Oscillations and Waves

Repeating Motion and Traveling Disturbances

Many systems oscillate — repeat a motion around an equilibrium. The ideal model is simple harmonic motion (SHM), which arises whenever the restoring force is proportional to the displacement (a mass on a spring: $F = -kx$, Hooke's law). SHM traces a smooth sinusoid in time, described by:

• Amplitude $A$ — the maximum displacement from equilibrium.
• Period $T$ — the time for one full cycle.
• Frequency $f = \frac{1}{T}$ — cycles per second, in hertz. Period and frequency are reciprocals.

A defining feature of SHM: the period does not depend on the amplitude (for an ideal spring or small-swing pendulum). A big swing and a small swing take the same time — the property that makes pendulums good clocks.

A wave is a disturbance that travels through space, carrying energy without transporting matter. Key quantities are wavelength $\lambda$ (distance between repeats), frequency $f$, and wave speed $v$, tied by the fundamental relation: $$v = f\lambda.$$ Waves come in two types: transverse (oscillation perpendicular to travel, like light or a plucked string) and longitudinal (oscillation along the travel direction, like sound). Waves also interfere — overlapping waves add, giving constructive (reinforcing) or destructive (cancelling) interference — the basis of everything from noise-cancelling headphones to the diffraction limits of chip manufacturing.

Common pitfall: thinking a wave carries matter along with it, and confusing frequency with period. A wave transports energy, not the medium — a cork on water bobs in place as waves pass; the water does not travel with the wave. And frequency and period are reciprocals ($f = 1/T$), not the same thing: high frequency means short period, so a large frequency pairs with a small period.

A transverse wave with wavelength marked between crests and a single accent point oscillating up and down in place as the wave travels rightward — energy moving, matter staying.

$$v = f\lambda,\quad f = 1/T$$