Selected slides from lectures ofFebruary 5 and February 7

two tones of nearly the same frequency - beats.

Beat frequency = f1 - f2.

Superposition of sounds Beats

2/5/2002

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- + - + - + - + - + - + - + - + - + - +

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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

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.

10 ms

200 Hz - period = 5 msec

300 Hz - period = 3.33 msec

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

t (msec)

Superposition of 200 Hz + 300 Hzrepetition 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

Time = period = 2L/s

Vibration of Strings:

Travel time along string and back = period of oscillation

Fundamental frequency f1

longer string -> lower f (inverse proportion)

f1 =s

2L=

12L

Tρ

higher tension(T) - higher f (square-root proportion)

more massive string () - lower f (square-root proportion)

s=

Tρ

=T

mlRemember:

“Voicing formula”:

2/7/2002

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)

Typical String Tension:Violin strings (G3- D4 - A4 - E5 )

tension 35-62 N for D-string 72-81 N for E-stringdownward force on bridge about 90 NPiano (grand) up to 1000 N/string

HIGHER MODES OF STRING

An oscillation is called a “MODE” if each point makes simple harmonic motion

demo: modes of stringexample: find frequencies of modes

oscillations called “harmonics” if frequencies are exact multiple of fundamental

mode freq

1st fundamental 1st partial f12nd 1st overtone 2nd partial f2=2f13rd 2nd overtone 3rd partial f3=3 f1

Actual string motion: SUPERPOSITION of MODES

Demo- click here: Modes

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 stringWhat 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