Sound
Physics 123
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Lecture IV
1
Sound
• Wave nature of sound
• Intensity of sound
• Standing sound waves
– String instruments
– Pipes
• Interference and beats.
• Doppler effect
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2
Sound = longitudinal wave in air
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Speed of sound
• Wave characteristics:
– Wave length – l (m)
– Frequency – f(Hz) - pitch
– Wave velocity - v=l f, m/s
• Wave speed – property of
material  one – to – one
correspondence of frequency
and wave length in a given
medium:
f 
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Lecture IV
v
l
4
Intensity of sound
• Intensity of sound: I=10-12  102 W/m2 – 14 orders of magnitude
• Measure of loudness b in Decibel: b(in dB)=10 log (I/I0)
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I0
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Sensitivity of human ear
Audible range (really good speakers) : 20Hz – 20 kHz
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Physics of a guitar
• Guitar = strings + sounding
box (resonator)
• Strings force resonance in
the sounding box
• Fundamental frequency
• Strings
• Tuning
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Physics of a guitar
v
string tension

mass per unit length
FT

m/l
• Standing wave
• Fundamental frequency:
– L=l1 /2
l1=2L
– f1=v/l1 f1=v /(2L)
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Lecture IV
String theory:
Thicker string  higher m/l 
lower v lower frequency f
Tuning:
Increase tension (FT) 
increase v  increase
frequency f.
Fingered string:
Decrease L  decrease l 
increase f.
8
Wave velocity vs particle velocity

•
•
•
•
•
•
w2pf – cyclic frequency, k=2p/l –wave vector
D=D0sin(kx-wt)
Riding the wave kx-wtconst
kx-wt=c
x=c/k+(w/k)t = x0+vt
Thus, wave velocity v=w/k=2pf/ (2p/l)fl  l/T
D=D0sin(kx-wt) medium displacement at point x at time t
Particle velocity:
– vp=dD/dt=-wD0cos(kx-wt)=-vmaxcos(kx-wt)
– vmax=wD0
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Lecture IV
9
Physics of an organ
• Open and closed pipes - resonators
• Boundary conditions (imagine yourself in a
crowded room) :
• Open end (next to an open door)
• Displacement (freedom to move):
Dx = max
• Pressure = Atmospheric P:
DP=0
• Closed end (pushed against a wall)
• Displacement
Dx = 0
• Pressure variation – max
DP=max
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Organ pipe
f n  nf1
Dx  max; DP  0
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only odd harmonics :
f 2 n 1  (2n  1) f1
Organ pipe
Dx  0; DP  max
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Interference
Two waves of the same frequency
C: Constructive interference
A+A=2A  I =4I0
Dx=0+nl; dsinqnl
D: Destructive interference
A-A=0  I =0
Dx=l/2+nl
dsinq=(n+1/2)l
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Beats
Two waves of the similar frequencies: f1 and f2.
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Doppler effect
•
•
•
•
sound source moving with velocity vs
Distance between crests l’=l-vsT=l-vsl/v=l(1-vs/v)
Frequency f’=f/(1-vs/v)
Moving towards you vs – positive  divide by a
number <1  f’>f – higher pitch
• Moving away from you vs – negative  divide by
a number >1  f’<f – lower pitch
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Demo data
• Open-closed end pipe
• f=512 Hz
• v=343m/s (maybe less,
cold)
• l=v/f=.67m
 l=4l1
• l1=l/4=0.17m
• l3=3l/4=0.51m
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Intensity of waves
• Energy of oscillation E is
proportional to amplitude squared A2
EA
2
• Intensity – I, W/m2
1
A
r
energy / time power
I

area
area
• Intensity I is proportional to amplitude
squared A2, inversely proportional to r2:
IA
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I 2
r
17