Beam break-up with tune chirp for an arbitrary wakefield

The asymptotic form of the beam break-up instability is computed up to quadrature for an arbitrary wakefield, in the presence of a linear variation in betatron tune from head to tail along the beam. For illustration, the result is applied to a broadband impedance (resistive wall) and a narrow band impedance (single resonator) and benchmarked against simulation, for parameters of interest in induction linacs.