Chapter 4 - Resonance

Frequency and Associated Harmonics

Questions to Consider

1) What happens if the Tension of the string is raised?

2) What happens if the length of the string is made longer?

3) Will the 'mu' of a string change between a steel string and a nylon string?

4) What are the first three harmonics of a 60 centimeter nylon string?

5) Can you deduce the Tension from the information given in question 4?

6) How would you make a musical instrument from pipes and water?

Standing Waves on a String
(please disregard page 57)
    On page 57, in the book, the author confuses the issue of Nodes and Anti-nodes. The author even states this in the second paragraph. Nodes occur at minimum displacement of the medium. The author mixes pressure nodes with displacement nodes. It's all so confusing. Transverse waves (strings) are different than Longitudinal waves (sound). Transverse waves are easily represented by sine and cosine waves. Longitudinal waves are NOT easily represented by sine and cosine waves. Enough said.

Guitar Example
    Consider a 80-cm long guitar string which has a fundamental frequency (1st harmonic) of 400 Hz. For the first harmonic, the wavelength of the wave pattern would be two times the length of the string; thus, the wavelength is 160 cm or 1.60 m. The speed of the standing wave can now be determined from the wavelength and the frequency. The speed of the standing wave is
speed = frequency * wavelength
speed = 400 Hz * 1.6 m
speed = 640 m/s
    This speed of 640 m/s corresponds to the speed of any wave within the guitar string. Since the speed of any wave is dependent upon the properties of the medium (and not upon the properties of the wave), every wave will have the same speed in this string regardless of its frequency and its wavelength. So the standing wave pattern associated with the second harmonic, third harmonic, fourth harmonic, etc. will also have this speed of 640 m/s. A change in frequency or wavelength will NOT cause a change in speed.

Slinky Demo is worth a thousand words!
Another good page SINES and WONDERS

Sympathetic Vibrations

System A vibrations caused by System B vibrating at the same resonant frequency of System A. Example: If you bring a vibrating tuning fork near a piano string board then one string that resonates at the frequency of the tuning fork will begin to vibrate. It's a cool experiment.
    After all this time, I have asked you all about the frequency of the C note on the piano. Have you discovered this, yet? Maybe, it would be worth your time to understand this setup.Look here.