1. Selected slides from lectures of February 5 and February 7

2. two tones of nearly the same frequency - beats. Beat frequency = f 1 - f 2. Superposition of sounds Beats 2/5/2002

3. + - + - + - + - + - + - + - + - + - + - 3. INTERFERENCE OF TWO SOUND SOURCES resulting sound is loud or soft depending on difference in distance to the source soft (opposite phase) loud (in phase) demo

4. Superposition - Another Special Case: Two pure tones with "simple" frequency ratio like 2:1 or 3:2 Simple frequency ratios = HARMONY e.g. 2:1 ratio = octave; 3:2 ratio = fifth. resulting tone is periodic find frequency of combined tone! Example: 300Hz +200Hz (frequency ratio 3:2) find largest common multiplier: 100 Hz - why? after 1/100 sec, 300 Hz made 3 full oscillations 200 Hz made 2 full oscillations thus waveform repeats exactly after 1/100 sec.

5. 10 ms 200 Hz - period = 5 msec 300 Hz - period = 3.33 msec

6. Black curve: sum (superposition) of 200Hz and 300 Hz

7. t (msec) Superposition of 200 Hz + 300 Hz repetition frequency 100 Hz = “largest common multiplyer” other examples: 150 Hz and 250 Hz (25/15 = 5/3) rep freq: 50 Hz 120 Hz and 160 Hz (ratio 16/12 = 4/3) 40 Hz

8. Time = period = 2L/s Vibration of Strings: Travel time along string and back = period of oscillation Fundamental frequency f 1 longer string -> lower f (inverse proportion) higher tension(T) - higher f (square-root proportion) more massive string (  ) - lower f (square-root proportion) Remember: “Voicing formula”: 2/7/2002

9. Example 4: string frequency 300 Hz for T= 40 N. Find frequency for T = 50 N. hint: use proportions! (Answ: 335 m/s) Example 2: piano string 80 cm long, mass 1.4g find frequency if tension is 120N. (Answ: 164 Hz) Example 3: guitar string 60 cm long. Where must one place a fret to raise the frequency by a major fourth (4:3 ratio) (Answ: 45 cm) EXAMPLES Example 1: the A string (440 Hz) of a violin is 32 cm long. Find the speed of wave propagation on this string. (Answ: 282 m/s)

10. Typical String Tension: Violin strings (G 3 - D 4 - A 4 - E 5 ) tension 35-62 N for D-string 72-81 N for E-string downward force on bridge about 90 N Piano (grand) up to 1000 N/string

11. HIGHER MODES OF STRING An oscillation is called a “MODE” if each point makes simple harmonic motion demo: modes of string example: find frequencies of modes oscillations called “harmonics” if frequencies are exact multiple of fundamental

12. Actual string motion: SUPERPOSITION of MODES Demo- click here: Modes

13. Playing Harmonics in Strings (flageolet tones) Example: 800 Hz string. Place your finger lightly at a point exactly 1/4 from the end of the string What frequencies will be present in the tone? Answer: those modes of the 800 Hz string which have a node where you place the finger- all other oscillations are killed by finger. What are they? Fourth mode, eighth mode, twelfth mode 3,200 Hz 6,400 Hz, 9,600Hz. (Explain on blackboard) Used in composition (e.g. Ravel) Homework # 4 Related comment: where you pluck or bow affects mix of partials