  We study collective modes in a classical system of particles with repulsive
inverse-power-law (IPL) interactions in the fluid phase, near the fluid-solid
coexistence (IPL melts). The IPL exponent is varied from $n=10$ to $n=100$ to
mimic the transition from moderately soft to hard-sphere-like interactions. We
compare the longitudinal dispersion relations obtained using molecular dynamic
(MD) simulations with those calculated using the quasi-crystalline
approximation (QCA) and find that this simple theoretical approach becomes
grossly inaccurate for $n\gtrsim 20$. Similarly, conventional expressions for
high-frequency (instantaneous) elastic moduli, predicting their divergence as
$n$ increases, are meaningless in this regime. Relations of the longitudinal
and transverse elastic velocities of the QCA model to the adiabatic sound
velocity, measured in MD simulations, are discussed for the regime where QCA is
applicable. Two potentially useful freezing indicators for classical particle
systems with steep repulsive interactions are discussed.
