  We investigate the effect of the cuspiness of scalar potentials on the
production of gravitational waves during oscillon formation after inflation. We
consider a more general form of potentials with a mass parameter $M$, which
repoduce cuspy potentials for fields much larger than $M$, and smooth
potentials in the opposite limit. For cuspy potentials, nonsmooth oscillations
of the inflaton induce an amplification of the inflaton fluctuations at the
bottom of the potential, so that oscillons copiously form, which leads to a
significant stochastic gravitational wave background with a double-peak
spectrum. By varying the parameter $M$, we find that cuspy potentials yield
stronger signals of gravitational waves and the generation of gravitational
waves disappears for smooth potentials. Moreover, we calculate the equation of
state after inflation and find the presence of a quasi matter-dominated stage
right before the transition to the radiation-dominated stage.
