Elastic wave propagation is highly complex when P- and S-waves scatter away and convert to each other at the discontinuities in the medium. Conventional elastic wave simulation methods often model particle displacement/velocity fields due to both P- and S-wave propagation, and require excessive computation costs. We present two separated, yet coupled second-order wave equations for P- and S-wave propagations, from which the wave-medium interactions can be interpreted directly. Governing relations between each medium parameter (Vp, Vs, and ρ) and the body wave propagation provide important insights for the modeling and inversion. To reduce the computational cost for elastic wave simulation, which is imbedded in each elastic waveform inversion iteration, we present an approximation to the full elastic wave equations based on a set of coupled scalar wave equations. Through numerical simulations, we show that although these approximated solutions sacrifice their accuracy of the modeled amplitudes, they accurately capture the kinematics of the full elastic solution for both P- and S-waves. We suggest these modified wave equations be used for elastic reverse time migration, migration velocity analysis, and amplitude-insensitive full waveform inversion to save the computational cost and to avoid the undesired crosstalk artifacts.