Final answer:
The displacement D(t) of a damped harmonic oscillator can be expressed using a cosine or sine function depending on the initial phase, with the damped harmonic motion commonly represented as Ae^{-βt} cos(ωt + φ).
Step-by-step explanation:
Determining the displacement, D(t), for a damped harmonic oscillator involves using a function that takes into account the amplitude, damping factor, angular frequency, and time. The displacement in simple harmonic motion (SHM) can be represented as x(t) = X cos(2πt/√(k/m)X) or a similar function using the sine wave, as both cos and sin only differ by a phase shift. For damped harmonic motion, the general form of the displacement equation is typically Ae^{-βt} cos(ωt + φ), where A is the amplitude, β is the damping factor, ω is the angular frequency, and φ is the phase shift. As mentioned in the reference material, the choice between a cosine or sine function depends on the conditions of the system, such as the phase at t=0. In this case, considering the cosine function was initially chosen, we stick with the cosine function unless a phase shift indicates otherwise.