The main purpose of the paper is to illustrate the analogy between the radiation from a guiding surface which is bent or curved, and the radiation from an electron which is deflected (brems-strahlung) or moves in a curved path in a magnetic field (synchrotron radiation). In developing this analogy the significance of the “characteristic radius” associated with a cylindrical or spherical wave system, defined as the radius at which the argument of the radial wavefunction is equal to its order, is examined. A small diversion is made to illustrate the relevance of this concept to atomic radiation, supergain antennas, and wave-mechanical scattering.