Oscillations and Waves

Oscillations are repeated variations of a quantity around an equilibrium; waves are the propagation of such a disturbance through space. They occur whenever inertia, a restoring force, and little dissipation combine, making systems exchange energy between motion and stored configuration. Oscillations and waves are relevant to clocks, vibration isolation, resonance, electrical filters, molecular vibration, sound, seismic activity, water waves, and string instruments. The fundamental building block is the Harmonic oscillator, from which all other oscillation types derive.

Details

The baseline model is a mass $m$ displaced from equilibrium by $x$ . For small displacement, the restoring force is approximately linear: $F = -kx$ . Newton’s law gives $m\ddot{x} + kx = 0$ , whose solution is $x(t) = A\cos(\omega t + \phi)$ with $\omega = \sqrt{k/m}$ .

Physical interpretation. The system exchanges energy between kinetic ( $\tfrac{1}{2}m\dot{x}^2$ ) and potential ( $\tfrac{1}{2}kx^2$ ) forms. The period $T = 2\pi/\omega$ is independent of amplitude—a hallmark of linearity.

Units. $k$ has units N/m (or equivalently kg/s²); $\omega$ is rad/s; $A$ has the same units as $x$ .

When the linear approximation breaks down. For a pendulum, the linear model $\ddot{\theta} + (g/L)\theta = 0$ is valid only for $|\theta| \ll 1$ rad. Beyond that, the $\sin\theta$ term introduces amplitude-dependent period (see Nonlinear oscillations).

Brief history

Galileo recognized pendulum isochronism.
Hooke associated a linear restoring force with displacement.
Newton’s laws made small oscillations quantitatively predictable.
Fourier analysis connected oscillations to superposition and normal modes.

Categorization

Each class isolates one physical mechanism that modifies or extends the ideal harmonic oscillator.

Page	Physical mechanism isolated	Typical behavior	Where it matters
Harmonic oscillator	Linear restoring force (baseline)	Constant period, sinusoidal motion	Clocks, reference models
Damped oscillations	Dissipation: energy loss to environment	Decaying amplitude, quality factor	Shock absorbers, pendulums, circuits
Driven oscillations	External energy input (force or parameter modulation)	Resonance, steady-state response, parametric growth	Filters, tuning, bridges, parametric amplifiers
Coupled oscillations	Multiple interacting degrees of freedom	Normal modes, beat frequencies	Molecules, crystals, coupled circuits
Nonlinear oscillations	Higher-order force terms beyond linear	Amplitude-dependent period, chaos	Metronomes, large pendulums, nonlinear circuits
Wave propagation	Continuous-media (spatial) limit of coupled oscillators	Wave equation, dispersion, interference	Sound, seismic waves, water waves, strings

Authoritative sources

OpenStax University Physics Volume 1, Chapter 15: Oscillations

Oscillations and Waves §

Brief history §

Categorization §

Oscillations and Waves

Brief history

Categorization