Course:PHYS341/2018/Calendar/Lecture 13
Phys341 Lecture 13: Summary and web references
2018.01.31
Textbook 11.1-11.2
Slide List
- String instruments
- 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.
- 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.
- 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 ).
- 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.
- 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.
- 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.
- 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).
- String plucked close to one end https://www.youtube.com/watch?v=_X72on6CSL0