Envelope equations for transients in linear chains of resonators

Transient signals in strings of resonators consist of regimes with different time constants: high frequency oscillations, beat signals and exponentials. If one is interested only in the signals envelope one can transform the system of second order differential equations into a system of first order differential equations. The latter carries fast varying terms, which are averaged out, and slowly varying terms. The resulting equations are well behaving and can be integrated numerically. Results are shown for the filling process under beam loading of the superconducting nine-cell TESLA cavity