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Exploring Sound Properties Through Mathematical Analysis and Wave Behavior

Dive into the world of sound with a mathematical excursion, learning how to interpret graphs, understand sound pressure variations, and differentiate between periodic and noise sounds. Explore the concepts of waves, harmonics, interference, and the Doppler effect, with practical applications such as Doppler radar and sonograms. Discover how sound waves combine and interfere, and learn about diffraction's role in wave behavior. Ultrasound and sonograms are also discussed, offering a comprehensive overview of sound phenomena.

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Exploring Sound Properties Through Mathematical Analysis and Wave Behavior

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  1. double the distance reduce intensity by a factor of 4 BUT Half of intensity does not “feel” like half as loud !

  2. Before going into other properties like interference, diffraction, … we will make a mathematical excursion (basically covered in Chapter 1 of Berg&Stork)

  3. How to read graphs • figure what’s in the horizontal axis (w/ units) • figure what’s in the vertical axis (w/ units) • find the value of “savings” at any particular time

  4. visual analytical

  5. WATCH !!! same information

  6. Let us watch now the graph of sound pressure variation as a function of time for some real sounds • Which sounds are periodic ? • What distinguishes noise from “musical” sounds ? • What makes a sound louder ? • What makes a sound lower or higher in pitch ?

  7. Wavetools http://www.physics.mcgill.ca/~grant/225B/Wavetools/Computerstuff.html

  8. A very “pure” (but annoying) sound:

  9. “Musical” sounds (“tones”) are periodic amplitude (A) period (T) frequency (f) = 1/T, T=1/f

  10. The mathematics of periodic waves wavelength: distance between two crests period: time between two crests frequency: how many crests per second

  11. Period (T) = time for one cycle (measured in s, …) • Frequency (f) = number of cycles per second (measured in 1/s = Hz) • Example

  12. With a few qualifications … amplitude = loudness frequency = pitch shape = timbre

  13. It is a good time now to read Berg & Stork, Chapter 1

  14. “Pure” tones are sine waves Harmonic oscillator

  15. [Using a wave generator, find the lowest and highest audible frequencies and calculate their period and wavelength]

  16. phase amplitude frequency

  17. How sounds combine: adding two waves 300 Hz 450 Hz 300 Hz + 450 Hz

  18. Beats

  19. Tartini tones

  20. Back to sound (wave) properties … • reflection • refraction • 1/r2 law • diffraction • interference • Doppler effect

  21. Waves interfere: • constructively (amplitudes have the same sign and enhance the wave) • destructively (amplitudes have opposite sign and partially or completely cancel)

  22. Interference movie http://www.colorado.edu/physics/2000/schroedinger/big_interference.html Interference applet http://www.falstad.com/ripple/ex-2source.html

  23. Doppler effect

  24. Doppler effect applethttp://www.lon-capa.org/~mmp/applist/doppler/d.htm

  25. [go to the blackboard and do some real physics] • Applications: • Doppler radar • Doppler sonogram • Expansion of the Universe

  26. mostly constructive interference mostly destructive interference l l

  27. Diffraction • Larger wavelength (~ obstacles) : more diffraction • Shorter wavelength (<< obstacles) : less diffraction

  28. Ultrasound and sonograms Typical frequencies: 300 MHz

  29. It is a good time now to read Berg & Stork, Chapter 2.4, 2.5, 2.6, 2.7, 2.8, 2.9

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