  The (third-order in time) JMGT equation \cite{Jordan2,HCP} is a nonlinear
(quasi-linear) Partial Differential Equation (PDE) model introduced to describe
a nonlinear propagation of sound in an acoustic medium. The important feature
is that the model avoids the infinite speed of propagation paradox associated
with a classical second order in time equation referred to as Westervelt
equation. Replacing Fourier's law by Maxwell-Cattaneo's law gives rise to the
third order in time derivative scaled by a small parameter $\tau >0$, the
latter represents the thermal relaxation time parameter and is intrinsic to the
medium where the dynamics occur. In this paper we provide an asymptotic
analysis of the third order model when $\tau \rightarrow 0 $. It is shown that
the corresponding solutions converge {\it in a strong topology of the phase
space } to a limit which is the solution of Westervelt equation. In addition,
rate of convergence is provided for solutions displaying higher order
regularity. This addresses an open question raised in \cite{kaltev2}, where a
related JMGT equation has been studied and {\it weak star } convergence of the
solutions when $\tau \rightarrow 0$ has been established. Thus, our main
contribution is showing {\it strong convergence on infinite time horizon,}
along with related rates of convergence valid on a finite time horizon. The key
to unlocking the difficulty owns to a tight control and propagation of the
"smallness" of the initial data in carrying the estimates at three different
topological levels. The rate of convergence allows one then to estimate the
relaxation time needed for the signal to reach the target. The interest in
studying this type of problems is motivated by a large array of applications
arising in engineering and medical sciences.
