2.3 Damped oscillations

Real oscillators lose energy. Adding a velocity-dependent drag $-b \dot x$ to the spring force gives Newton’s second law

m \ddot x \;=\; -k x - b \dot x.

Divide by $m$ and define $\omega_0^2 = k/m$ and $\gamma = b/(2m)$ . The equation in canonical form is

\ddot x + 2\gamma \dot x + \omega_0^2 x \;=\; 0.

The parameter $\gamma$ has units of inverse time; it controls how fast amplitude decays.

Three regimes

Try $x(t) = e^{\lambda t}$ . The characteristic equation is

\lambda^2 + 2\gamma \lambda + \omega_0^2 \;=\; 0,

with roots

\lambda_\pm \;=\; -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}.

The sign of $\gamma^2 - \omega_0^2$ sets the regime:

Underdamped ( $\gamma < \omega_0$ ): roots are complex, $\lambda_\pm = -\gamma \pm i \omega_d$ with $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ . The motion oscillates at frequency $\omega_d$ (slightly slower than $\omega_0$ ) inside a decaying envelope $e^{-\gamma t}$ .

x(t) \;=\; X_0\, e^{-\gamma t}\, \cos(\omega_d t + \varphi).

Critically damped ( $\gamma = \omega_0$ ): one repeated real root, $\lambda = -\gamma$ . The solution is $x(t) = (A + B t)\, e^{-\gamma t}$ — no oscillation, fastest possible non-oscillating return to equilibrium.
Overdamped ( $\gamma > \omega_0$ ): two distinct real roots, both negative. The solution is $x(t) = A e^{-\lambda_+ t} + B e^{-\lambda_- t}$ — a sluggish, monotonic return.

▶ Where $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ comes from

Inserting $x = e^{\lambda t}$ into the ODE produces the quadratic $\lambda^2 + 2\gamma\lambda + \omega_0^2 = 0$ . The quadratic formula gives $\lambda = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$ . In the underdamped regime, the discriminant is negative; write $\gamma^2 - \omega_0^2 = -\omega_d^2$ with $\omega_d^2 = \omega_0^2 - \gamma^2 > 0$ . Then $\lambda = -\gamma \pm i \omega_d$ , and the real solution is $e^{-\gamma t}[A \cos(\omega_d t) + B \sin(\omega_d t)]$ .

Energy decay

For the underdamped solution, the mechanical energy is roughly

E(t) \;\approx\; \tfrac12 k X_0^2 \, e^{-2\gamma t},

decaying as $e^{-2\gamma t}$ (the factor of 2 because energy is amplitude squared). The 1/e time for amplitude is $\tau_{\text{amp}} = 1/\gamma$ ; the 1/e time for energy is $\tau_{\text{en}} = 1/(2\gamma)$ . We will see in lesson 2.5 that the quality factor $Q \equiv \omega_0 / (2\gamma)$ is the number of radians of oscillation in which the energy decays by $1/e$ — a useful intuition.

Where damping comes from

In a mechanical mass-spring, damping comes from air drag, friction at the spring’s pivots, or internal hysteretic losses in the spring material. In an acoustic mode of a tube or cavity, damping comes from viscous losses at the walls, thermal conduction into the walls, and radiation out of the open end. We will revisit each of these mechanisms in detail in chapter 10 (attenuation). For now, $\gamma$ is just a lump that subsumes them all.

What this gives us

The free, damped oscillator is the homogeneous solution of the second-order linear ODE. Every transient response of every linear acoustic system contains a piece of this form — a sum of damped exponentials at the natural frequencies — alongside whatever the forcing is doing. In the next lesson we add the forcing and discover resonance.