What is the expression of an arbitrary curved line source wave?

Say that we have a point source. And at some distance away from the source point, the wave front is a non-closure curved line. Say somehow we absorbed all other "waves", now we treat this curved line as a new source, and it is incident on some cylinder scatterer.
If the curve is given: $f(x,y)=0$
Then what is the expression of this kind of incident wave? Can we expand it in cylindrical coordinates with Bessel functions?
Or do I need further assumptions?