Oscillations

Oscillations are repetitive variations of a system about an equilibrium position or between two states. They can be periodic, repeating at regular intervals, or quasi-periodic. Common examples include a mass on a spring, a swinging pendulum, and electrical LC circuits. Oscillations arise when a restoring force tends to return the system to equilibrium and there is often some energy exchange with a reservoir or between potential and kinetic energy.

A standard mathematical model is the harmonic oscillator. For a mass-spring-damper system, m x'' + c x' + k x = F_ext(t), where x is displacement, m is mass, k is spring stiffness, c is damping, and F_ext is an external force. Defining ω0 = sqrt(k/m) as the natural frequency, the undamped case (c = 0, F_ext = 0) reduces to x'' + ω0^2 x = 0, with solutions x(t) = A cos(ω0 t + φ). Damping modifies the behavior: x'' + 2β x' + ω0^2 x = 0, with β = c/(2m). If β < ω0, the system is underdamped and oscillations decay in amplitude; if β = ω0, it is critically damped; if β > ω0, it is overdamped and returns to equilibrium without oscillating.

When driven by a periodic force F_ext(t) = F0 cos(ωt), the system exhibits forced oscillations. After transients decay, the steady-state response has amplitude A(ω) and phase lag relative to the drive, governed by the system’s parameters. Resonance occurs near the natural frequency with light damping, producing large amplitudes.

Energy in oscillations continually shifts between kinetic and potential forms, with damping converting some energy to heat. The quality factor Q measures how much energy remains stored relative to dissipation per cycle and is high for sharply tuned systems. Oscillations appear across physics and engineering, from mechanical devices to electrical circuits and biological rhythms.