How oscillations repeat and waves carry energy
Simple harmonic motion occurs when the restoring force is proportional to displacement and directed towards the mean position, F = −kx. The displacement varies as x = A sin(ωt + φ), with time period T = 2π/ω. For a spring–mass system T = 2π√(m/k), and for a simple pendulum T = 2π√(l/g) for small angles. Energy keeps changing between kinetic and potential, while the total stays constant in the absence of damping.
A wave carries energy without carrying matter. A travelling wave can be written as y = A sin(kx − ωt), with speed v = fλ. Transverse waves on a string travel at v = √(T/μ), and sound in a gas at v = √(γP/ρ). When waves overlap, the principle of superposition adds their displacements. Two identical waves moving in opposite directions form standing waves with nodes and antinodes; on a string fixed at both ends the allowed frequencies are nv/2L. Beats occur when two close frequencies combine, with beat frequency f₁ − f₂.
Why it matters
Pendulum clocks, car suspension springs, musical instruments and earthquake-resistant buildings all depend on oscillations and resonance. Musicians tune instruments using beats, and engineers avoid resonance that could damage bridges. Wave equations introduced here are used again for sound, light, electromagnetic waves and quantum physics.
Key terms
- Simple harmonic motion
- Periodic motion in which acceleration is proportional to displacement and directed towards the mean position.
- Superposition
- When waves overlap, the resulting displacement is the sum of individual displacements.
- Node
- A point on a standing wave where the displacement is always zero.
- Resonance
- Large oscillations produced when a system is driven at its natural frequency.