Periodic Waves

Summary

• 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; for example…
      • meter [m] for displacement
      • pascal [Pa] for pressure
      • volt per meter [V/m] or newton per coulomb [N/C] for electric field, etc.
  • Period (T)
    • The time between successive cycles of a repeating sequence of events
    • The ratio of time to the number of cycles

      T =	t
      	n

    • 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
    • The ratio of the number of cycles to time

      f =	n
      	t

    • 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]

  • Wavelength (λ, the Greek letter lambda)
    • 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
      • the state of the medium.
    • The rate of change of distance with time, by definition

      v =	∆s
      	∆t

    • The product of frequency and wavelength for periodic waves

      v = fλ

    • Frequency and wavelength are inversely proportional.

      f ∝	1
      	λ

      • 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].
• Phase (φ, the Greek letter phi)
  • 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 (symbol rad), which is itself a unitless ratio.

    [rad = m/m = Pa/Pa = (V/m)/(V/m) = etc.]

  • The degree is a non-SI unit that is acceptable for phase.

    [360° = 2π rad = 1 cycle]

• One-dimensional, periodic wave function
  • linear version

    f(x,t) = A sin(2π(x/λ − ft) + 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 =	angular wave number (spatial frequency), k = 2π/λ
    φ =	phase