Wave

4 min read#waves

A disturbance that travels through a medium while each point of the medium merely oscillates in place — carrying energy without carrying matter.

Contents

Wave

A wave is a self-propagating disturbance: a pattern that moves through a medium while the medium itself stays put. Drop a pebble in a pond and ripples race outward, but a floating leaf only bobs up and down — it does not travel with the ripple. That is the defining paradox of wave motion. Energy and shape propagate; matter oscillates in place. Each particle is just a Simple Harmonic Oscillator, and the wave is the choreography linking one oscillator to the next.

Anatomy of a wave

A pure traveling sine wave is described by

u(x,t)=A\sin\!\left(kx-\omega t\right),

and three numbers fix everything about it:

  • Amplitude A — the peak displacement, setting how much energy the wave carries (energy \propto A^2).
  • Wavelength \lambda — the distance between successive crests; the spatial period. The wavenumber is k=2\pi/\lambda.
  • Frequency f — how many crests pass a fixed point per second; the temporal period is T=1/f and the angular frequency is \omega=2\pi f.

These are tied together by the single most important relation in the subject, linking how often the medium wiggles to how fast the pattern moves:

v = f\lambda = \frac{\omega}{k}
(1)

The speed v is usually a fixed property of the medium (tension and density for a string, stiffness for sound). So if you raise the frequency, the wavelength must shrink to keep the product constant.

A traveling sine wave moving left to right. The orange dot is a single particle of the medium — it only moves up and down, tracing simple harmonic motion as the pattern slides past. Move the mouse left/right to change frequency, up/down to change amplitude.

Transverse and longitudinal

Waves come in two flavors, distinguished by the direction the medium moves relative to the direction the wave travels:

  • In a transverse wave the particles oscillate perpendicular to the direction of travel — a wave on a string, light, the ripples above. Crests and troughs.
  • In a longitudinal wave the particles oscillate along the direction of travel — sound in air, a compression pulse down a slinky. Compressions and rarefactions instead of crests and troughs.

Both obey the same mathematics; only the geometry of the displacement differs. Either way, the next step is the equation that governs how the shape propagates: the Wave Equation.

See also

Related topics

  • OscillationsHow systems swing, settle, and sustain rhythm — from a single bob on a spring to whole populations of synchronizing oscillators.
  • Wave EquationThe partial differential equation whose solutions are anything that travels left or right at a fixed speed without changing shape.
  • WavesFrom a single swinging oscillator to ripples that travel, interfere, and carry frequency through space — the bridge between Oscillations and Fourier Analysis.
  • Simple Harmonic OscillatorA mass pulled back by a force proportional to its displacement — the linear ideal whose sinusoidal motion underlies every other oscillator.
  • Superposition PrincipleWhen two waves meet, the medium's displacement is simply the sum of what each wave would do alone — the linearity that underlies all of wave physics.
  • InterferenceTwo overlapping waves reinforce where their crests align and cancel where crest meets trough, painting a fixed pattern of bright and dark fringes.

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