The Nature of Waves

Summary

Properties
- A wave is a disturbance that propagates through a medium.
- Propagation describes the spreading of a disturbance
- 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.
  (The existence of gravitational waves has not yet been confirmed.)
Classifying Waves by Orientation
- Transverse Waves: disturbance is perpendicular to the direction of propagation
  - All electromagnetic waves are transverse. This includes light.
  - Crest: a point of maximal displacement in the positive direction
  - Trough: a point of maximal displacement in the negative direction
- Longitudinal Waves: disturbance is parallel to the direction of propagation
  - Sound is a longitudinal wave
  - Compression or Condensation: a region where the medium is under compression
  - Rarefaction or Dilation: a region where the medium is under tension
- Surface Waves or Complex Waves: a combination transverse-longitudinal wave, forms near the surface of some media
- Torsional Waves: disturbance causes the medium to twist
Classifying Waves by Duration
- adj. episodic; noun pulse: disturbance is momentary and sudden
- adj. periodic, harmonic; noun wave train: disturbance repeats at regular intervals
Classifying Waves by Appearance
- Traveling Waves: appear to move
- Standing Waves: do not appear to move
Characteristics
- Amplitude (A) is 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) is 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) is 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) is 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) is the distance between any point on a periodic wave and the next point corresponding to the same portion of the wave measured along the path of propagation.
  - Wavelength is measured between adjacent points in phase.
  - The SI unit of wavelength is the meter [m].
- Speed (v) is …
  - 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 and
  - 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
- f (x, t) = A sin (2π (ft − x/λ) + ϕ)
  - A, amplitude
  - f, frequency
  - λ, wavelength
  - ϕ, phase
- f (x, t) = A sin (ωt − kx + ϕ)
  - A, amplitude
  - ω, angular frequency, ω = 2πf
  - k, wave number k = 2π/λ
  - ϕ, phase

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