## 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.