Directional pulse propagation in beam, rod, pipe, and disk geometries

I derive directional wave equations useful for pulses propagating in beam, rod, pipe, and disk geometries by using a cylindrical coordinate system; the scheme works equally well for either long multi-cycle or single-cycle ultrashort pulses. This is achieved by means of a factorization procedure that conveniently generates exact bi-directional and first order wave equations after the selection of propagation direction - either axial, radial, or even angular. I then discuss how to reduce these to a uni-directional form, and discuss the necessary approximation, which is essentially a paraxial approximation as appropriately generalized to the specific geometry.