Waves

3 min read#waves

From a single swinging oscillator to ripples that travel, interfere, and carry frequency through space — the bridge between Oscillations and Fourier Analysis.

Contents

Waves

Set one oscillator going and it traces a rhythm in time. Now line up a chain of them, each nudging its neighbor, and that rhythm starts to travel: a disturbance in one place reappears, an instant later, in the next. That propagating disturbance is a wave. This section follows a single arc — oscillation → wave → interference → frequency — that turns the local, in-place motion of the Oscillations chapter into something that moves, spreads, overlaps, and ultimately decomposes into a spectrum.

A wave is what a Simple Harmonic Oscillator becomes when you give it room to spread. Every point of the medium oscillates just as a mass on a spring does; the wave is the pattern those oscillations make across space. That is the thread connecting this chapter backward to Oscillations and forward to Fourier Analysis, where we learn that any wave is a sum of pure sinusoidal oscillations.

The arc of this section

  • Wave — anatomy of a moving disturbance: wavelength, frequency, amplitude, and the master relation v=f\lambda.
  • Wave Equation — the single partial differential equation \partial_{tt}u=c^2\partial_{xx}u that every non-dispersive wave obeys, and d'Alembert's left- and right-movers.
  • Superposition Principle — the linearity that lets waves pass through one another and add up cleanly. Everything else is a consequence of it.
  • Standing Wave — what happens when a wave is trapped: nodes, antinodes, and the harmonic ladder f_n=nv/2L.
  • Interference — two sources, one pattern of bright and dark fringes set by path difference.
  • Beats — interference in time: two near frequencies producing a slow throb.
  • Doppler Effect — how motion of the source or observer shifts the frequency you receive.
  • Dispersion — when the speed depends on frequency, and phase and group velocities part ways.
  • Wave Packet — a localized lump of wave, the object that carries both energy and information.

Where it connects

See also

Related topics

  • Fourier AnalysisAny signal is a sum of pure sinusoids — and reading it as such turns the tangled language of time into the clean language of frequency.
  • Systems, Signals & ComputationAn interactive field guide to how things change, oscillate, propagate, and compute — from a swinging pendulum to chaos, waves, networks, and learning machines.
  • 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.
  • Standing WaveTrap a wave between two fixed ends and it stops traveling — freezing into a pattern of motionless nodes and violently swinging antinodes.
  • Wave EquationThe partial differential equation whose solutions are anything that travels left or right at a fixed speed without changing shape.
  • Fourier TransformThe generalization of the Fourier series to non-periodic signals — an integral that resolves any signal into a continuous band of frequencies.

Referenced by