Waves & Oscillations Notes

Last reviewed 16 Sept 2026 · 9 min read

Simple harmonic motion (SHM)

SHM — periodic motion in which the restoring force (acceleration) is proportional to displacement and directed towards the mean position:

a = - ω^{2} x

FormulaSHM relations

Displacement $x = A sin (ω t + ϕ)$ Velocity $v = ω A^{2} - x^{2}$ — maximum $A ω$ at mean position Acceleration $a = - ω^{2} x$ — maximum $A ω^{2}$ at extreme positions Period $T = \frac{2 π}{ω}$ ; frequency $f = 1/ T$

Energy: $E = \frac{1}{2} k A^{2} = \frac{1}{2} m ω^{2} A^{2}$ (constant); KE = PE at $x = A / 2$

Spring–mass system: $T = 2 π \frac{m}{k}$ (independent of g) Simple pendulum (small amplitude): $T = 2 π \frac{l}{g}$ (independent of mass) Springs: series $\frac{1}{k} = \frac{1}{k _{1}} + \frac{1}{k _{2}}$ ; parallel $k = k_{1} + k_{2}$

Damped oscillations

Real oscillations lose energy (friction, air resistance) — amplitude decays exponentially: $A (t) = A_{0} e^{- b t /2 m}$ .
Underdamped (oscillatory decay), critically damped (fastest return without oscillation — shock absorbers, instrument pointers), overdamped (slow return).
See Structural Dynamics for damping ratio and logarithmic decrement.

Forced oscillations and resonance

A system driven by a periodic force oscillates at the driving frequency.
Resonance — amplitude becomes maximum when the driving frequency equals the natural frequency; the peak is sharper for low damping.
Engineering relevance:
- Machine foundations must avoid resonance with operating speeds.
- Soldiers are asked to break step while crossing bridges.
- Footbridges may vibrate under synchronised pedestrian loads (e.g. lateral vibrations).
- Wind-induced oscillations of slender structures; the Tacoma Narrows Bridge (1940) collapse is a famous example of wind-induced (aeroelastic) oscillation.
- Earthquake ground motion amplifies buildings whose periods match predominant ground periods.
- Useful resonance: tuning circuits, musical instruments, tuned mass dampers.

Wave motion

A wave transfers energy (not matter) through a medium or space.

Type	Particle motion	Examples
Transverse	Perpendicular to direction of propagation (crests and troughs)	Waves on strings, light and EM waves, S-waves
Longitudinal	Parallel to propagation (compressions and rarefactions)	Sound in air, P-waves

Mechanical waves need a medium; electromagnetic waves do not.

FormulaWave relations

v = f λ, y = A sin (k x - ω t), k = \frac{2 π}{λ}, ω = 2 π f

Wave on a stretched string: $v = \frac{T}{μ}$ (T = tension, μ = mass per unit length)

Speed of sound:

In solids: $v = \frac{E}{ρ}$ ; in fluids: $v = \frac{K}{ρ}$
In gases (Newton–Laplace): $v = \frac{γ p}{ρ}$
In air: about 331 m/s at 0 °C, increasing by about 0.6 m/s per °C (≈ 343 m/s at 20 °C); independent of pressure at constant temperature; increases with humidity
Sound travels faster in water (~~1480 m/s) and steel (~~5000–6000 m/s)

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Simple harmonic motion (SHM)

Damped oscillations

Forced oscillations and resonance

Wave motion

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