An analytical investigation is made of the transient response of a harmonic oscillator with damping proportional to the square of the velocity. The input motions considered are the half-cycle velocity pulse and the half- cycle displacement pulse. The response quantities studied are the maximum displacement, velocity, arid acceleration of the system. Results were obtained by integrating the equation numerically and are summarized in the form of response spectra. The results are compared to those obtained with linear damping; a method for obtaining equivalent linear damping coefficients is given. The study indicates that quadratic damping is most effective in reducing responses in the range of frequencies for which the responses are the largest. For lightly damped systems, quadratic damping was found to be equally effective in reducing each of the three responses studied. But for heavily damped systems it is most effective in reducing velocities.