The Nature of Waves

Summary

• Properties
  • A wave is a disturbance that propagates through a medium.
    • verb propogate; noun propogation refers to the transmission of a disturbance from one location to another.
    • sing. medium plu. media refers to the intervening substance(s) through which a disturbance is transmitted.
  • Waves transfer energy, momentum, and information, but not mass.
• Classifying waves by medium
  • Mechanical Waves
    • Matter is the medium.
    • Sound is a mechanical wave.
  • Electromagnetic Waves
    • Electric and magnetic fields are the media.
    • Light is an electromagnetic wave.
  • Gravitational Waves
    • The gravitational field is the medium.
• Classifying waves by orientation
  • Transverse Waves
    • The disturbance is perpendicular to the direction of propagation.
    • All electromagnetic waves are transverse. This includes light.
    • A crest is a point of maximal change in the positive direction.
    • A trough is a point of maximal change in the negative direction.
  • Longitudinal Waves
    • The disturbance is parallel to the direction of propagation.
    • Sound is a longitudinal wave.
    • A compression or condensation is a region where the medium is under compression.
    • A rarefaction or dilation is a region where the medium is under tension.
  • Surface Waves or Complex Waves
    • A combination transverse-longitudinal wave that forms near the surface of some media.
    • Deep water waves are surface waves.
  • Torsional Waves
    • The disturbance causes the medium to twist.
• Classifying waves by duration
  • adj. episodic; noun pulse
    • The disturbance is momentary and sudden.
  • adj. periodic, harmonic; noun wave train
    • The disturbance repeats at regular intervals.
• Classifying waves by appearance
  • Traveling Waves
    • Appear to move.
  • Standing Waves
    • Do not appear to move.
• Characteristics of periodic waves
  • Amplitude (A)
    • The maximum absolute value of a periodically varying quantity.
    • Amplitude has the unit of the quantity that is changing (ex. displacement, pressure, field strength, etc.)
  • Period (T)
    • The time between successive cycles of a repeating sequence of events.
    • T = t/n (time per number of cycles)
    • The SI unit of period is the second [s].
  • Frequency (f)
    • The number of cycles of a repeating sequence of events in a unit interval of time.
    • f = n/t (number of cycles per time)
    • Frequency and period are reciprocals (or inverses) of one another: f = 1/T.
    • The SI unit of frequency is the hertz [Hz = 1/s = s⁻¹].
  • Phase (ϕ)
    • The stage of development of a periodic process.
    • Two points on a wave with the same phase have the same…
      • quantity of disturbance (ex. displacement) and
      • rate of change of disturbance (ex. velocity).
    • Phase is an angular quantity.
      • Adjacent points in phase are separated by one complete cycle.
      • Adjacent points out of phase are separated by half a cycle.
    • The SI unit of phase is the radian, which is itself a unitless ratio [rad = m/m = Pa/Pa = (V/m)/(V/m) = etc.].
  • Wavelength (λ)
    • The distance between any point on a periodic wave and the next nearest point corresponding to the same portion of the wave.
    • Wavelength is measured between adjacent points in phase.
    • The SI unit of wavelength is the meter [m].
  • Speed (v)
    • Waves propagate with a finite speed (sometimes called the wave speed) that depends upon…
      • the type of wave,
      • the composition of the medium, and
      • the state of the medium
    • v = Δs/Δt the rate of change of distance with time by definition.
    • v = fλ the product of frequency and wavelength for periodic waves.
    • Frequency and wavelength are inversely proportional.
      • Lower frequency waves have longer wavelengths.
      • Higher frequency waves have shorter wavelengths.
    • The speed of a wave is sometimes known as its wave speed.
    • The SI unit of speed is the meter per second [m/s].
• One-dimensional wave equation
  • linear version

    f(x, t) = A sin(2π(ft − x/λ) + ϕ)

    where

    A =	amplitude
    f =	frequency
    λ =	wavelength
    ϕ =	phase

  • angular version

    f(x, t) = A sin(ωt − kx + ϕ)

    where

    A =	amplitude
    ω =	angular frequency (temporal frequency), ω = 2πf
    k =	wave number (spatial frequency), k = 2π/λ
    ϕ =	phase