Course:PHYS341/2018/Calendar/Lecture 13

Phys341 Lecture 13: Summary and web references

2018.01.31

Textbook 11.1-11.2

Slide List

1. String instruments
   • Earth bow: http://www.baka.co.uk/baka/instr_angbindi.html
2. Normal modes
   • A “normal mode” is a standing wave in which all parts of the object are moving with one frequency.
   • A string held tightly between two fixed end-points can be set into a normal mode of vibration if it excited at one of particular set of frequencies.
   • These frequencies are the ones corresponding to modes that have nodes at the end-points.
3. Normal modes of a string
   • Here are the first three normal modes of a string held tightly at both ends.
   • Mode 2 vibrates at twice the frequency of mode 1; i.e. an octave higher.
   • Mode 3 vibrates at three times the frequency of mode 1, i.e. an octave and a fifth higher.
   • Higher modes in simple integer multiples of the lowest (the fundamental) are called harmonics or harmonic partials.
4. Building real waveforms from harmonics
   • https://phet.colorado.edu/en/simulation/fourier
   • When thinking about waves on a string, consider just the right hand half of the middle and lower plot, i.e. only the positive values of ).
5. String plucked in centre
   • Here is an ideal string (no damping) plucked in the centre.
   • The modes contributing to the triangular shape are odd-numbered, rapidly falling off in size.
   • Hence one would expect to hear the fundamental, but not the octave above.
6. Add some damping
   • Here is a damped string plucked in the centre.
   • Note the triangular shape is not preserved for long, and the higher modes damp out faster than the fundamental.
   • Hence one would expect to hear the fundamental more prominently as time progresses.
7. Real modes https://www.youtube.com/watch?v=Qr_rxqwc1jE
   • When a string is plucked (qin, guitar), bowed (violin, erhu), or struck (piano, santur) the vibrating shape of the string is generally not a sinusoid like one of the normal modes.
   • If the string is plucked or bowed, the shape starts as triangular.
   • If the string is struck, the initial disturbance spreads out like a pebble dropped in a pond
   • However, whatever the shape, it can be considered to be a summation of many normal modes in different quantities.
8. Plucking string in different places
   • A string length L plucked in the centre, i.e. L/2 from one end is missing even numbered harmonics, i.e. harmonics 2, 4, 6, ... The missing harmonics have an node in the centre and so cannot be excited.
   • A string length L plucked L/3 from one end is missing harmonics 3, 6, 9, .... (for the same reason).
   • A string length L plucked L/n from one end is missing harmonics n, 2n, 3n, ....
   • A favoured place to pluck a string is at L/7, because the 7th harmonic is discordant with the fundamental (two octaves minus a full tone).
9. String plucked close to one end https://www.youtube.com/watch?v=_X72on6CSL0