  A theoretical model of an underdamped harmonic oscillator (UHO) driven by
periodic short pulses may find plenty of applications in classical,
semiclassical, and quantum physics. We present here two different forms of
analytical solutions: {\it time-periodic solutions} and {\it harmonic
solutions} for one-dimensional classical UHO driven by three different trains
of short pulses. They are a Dirac comb, a train of square pulses, and a train
of Gaussian pulses with the same pulse-to-pulse time interval $T$ and pulse
width $2\tau$. Two solutions for square and Gaussian pulses approach to that of
the Dirac comb when the pulse width $2\tau \rightarrow 0$ as expected. In
particular, the harmonic solutions for Dirac comb and Gaussian pulses could be
expressed approximately with harmonic terms of the repetition frequency
$\omega_{\rm R} = 2\pi/T$ up to the second order. The presented analytical
solutions would provide a practical way to determine experimentally the system
parameters such as the underdamped oscillation frequency $\omega =
\sqrt{\omega_0^2-\gamma^2}$, the natural frequency $\omega_0$, and the damping
rate $\gamma$, by nonlinear curve fitting procedures for different driving
force parameters of $T$ and $2\tau$.
