  We establish the jump conditions for the wavefunction and its derivatives
through the formal solutions of the wave equation. These conditions respond to
the requirement of continuity of the perturbations at the position of the
particle and they are given for any mode at first order. Using these jump
conditions, we then propose a new method for computing the radiated waveform
without direct integration of the source term. We consider this approach
potentially applicable to generic orbits.
