We use electrostatic partice-in-cell (PIC) simulations and theory to study the damping of 1D plasma waves. We consider the linear regime where the asymptotic damping rate is much bigger than the bounce frequency. In this regime the waves are typically very small and often below the thermal noise in simulations and experiments. These waves can be studied using a subtraction technique in which two simulations with identical random number generation seeds are carried out. In the first, a small amplitude wave is excited. In the second simulation no wave is excited. The results from each simulation are subtracted providing a clean linear wave that can be studied. Since the Landau derivation does not provide a description of damping in terms of individual particle trajectories, we analyze Landau damping using a Lagrangian approach based on energy conservation and the linearized particle trajectories. This method provides a time-dependent resonance curve and the energy transfer of the particles in the damping process. The time-dependent resonant width measured in the simulations is compared with the theoretical prediction. Simulations in which particles within the resonance width are removed are also presented.