The problem of propagating nonlinear acoustic waves is considered; the solution to which, both with and without damping, having been obtained to-date starting from the Navier–Stokes–Duhem equations together with the continuity and thermal conduction equation. The novel approach reported here adopts instead, a discontinuous Lagrangian approach, i.e. from Hamilton’s principle together with a discontinuous Lagrangian for the case of a general viscous flow. It is shown that ensemble averaging of the equation of motion resulting from the Euler–Lagrange equations, under the assumption of irrotational flow, leads to a weakly nonlinear wave equation for the velocity potential: in effect a generalisation of Kuznetsov’s well known equation with an additional term due to thermodynamic non-equilibrium effects. •Viscous flow with thermal conduction is deducible from a discontinuous Lagrangian.•Non-classical effects occur beyond thermodynamic equilibrium.•By ensemble averaging a non-classical equation of motion is derived.•In the irrotational and weakly nonlinear case a generalised Kuznetsov equation is obtained.