Selected slides from lectures of
February 5 and February 7
Superposition of sounds
2/5/2002
Beats
two tones of nearly the same frequency - beats.
Beat frequency = f1 - f2.
3. INTERFERENCE OF TWO SOUND SOURCES
resulting sound is loud or soft depending on
difference in distance to the source
soft
(opposite phase)
demo
loud
(in phase)
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.
300 Hz - period = 3.33 msec
200 Hz - period = 5 msec
10 ms
Black curve: sum (superposition) of 200Hz and 300 Hz
Superposition of 200 Hz + 300 Hz
repetition frequency 100 Hz = “largest common multiplyer”
t (msec)
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
Vibration of Strings:
2/7/2002
Travel time along string and back = period of oscillation
Time = period = 2L/s
“Voicing formula”:
Fundamental frequency f1
Remember: s 
s
1
f1 

2L 2L
T


T
m
T

longer string -> lower f (inverse proportion)
higher tension(T) - higher f (square-root proportion)
more massive string () - lower f (square-root proportion)
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)
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)
Example 4: string frequency 300 Hz for T= 40 N.
Find frequency for T = 50 N.
hint: use proportions!
(Answ: 335 m/s)
Typical String Tension:
Violin strings (G3- D4 - A4 - E5 )
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
HIGHER MODES OF STRING
An oscillation is called a “MODE” if
each point makes simple harmonic motion
mode
1st
2nd
3rd
freq
fundamental
1st overtone
2nd overtone
1st partial f
1
2nd partial f =2f
2
1
3rd partial f =3 f
3
1
demo: modes of string
example: find frequencies of modes
oscillations called “harmonics” if frequencies
are exact multiple of fundamental
Actual string motion:
SUPERPOSITION of MODES
Demo- click here: Modes
Playing Harmonics in Strings (flageolet tones)
Homework # 4
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 fingerall 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)
Related comment: where you pluck or bow affects mix of partials