waves in the nonlinear laminates they studied. @andrianov2014numerical later showed this result analytically for a different nonlinear model, using an asymptotic homogenization method [@andrianov2013dynamic]. Here, we generalize their conclusion to the present vectorial problem, where there are two coupled components of the displacement field and two coupled wave polarizations, using the limiting velocities of the quasi-shear and quasi-pressure waves. To this end, we follow @santosa1991dispersive, who rigorously showed that the velocity of the leading term in the Bloch expansion is simply the root square of the harmonic average of the stiffness over the weighted average of the mass density, and further showed that this result agrees with multi-scale asymptotics or homogenization. To carry out this calculation in the linear limit of our problem, we evaluate the instantaneous stiffness of the phases in the pre-strained configuration [@ogden97book; @LUSTIG2018jmps], and use the initial mass density. We denote the velocities that are calculated using the resultant average for the quasi-pressure and quasi-shear waves by $c_{+}^{\left(LL\right)}$ and $c_{-}^{\left(LL\right)}$, respectively. These values at each pre-strain are given by the dashed lines in Fig.$\ $\[fig:velocities\], confirming they are lower bounds. This observation implies that quasi-pressure and quasi-shear vector solitary waves are supersonic, thereby generalizing the results 