The discontinuous Lagrangian approach, allowing for a variational description of irreversible phenomena in continuum theory such as viscosity and thermal conductivity, is utilised for the analysis of damped acoustic waves. Starting from a Lagrangian for general viscous flow theory, by linearisation of the resulting Euler–Lagrange equations and performing an ensemble average, a single wave equation for the density perturbations is obtained, being the one resulting from classical Navier–Stokes theory with an additional term due to thermodynamic non-equilibrium. By considering harmonic waves, the respective non-classical dispersion relation and its implications are analysed. •Viscous flow with thermal conduction is deducible from a discontinuous Lagrangian.•Non-classical effects occur beyond thermodynamic equilibrium.•By linearisation and ensemble averaging a non-classical wave equation is derived.•The corresponding dispersion relation contains non-equilibrium terms.