From UBC Wiki
Revision as of 15:28, 5 February 2018 by Chris Waltham (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Phys341 Lecture 15: Summary and web references


Textbook 12.1-12.3

Slide List

  1. Some terminology
    • For nearly all musical instruments, one of the most important factors is the contained air.
    • This is true for a violin as for a flute.
    • Air has pressure, density, velocity.
    • For our purposes, pressure and density go hand-in-hand: if you compress air the density goes up (same mass, less volume).
    • Acoustic pressure is the rapidly oscillating difference between measured air pressure (say, inside an instrument at a certain time and position), and ambient air pressure; acoustic pressure can be positive or negative.
    • Acoustic pressure is measured with a microphone.
    • Acoustic velocity is the rapidly oscillating velocity of the air; acoustic velocity can be positive or negative. It is hard to measure directly.
    • Acoustic velocity is NOT the random velocity of individual air molecules, and it is NOT the speed of flow down a pipe (e.g. your breath exiting your mouth while talking, singing or whistling).
  2. Flow and standing wave can exist together https://www.youtube.com/watch?v=YnyM9SUipfE
  3. Normal modes in a pipe
    • A “normal mode” is a standing wave in which all parts of the object are moving with one frequency.
    • Strings tend to be held tightly between two fixed end-points (a loose end is musically useless).
    • With air in a pipe there are more possibilities for the end points than with a string. The pipe ends can be open or closed.
    • Closed pipe means the air velocity is fixed at zero (the air is up against a hard boundary).
    • Open pipe means the acoustic pressure is fixed at zero* (contact with free air).
    • Three (extreme) possibilities: closed-closed, closed-open, open-open.
    • The shapes of the modes are very similar to those seen on strings.

*Not quite true; see later

  1. 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.
    • Here, y is the transverse displacement of the string.
  2. Normal modes in a pipe
    • Here are the first three normal modes of a pipe closed at both ends.
    • The shapes look the same as those for standing waves on a string.
    • Here, v is longitudinal acoustic velocity of the air, pinned at zero by the ends.
    • The mode frequencies are still in ratio 1:2:3...
    • Note the tiny amplitude of the velocity, ~ 0.2 mm/s
    • (for a pressure amplitude of 1 Pa which corresponds to a SPL of 91 dB – loud!)
  3. Normal modes in a pipe I
    • Here are the first three normal modes of a pipe closed at both ends.
    • Imagine the dots to be air molecules, jiggled with random thermal motion, and also moving with the acoustic wave.
    • Let your eye distinguish the jiggling from the larger collective motion.
    • The size of the displacement is hugely exaggerated, or you would not see it.
  4. Normal modes in a pipe II
    • Here we look at the acoustic pressure in the pipe (closed at both ends).
    • The shapes are in some sense opposite to the velocity shapes:
    • nodes in velocity → antinodes in pressure
    • antinodes in velocity → nodes in pressure
    • As the air hits the end walls, the pressure varies maximally, while the longitudinal velocity is constrained the be zero (the air cannot go through the walls).
  5. A different view – pressure contours
  6. Open one end of the pipe
    • A pipe closed at both end is musically useless, so open one end – like a resonator on a marimba.
    • Here we look at velocity:
    • Zero at the wall (left)
    • Maximal at the open end (right)
  7. Open one end of the pipe
  8. Pipe open at one end
  9. But...
    • With an open end, sound is radiated to the environment: we have the makings of a musical instrument.
    • Simulation:
    • Pipe closed at bottom
    • 3rd mode
    • Radiated wave greatly exaggerated for visibility
    • Note: if we could place our ears inside a flute, we would be deafened.
  10. Pipe open at both ends
    • A pipe open at both ends has the same spectrum of modes as a pipe closed at both ends.
    • Differences:
    • Ends are pressure nodes (open to the environment)
    • Ends are velocity (and displacement) antinodes
    • Radiates from both ends
    • Looking ahead, instruments with air reeds (recorders, organ pipe, xiaos etc.) behave approximately like pipes with two open ends.
  11. Pipe summary
    • Pipes open at both ends have a spectrum of modes 1:2:3...
    • For a length of 1m: 172 Hz, 344 Hz, 516 Hz... (F3, F4, C5...)
    • For a length of 0.5m, an octave higher (twice the frequency).
    • Pipes open at one end have a spectrum of modes 1:3:5...
    • For a length of 1m: 88 Hz, 264 Hz, 440 Hz... (F2, C4, A4...)
    • For a length of 0.5m, an octave higher (twice the frequency).
  12. Standing waves in a tube: Pressure amplitude scans