,

or, more compactly,

,

where is the natural frequency of the undamped oscillator and is the damping constant. With the initial conditions and , the equation can be solved in closed form to give ,

which represents damped oscillatory motion.

Linear damping, proportional to the first power of the velocity, is most often considered because it leads to an exactly solvable equation. In fact, this is usually a good approximation in the case of air resistance with laminar flow. However, for turbulent flow, quadratic damping, with , is a better approximation. Cubic damping, with , has been suggested in considerations of energy-harvesting devices. Fractional power-law functions sometimes appear in motion through dense fluids and in models for viscoelasticity. Thus motivated, we will consider the damping power laws with , , , and . Apart from , none of these cases can be solved analytically. We will therefore seek numerical solutions of the corresponding differential equations, written as

,

specifying, in all cases, the initial conditions and . The inclusion of the factor guarantees that the damping always instantaneously retards the motion of the oscillator.

The solutions to the above equation are displayed graphically. You may compare the behavior with that of the corresponding linear damped oscillator, shown in red, by checking the appropriate box. Alternatively, you may choose to fit the oscillation to an empirical formula , with parameters , , selected for the best visual fit. The corresponding plot is then shown in blue. Usually, the approximation is reasonable only for a limited range of .