  We are concerned with hyperbolic systems of order-one linear PDEs originated
on non-characteristic manifolds. We put forward a simple but effective method
of transforming such initial conditions to standard initial conditions (i.e.
when the solution is specified at an initial moment of time). We then show how
our method applies in fluid mechanics. More specifically, we present a complete
solution to the problem of long waves run-up in inclined bays of arbitrary
shape with nonzero initial velocity.
