Standing Waves

Summary

• Traveling waves…
  • appear to propagate (to be traveling)
• Standing waves or stationary waves…
  • appear not to propagate (to be standing in place)
  • are produced by the interference of two waves traveling in opposite directions with the same frequency and amplitude
• Parts of a standing wave
  • nodes
    • locations where the amplitude is zero (or at least a minimum)
    • always form at fixed ends (closed ends)
  • antinodes
    • locations where the amplitude is a maximum
    • always form at free ends (open ends)
• Resonance
  • is a noticeable increase in the amplitude of a wave
  • occurs when the driving frequency matches one of the natural frequencies of a system
• Standing waves and resonance
  • Standing waves form during resonance (but resonance does not always lead to the formation of standing waves)
  • A wave moving in a medium of finite length, can interfere with its own reflection to produce a standing wave
    • if it has the same frequency as one of the natural frequencies of the medium
    • if it has a wavelength that allows nodes to form at the fixed ends and antinodes to form at the free ends
• Harmonics
  • are the set of all possible standing waves in a system
  • are countably infinite in number (form a countable infinite set in the manner of whole numbers)
• Groups of harmonics
  • The harmonic with the lowest frequency and longest wavelength is called the fundamental frequency (sometimes shortened to the fundamental).
  • Harmonics other than the fundamental are called overtones because they have frequencies higher than the fundamental (they are above or over the fundamental)
• When standing waves form in a linear medium that has…
  • two fixed ends or two free ends…
    • a whole number of half wavelengths fit inside the medium
    • the overtones are whole number multiples of the fundamental frequency
  • one fixed end and one free end…
    • an odd number of quarter wavelengths fit inside the medium
    • the overtones are odd multiples of the fundamental frequency
    • the even numbered harmonics do not occur
• Higher-dimensional cases
  • You probably don't need to worry about it.