Physics 1251 The Science and Technology of Musical Sound

1. Physics 1251The Science and Technology of Musical Sound Unit 1 Session 7 Good Vibrations

2. Physics 1251 Unit 1 Session 7 Good Vibrations Foolscap Quiz: Two tuning forks that each, separately, oscillate at 128 Hz are joined together by one tine. How many modes of oscillation are there with them joined? Are the frequencies the same as before?

3. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20The number of normal or natural modes of vibration of an oscillator is equal to the number of simple harmonic oscillators that comprise the complex oscillator.

4. Physics 1251 Unit 1 Session 7 Good Vibrations Example: Two coupled masses on springs with same k and m. Separately: f1 = (1/2π)√k /m , f2 = (1/2π)√k /m . Coupled Normal Modes: fi = (1/2π)√k /2m :[m1 &m2 move as one], fii = (1/2π)√2k /m :[m1 &m2 move in opposition].

5. Physics 1251 Unit 1 Session 7 Good Vibrations Two Masses on Two Coupled Springs Spring ———→ Mass ————→ Spring ————→ Mass ————→ Mode 1 Mode 2

6. Physics 1251 Unit 1 Session 7 Good Vibrations The “Fingered” Bell • Bell is constructed of many “tuning forks.” • New normal modes appear when coupled

7. Physics 1251 Unit 1 Session 7 Good Vibrations 1′ Lecture: • Any waveform can be broken up into simple sinusoidal components. • Any waveform can be built up from single frequency components in the correct combination.

8. Physics 1251 Unit 1 Session 7 Good Vibrations Doctor M’s Stupendous, Colossal, Remarkable BBC Beer Bottle Choir

9. Physics 1251 Unit 1 Session 7 Good Vibrations Why don’t we see “Jug Bands” and “Beer Bottle Choirs” except as curiosities? • How would you describe the “timbre” (quality of the sound) of a Helmholtz resonator? • What gives a “jug” its timbre?

10. Physics 1251 Unit 1 Session 7 Good Vibrations Normal Modes Of Hand Bell • Bell is shaped to produce many normal modes of oscillation. • Timbre is determined by combination of normal modes.

11. Physics 1251 Unit 1 Session 7 Good Vibrations Peregrination in Existential Physics: • Move you hand in a circle. • Notice its height. • Now move your other hand up and down in a straight line at the same height. • This is sinusoidal (like a sine) motion.

12. Physics 1251 Unit 1 Session 7 Good Vibrations Peregrination in Existential Physics:

13. Physics 1251 Unit 1 Session 7 Good Vibrations Peregrination in Existential Physics: Sine of the angle

14. Physics 1251 Unit 1 Session 7 Good Vibrations Peregrination in Existential Physics: • Move you hand in a circle while waving it up and down. • Notice its height. • This is a complex motion that can be described as the sum of simple sinusoidal motions.

15. Transducer Electronics Physics 1251 Unit 1 Session 7 Good Vibrations 80/20A microphone converts pressure fluctuations into electrical signals.

16. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20An oscilloscope displays electrical signals versus time. Repeating oscillations can be viewed on the screen.

17. Argument A Physics 1251 Unit 1 Session 7 Good Vibrations 80/20The term “sine wave” describes anything that varies with time or position like the trigonometric function called “sine” (abbreviated sin). Sine (A)

18. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20A simple harmonic oscillator moves like a sine wave. Its motion is called “sinusoidal [=like a sine].”

19. Compression Rarefaction Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators In one place, the pressure produced by the sound from a simple harmonic oscillator varies with time as a sine wave.

20. Compression Rarefaction Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators At an instant of time, the pressure produced by the sound from a simple harmonic oscillator varies with position as a sine wave.

21. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20Complex sounds can be produced from simple sounds. Example: Play C4, C4+G4, C4+G4+E4 Observe oscilloscope.

22. Physics 1251 Unit 1 Session 7 Good Vibrations Fourier Analysis and Synthesis • Baron Jean Baptiste Joseph Fourier (1768-1837) • Mathematician • Fourier Analysis • Fourier Synthesis[Pronounced “Four-yeah”]

23. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20Fourier Analysis is the decomposition of a wave into the sine wave components from which it can be built up.

24. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20The lowest frequency component of a wave is equal to 1/Prepeat the reciprocal of the repeat period Prepeat Repeat Period Prepeat flowest = 1/Prepeat Time →

25. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20 A Fourier Analysis is a representation of all the components that comprise a waveform, amplitude versus frequency and phase versus frequency.

26. Physics 1251 Unit 1 Session 7 Good Vibrations Example: Synthesis of a Square wave fo 3fo 5fo 7fo Square Wave

27. Fundamental – 1st Harmonic 1st Overtone – 3rd Harmonic 2nd Overtone – 5th Harmonic Physics 1251 Unit 1 Session 7 Good Vibrations Example: Synthesis of a Square wave- The Fourier Spectrum Amplitude fo 3fo 5fo 7fo Frequency

28. Physics 1251 Unit 1 Session 7 Good Vibrations Summary: • Waveforms can be broken down into simple sine waves, called Fourier Analysis. • Waveforms can be built up from simple sine waves, called Fourier Synthesis. • The lowest frequency needed is 1/Prepeat. • A Fourier Spectrum is the recipe for building the waveform.