Abstract Characterizing the loss of coherence of an internal tide propagating through mesoscale turbulence has been a major challenge in oceanography, particularly due to its implications for the interpretation of satellite data. In this paper, we intend to study the correlations between a balanced motion and the incoherent part of a wave in an idealised configuration. We introduce a new modal decomposition technique, named broad-band proper orthogonal decomposition (BBPOD), which consists in performing a proper orthogonal decomposition (POD) on complex demodulated variables. After connecting BBPOD to the standard SPOD, we show that BBPOD, coupled with the extended POD technique enables us to associate the principal components of the incoherent field to the slow flow structures responsible of this loss of coherence through triadic interactions with the incident wave.