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

3. 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?

4. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20 The 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.

5. Physics 1251 Unit 1 Session 7 Good Vibrations Example: Two coupled masses on springs with same k and m. Separately: f 1 = (1/2π)√k /m, f 2 = (1/2π)√k /m. Coupled Normal Modes: f i = (1/2π)√k /2m :[m 1 & m 2 move as one], f ii = (1/2π)√2k /m :[ m 1 & m 2 move in opposition].

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

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

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

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

10. 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? How would you describe the “timbre” (quality of the sound) of a Helmholtz resonator? What gives a “jug” its timbre? What gives a “jug” its timbre?

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

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

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

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

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

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

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

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

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

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

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

22. Physics 1251 Unit 1 Session 7 Good Vibrations 80/20 Complex sounds can be produced from simple sounds. Example: Play C 4, C 4 +G 4, C 4 +G 4 +E 4 Observe oscilloscope.

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

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

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

26. 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.

27. Physics 1251 Unit 1 Session 7 Good Vibrations Example: Synthesis of a Square wave fofofofo 3f o 5f o 7f o Square Wave

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

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