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Waves

[toc]

Introduction

A wave is a propagating disturbance. Movement may happen across temporal and spatial dimensions.

An oscillation is a wave with a periodic disturbance.

  • Waves have a period T and a frequency (rate) f = 1/T.
  • Sinoids have an amplitude A and a phase offset δ.

An harmonic is a wave with a frequency that is a multiple of a given fundamental frequency.

Waves with unrelated frequencies produce noise.

Types

  • Traveling wave. Wave moves in one direction.
  • Standing wave. Multiple superimposed waves that travel in different directions?
    • Longitudinal wave. Through a different medium?
    • Transverse wave.

Fractals

Geometric patterns that exhibit self-similarity. They repeat a pattern over different time-scales. E.g. a snowflake.

Simulations

Traveling Waves

Harmony

Waves that are in tune produce an distinct or characteristic output (when superimposed). Unrelated waves produce noise.

One Spatial Dimension

f(x) = Σ A sin(f x + φ) for each amplitude A and frequency f.

Arbitrary or unrelated frequencies product noise.

plot-waves-random

Frequencies that are in tune produce a sawtooth wave.

plot-waves-saw

Frequencies with alternating phases produce a square-shaped wave.

plot-waves-square

Two Spatial Dimensions

Lissajous-curve.mov

Color

Using color to show the amplitude of a two-dimensional wave.

c(x, y) = sin(f * (ax + by))

Unrelated waves

plot-waves-2d-random

Tuned waves

plot-waves-2d-saw-square

Radial waves

c(x, y) = sin(f * (axx + byy)

plot-waves-2d-saw-square

Standing Waves

See holography.

Vibrating Membrane

drum.mp4

Non-linear Systems

Non-linear differential equations.

Predator-prey Model

src

Growth of two populations. Their growth is inversely related.

lotka-volterra

Lorenz System

src

lorenz

Diffusion Systems

Gray-scott diffusion systems.

gray-scott-jet.mp4

gray-scott.mp4

Navy-Stokes

...

Real World Systems

Periodic Systems

See periodic structures.

Unstable systems

See critical points, self-organized criticality.

  • Boom And Bust Cycle (years)
    • Expansion, crisis (critical point), recession, recovery
  • Pandemics
  • Forest fires
  • Avalanches
  • Lightning / blizzards
  • Pendulum / swing set