The Physics
Hypertextbook
Opus in profectus

# Standing Waves

## Summary

• Traveling waves
• appear to propagate (to be traveling)
• Standing 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.