The Science of Music: Waves and Resonance

1. 3.3 Waves and Stuff Science of Music 2007

2. Last Time • Dr. Koons talked about consonance and beats. • Let’s take a quick look & listen at what this means ….

3. Listen First

4. Recall what a single frequency tone sounds like • Play on Sound Generator A=440 Hz. • The Graph:

5. 440 Hz. and 450 Hz. Compared 440 Hz. 450 Hz.

6. 440 Hz. and 450 Hz. Together

7. 440 Hz. and 450 Hz. Togetherfor a longer time (0.1 second)

8. 440 Hz. and 450 Hz. Togetherfor an even longer time (0.5 second) Tbeat These are BEATS From this graph we see that Tbeat = 0.1 seconds. fbeat = 10 Hz..

9. Why?? In phase out of phase in phase again

10. The Concept • Two sounds start together (in phase). • After Tbeat seconds they get back together again. • One of the waves must have gone through N cycles while the other went through (N+1) complete cycles. • They are therefore together again!

11. The *(*@)$# math ! The beat frequency between two simultaneous tones is equal to The difference between the frequencies of the two tones!

12. Back to STRINGS

13. These are the frequencies at which the String RESONATES. The frequencies

14. RESONANCE STRINGS HAVE MORE THAN ONE RESONANT FREQUENCY

15. The RESONANT frequencies Difference???

16. Standing Wave • Produced by TWO waves traveling along a string in opposite directions. • Each wave reflects at the end of the string and then goes the other way. • Both waves travel with the same velocity over the same length of string. • Many pairs of waves may travel along the string at the same time. • More than one set of standing waves is possible on the string at the same time.

17. actual string shape

18. fundamental first overtone, second harmonic, second octave second overtone, third harmonic, fifth above second octave third overtone, fourth harmonic, second octave. etc.

19. Resonance of Strings Multiple Frequencies Determine the Timbre

21. What does sound look like? Repeats

22. Music • For short periods of time, a musical sound is PERIODIC. • It has a weird shape. • How do we produce a strange looking but periodic shape? • Answer: FOURIER

23. Fourier’s Theorem • Any periodic signal can be broken down into a sum of simple sine waves at different frequencies and sizes (amplitudes). • This theorem allows us to understand why different instruments sound different even when playing what we perceive as the same tone.

24. Square Wave

25. How do we create weird periodic shapes?? FOURIER THEOREM

26. For Example

27. The FOURIER spectra for each of these consists of a single frequency.

28. When you strike the string • All standing wave modes are excited at the same time. • “Non-Standing” waves die out quickly. • This is a Fourier thing. • If we could initially shape the string exactly to one of these modes, then this would be the only one that would be excited. • But we can’t do this .. can we???

29. We can! Sorta..

30. Two different plucks would require different sets of harmonics to create the shape. These will produce somewhat different instrument sounds.

31. Modes Place finger near the center of the string and the strike it. The odd overtones should be suppressed.

34. Plucked at Midpoint Touched at Midpoint

37. So – How does the guitar work? • Each string that is plucked will vibrate in one of its fundamental modes. The shape of the initial string stretch determines which modes will be excited. • Each mode is established by waves bouncing back and forth along the instrument.

38. More .. • The sound from the instrument depends on how and where along the string it is plucked. • The strings by themselves emit little sound. The connection to the bridge causes the sound box to move and resonate, a topic we will discuss later.

39. So now you know!