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The vast majority of Pulsar Wind Nebulae (PWNe) present in the Galaxy isformed by middle-aged systems characterized by a strong interaction of the PWNitself with the supernova remnant (SNR). Unfortunately, modelling these systemscan be quite complex and numerically expensive, due to the non-linearity of thePWN-SNR evolution even in the simple 1D / one-zone case when the reverse shockof the SNR reaches the PWN, and the two begin to interact (and reverberationstarts).  Here we introduce a new numerical technique that couples the numericalefficiency of the one-zone thin shell approach with the reliability of a full``lagrangian'' evolution, able to correctly reproduce the PWN-SNR interactionduring the reverberation and to consistently evolve the particle spectrumbeyond. Based on our previous findings, we show that our novel strategyresolves many of the uncertainties present in previous approaches, as thearbitrariness in the SNR structure, and ensure a robust evolution, compatiblewith results that can be obtained with more complex 1D dynamical approaches.Our approach enable us for the first time to provide reliable spectral modelsof the later compression phases in the evolution of PWNe. While in general wefound that the compression is less extreme than that obtained without suchdetailed dynamical considerations, leading to the formation of less structuredspectral energy distributions, we still find that a non negligible fraction ofPWNe might experience a super-efficient phase, with the optical and/or X-rayluminosity exceeding the spin-down one.

Reverberation of pulsar wind nebulae (III): Modelling of the plasma interface empowering a long term radiative evolution