In general, the nonlinear behavior of an elastic wave in isotropic hyperelastic solids with quadratic nonlinearity depends on five independent elastic constants, namely, the three third-order elastic constants and two second-order elastic constants. In this article, we show that such nonlinear behavior can be described fully by only three independent non-dimensional parameters if the wave motion is two-dimensional. Furthermore, if the motion is a plane wave, only two independent non-dimensional parameters are needed to fully describe the nonlinear behavior of the wave. These results are useful for conducting numerical simulations and for interpreting experimental measurement data.