  We analyze the behavior of third-order in time linear and nonlinear sound
waves in thermally relaxing fluids and gases as the sound diffusivity vanishes.
The nonlinear acoustic propagation is modeled by the
Jordan--Moore--Gibson--Thompson equation both in its Westervelt and in its
Kuznetsov-type forms, that is, including quadratic nonlinearities of the type
$(u^2)_{tt}$ and $ (u_t^2 + |\nabla u|^2)_t$. As it turns out, sufficiently
smooth solutions of these equations converge in the energy norm to the
solutions of the corresponding inviscid models at a linear rate. Numerical
experiments illustrate our theoretical findings.
