approximately , or exactly . Q factor: is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. Damping ratio definition The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator. In general, systems with higher damping ratios (one or greater) will demonstrate more of a damping effect. Underdamped systems have a value of less than one. Critically damped systems have a damping ratio of exactly 1, or at least very close to it. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: where the system's equation of motion is and the corresponding critical damping coefficient is or where is the natural frequency of the system. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. Derivation Using the natural frequency of a harmonic oscillator and the definition of the damping ratio above, we can rewrite this as: This equation is more general than just the mass–spring system, and also applies to electrical circuits and to other domains. It can be solved with the approach where C and s are both complex constants, with s satisfying Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: Undamped Is the case where corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like , as expected. This case is extremely rare in the natural world with the closest examples being cases where friction was purposefully reduced to minimal values. Underdamped If s is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like . This case occurs for , and is referred to as underdamped (e.g., bungee cable). Overdamped If s is a pair of real values, then the solution is simply a sum of two