The Hankel functions solves the Laplace equation for the velocity potential in cylindrical coordinates, so it describes a propagating circular wave.
How do I go about finding the expression for the surface displacement from the Hankel functions?
Is there no simpler way? I imagine the displacement must vanish at a rate proportional to 1/r, where r is the distance from the origin.
In electromagnetics, the Bessel function is a standing wave and the Hankel function is a travelling wave. The wave equation that would arise with a line sourc such as the OP's would be a Hankel function unless he required a reflecting boundary.Since Hankel functions are a combination of Bessel functions J and Y, with this case, there would be a singularity at r=0 for Y and thus the result should be a function of J. The only places I have sen Henkel functions arise is in situations where there is no interest in the r=0 case, i.e. annular ducts, incident flows or a pulsating source.
uart: You may want to check your equation for the asymptotic expansion of Jn. Your second cosine term should be [tex]\frac{m \pi}{2}[/tex]