
Why do the conformal mapping? What advantage does applying these transformations give us?
Consider a steepening wave by computing the flow and height fields by the method of characteristics.
Suppose there is a quiescent region to the right of $x=0$ of thickness $d_0$ such that the shallow water wave speed there is $c_0=0.1 \ m/s$. For $x<-2.5\ m$ $c = 1 \ m/s$. The wave speed varies linearly between these two points.
We define $u$ such that the negative characteristic value is the same every where, i.e. $u(x, 0) = 2 (c(x, 0) - c_0)$, so that the velocity is zero in the shallow region, and $1.8 \ m/s$ in the deep region, and linearly varying between the two regions.