SOUND WAVES

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waves_04
SOUND WAVES
flute
clarinet
click for sounds
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waves_04 MINDMAP SUMMARY - SOUND WAVES
Sound waves, ultrasound, compressional
(longitudinal) waves, pressure, particle
displacement, medical imaging, superposition
principle, stadning waves in air columns,
constructive interference, destructive
interference, boundary conditions, pipe open
and closed ends, nodes, antinodes, speed of sound
in air, period, frequency, wavelength,
propagation constant (wave number), angular
frequency, normal modes of vibrations, natural
frequencies of vibration, fundamental, harmonics,
overtones, harmonic series, frequency spectrum,
radian, phase, sinusoidal functions, wind musical
instruments, beats, beat frequency, Doppler
Effect, Doppler radar, shock waves
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SOUND WAVES IN AIR

• Longitudinal wave through any medium which can be
  compressed gas, liquid, solid
• Frequency range 20 Hz - 20 kHz (human hearing
  range)
• Ultrasound f gt 20 kHz
• Infrasound f lt 20 Hz
• Atoms/molecules are displaced in the direction of
  propagation about there equilibrium positions

For medical imaging f 1-10 MHz Why so large?
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SOUND WAVES IN AIR
Displacement of molecules
Pressure variation
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ULTRASOUND
Ultrasonic sound waves have frequencies greater
than 20 kHz and, as the speed of sound is
constant for given temperature and medium, they
have shorter wavelength. Shorter wavelengths
allow them to image smaller objects and
ultrasonic waves are, therefore, used as a
diagnostic tool and in certain treatments. Intern
al organs can be examined via the images produced
by the reflection and absorption of ultrasonic
waves. Use of ultrasonic waves is safer than
x-rays but images show less details. Certain
organs such as the liver and the spleen are
invisible to x-rays but visible to ultrasonic
waves. Physicians commonly use ultrasonic waves
to observe fetuses. This technique presents far
less risk than do x-rays, which deposit more
energy in cells and can produce birth defects.
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What is the physics of this image?
CP 495
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ULTRASOUND
Flow of blood through the placenta
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• Speed of sound wave in a fluid

The speed of a sound wave in a fluid depends on
the fluids compressibility and inertia.
B bulk modulus of the fluid
r equilibrium density of the fluid

• Speed of sound wave in a solid rod

Y Youngs modulus of the rod
r density of the fluid

• Speed of sound wave in air

v 343 m.s-1 at T 20oC
Above formulae not examinable
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Energy and Intensity of Sound waves

• Intensity level in decibel

• The loudest tolerable sounds have intensities
  about 1.0x1012 times
• greater than the faintest detectable sounds.

• The sensation of loudness is approximately
  logarithmic in the human
• ear. Because of that the relative intensity of
  a sound is called the
• intensity level or decibel level, defined by

I0 1.0x10-12 W.m-2 the reference
intensity the sound intensity at the threshold of
hearing
Threshold of hearing
Threshold of pain
Not examinable
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Problem 1 A noisy grinding machine in a
factory produces a sound intensity of 1.00x10-5
W.m-2. (a) Calculate the intensity level of the
single grinder.
(b) If a second machine is added, then
(c) Find the intensity corresponding to an
intensity level of 77.0 dB.
Not examinable
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Problem 2 A point source of sound waves emits a
disturbance with a power of 50 W into a
surrounding homogeneous medium. Determine the
intensity of the radiation at a distance of 10 m
from the source. How much energy arrives on a
little detector with an area of 1.0 cm2 held
perpendicular to the flow each second? Assume no
losses.
Solution P 50 W r 10 m I ? W.m-2 A
1.0 cm2 1.010-4 m2 W ? J I P / (4?
r2) 4.010-2 W.m-2 W I A 4.010-6 J
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Problem 3
A small source emits sound waves with a power
output of 80.0 W. (a) Find the intensity 3.00 m
from the source.
(b) At what distance would the intensity be
one-fourth as much as it is at r 3.00 m?
(c) Find the distance at which the sound level
is 40.0 dB?
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BEATS

• Loud-soft-loud modulations of intensity are
  produced when waves of slightly different
  frequencies are superimposed.
• The beat frequency is equal to the difference
  frequency fbeat f1 - f2

1 beat
Used to tune musical instruments to same pitch
CP 52
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• BEATS two interfering sound waves can make
  beat

Two waves with different frequency create a beat
because of interference between them. The beat
frequency is the difference of the two
frequencies.
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BEATS
Superimpose oscillations of equal amplitude, but
different frequencies
Modulation of amplitude
Oscillation at the average frequency
frequency of pulses is f1-f2
CP 527
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BEATS interference in time Consider two sound
sources producing audible sinusoidal waves at
slightly different frequencies f1 and f2. What
will a person hear? How can a piano tuner use
beats in tuning a piano? If the two waves at
first are in phase they will interfere
constructively and a large amplitude resultant
wave occurs which will give a loud sound. As time
passes, the two waves become progressively out of
phase until they interfere destructively and it
will be very quite. The waves then gradually
become in phase again and the pattern repeats
itself. The resultant waveform shows rapid
fluctuations but with an envelope that various
slowly. The frequency of the rapid fluctuations
is the average frequencies The frequency of
the slowly varying envelope
Since the envelope has two extreme values in a
cycle, we hear a loud sound twice in one cycle
since the ear is sensitive to the square of the
wave amplitude. The beat frequency is
CP 527
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click for sound
f1 100 Hz f2 104 Hz frapid 102 Hz
Trapid 9.8 ms fbeat 4 Hz Tbeat 0.25 s
(loud pulsation every 0.25 s)
CP 527
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click for sound
f1 100 Hz f2 110 Hz frapid 105 Hz
Trapid 9.5 ms fbeat 10 Hz Tbeat 0.1 s
(loud pulsation every 0.1 s)
CP 527
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24 25
click for sound
f1 100 Hz f2 120 Hz frapid 110 Hz
Trapid 9.1 ms fbeat 20 Hz Tbeat 0.05 s
(loud pulsation every 0.05 s)
CP 527
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DOPPLER EFFECT - motion related frequency
changes Doppler 1842, Buys Ballot 1845 -
trumpeters on railway carriage
Source (s) Observer (o)
formula different to textbook
Applications police microwave speed units, speed
of a tennis ball, speed of blood flowing through
an artery, heart beat of a developing fetous,
burglar alarms, sonar ships submarines to
detect submerged objects, detecting distance
planets, observing the motion of oscillating
stars, weather (Doppler radar), plaet detection
note formula is very different to textbook
CP 495
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DOPPLER RADAR
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DOPPLER EFFECT

• Consider source of sound at frequency fs, moving
  speed vs, observer at rest (vo 0)
• Speed of sound v
• What is frequency fo heard by observer?

v ? f

• On right - source approaching
• source catching up on waves
• wavelength reduced
• frequency increased

• On left - source receding
• source moving away from waves
• wavelength increased
• frequency reduced

CP 495
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Source frequency fs 1000 Hz
fo gt 1000 Hz
fo lt 1000 Hz
click for sounds
CP 495
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source vs observer vo observed frequency fo
stationary stationary fs
stationary receding lt fs
stationary approaching gt fs
receding stationary lt fs
approaching stationary gt fs
receding receding lt fs
approaching approaching gt fs
approaching receding ?
receding approaching ?
CP 595
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SHOCK Waves supersonic waves
bullet travelling at Mach 2.45
CP 506
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Shock Waves supersonic waves
CP 506
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DOPPLER EFFECT
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Stationary Sound Source
Source moving with vsource lt vsound ( Mach 0.7 )
Source moving with vsource vsound ( Mach 1 -
breaking the sound barrier )
Source moving with vsource gt vsound (Mach 1.4 -
supersonic)
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Problem 4 A train whistle is blown by the
driver who hears the sound at 650 Hz. If the
train is heading towards a station at 20.0 m.s-1,
what will the whistle sound like to a waiting
commuter? Take the speed of sound to be 340 m.s-1.
Solution fs 650 Hz vs 20 m.s-1 vo 0
m.s-1 v 340 m.s-1 fo ? Hz (must be higher
since train approaching observer).
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Problem 5 The speed of blood in the aorta is
normally about 0.3000 m.s-1. What beat
frequency would you expect if 4.000 MHz
ultrasound waves were directed along the blood
flow and reflected from the end of red blood
cells? Assume that the sound waves travel
through the blood with a velocity of 1540 m.s-1.
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Solution 5
Doppler Effect
Beats
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Blood is moving away from source ? observer
moving away from source ? fo lt fs
Wave reflected off red blood cells ? source
moving away from observer ? fo lt fs
Beat frequency 4.00 3.998442 ?106 Hz
1558 Hz In this type of calculation you must
keep extra significant figures.
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Problem 6
An ambulance travels down a highway at a speed of
33.5 m.s-1, its siren emitting sound at a
frequency of 4.00x102 Hz. What frequency is heard
by a passenger in a car traveling at 24.6 m.s-1
in the opposite direction as the car and
ambulance (a) approach each other and (b) pass
and move away from each others? Speed of sound
in air is 345 m.s-1.
Solution 6
(a)
(b)
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Problem 7 An ultrasonic wave at 8.000?104 Hz is
emitted into a vein where the speed of sound is
about 1.5 km.s-1. The wave reflects off the red
blood cells moving towards the stationary
receiver. If the frequency of the returning
signal is 8.002?104 Hz, what is the speed of the
blood flow? What would be the beat frequency
detected and the beat period? Draw a diagram
showing the beat pattern and indicate the beat
period.
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Solution fs 8.000104 Hz fo 8.002104 Hz
v 1.5103 m.s-1 vb ? m.s-1
Need to consider two Doppler shifts in frequency
blood cells act as observer and than as source.
Red blood cells (observer) moving toward source
Red blood cells (source) moving toward observer
fbeat f2-f1 (8.002-8.000)104 Hz 20
Hz Tbeat 1/fbeat 0.05 s
tbeat
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STANDING WAVES IN AIR COLUMNS (PIPES)

• If we try to produce a traveling harmonic wave in
  a pipe, repeated reflections from an end produces
  a wave traveling in the opposite direction - with
  subsequent reflections we have waves travelling
  in both directions
• The result is the superposition (sum) of two
  waves traveling in opposite directions
• The superposition of two waves of the same
  amplitude travelling in opposite directions is
  called a standing wave

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Flute clarinet same length, why can a much
lower note be played on a clarinet?
STANDING WAVES IN AIR COLUMNS (PIPES)
Closed at both ends Closed at one end open at
the other Open at both ends
Closed end displacement zero (node), pressure
max (antinode) Open end displacement max
(antinode), pressure zero (node)
CP 516
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Organ pipes are open at both ends
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STANDING WAVES IN AIR COLUMNS (PIPES)
Sound wave in a pipe with one closed and one open
end (stopped pipe)
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STANDING WAVES IN AIR COLUMNS (PIPES)
CP 516
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Search google or YouTube for Rubens or Rubins
tube
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STANDING WAVES IN AIR COLUMNS (PIPES)
CP 516
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STANDING WAVES IN AIR COLUMNS (PIPES)
Normal modes in a pipe with an open and a closed
end (stopped pipe)
Fundamental
3rd harmonic or 2nd overtone
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CP 523
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Musical instruments wind An air stream
produced by mouth by blowing the instruments
interacts with the air in the pipe to maintain a
steady oscillation. All brass instruments are
closed at one end by the mouth of the player.
Flute and piccolo open at atmosphere and
mouth piece (embouchure) covering holes L ?
? ? ? ? f ? Trumpet open at atmosphere
and closed at mouth covering holes adds loops
of tubing into air stream L ? ? ? ? ?
f ? Woodwinds vibrating reed used to produce
oscillation of the air molecules in the pipe.
CP 516
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Woodwind instruments are not necessarily made of
wood eg saxophone, but they do require wind to
make a sound. They basically consist of a tube
with a series of holes. Air is blow into the top
of the tube, either across a hole or past a
flexible reed. This makes the air inside the tube
vibrate and give out a note. The pitch of the
note depends upon the length of the tube. A
shorter tube produces a higher note, and so holes
are covered. Blowing harder makes a louder sound.
To produce deep notes woodwind instruments have
to be quite long and therefore the tube is
curved. Brass instruments (usually made of
brass) consist of a long pipe that is usually
coiled and has no holes. The player blows into a
mouthpiece at one end of the pipe, the vibration
of the lips setting the air column vibrating
throughout the pipe. The trombone has a section
of pipe called a slide that can be moved in and
out. To produce a lower note the slide is moved
out. The trumpet has three pistons that are
pushed down to open extra sections of tubing. Up
to six different notes are obtained by using
combinations of the three pistons.
CP 516
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Natural frequencies of vibration (open closed
air column) Speed of sound in air (at room
temperature v 344 m.s-1) v f ?
Boundary conditions Reflection of sound wave at
ends of air column Open end a compression is
reflected as a rarefaction and a rarefaction as a
compression (? phase shift). Zero phase change at
closed end.
odd harmonics exit f1, f3, f5, f7 ,
CP 516
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Problem 8 A narrow glass tube 0.50 m long and
sealed at its bottom end is held vertically just
below a loudspeaker that is connected to an audio
oscillator and amplifier. A tone with a gradually
increasing frequency is fed into the tube, and a
loud resonance is first observed at 170 Hz. What
is the speed of sound in the room?
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f 170 Hz
Solution 8
pressure node
Pressure distribution in tube for fundamental mode
L 0.50 m
v ?0 m.s-1
pressure antinode
speed of wave
N
A
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Problem 9 What are the natural frequencies of
vibration for a human ear? Why do sounds (3000
4000) Hz appear loudest? Solution Assume the
ear acts as pipe open at the atmosphere and
closed at the eardrum. The length of the auditory
canal is about 25 mm. Take the speed of sound in
air as 340 m.s-1. L 25 mm 0.025 m v
340 m.s-1 For air column closed at one end and
open at the other L ?1 / 4 ? ?1 4 L ? f1
v / ?1 (340)/(4)(0.025) 3400 Hz When the
ear is excited at a natural frequency of
vibration ? large amplitude oscillations
(resonance) ? sounds will appear loudest (3000
4000) Hz.
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• RESONANCE

• When we apply a periodically varying force to a
  system that can
• oscillate, the system is forced to oscillate
  with a frequency equal
• to the frequency of the applied force (driving
  frequency) forced
• oscillation. When the applied frequency is
  close to a characteristic
• frequency of the system, a phenomenon called
  resonance occurs.

• Resonance also occurs when a
• periodically varying force is applied
• to a system with normal modes.
• When the frequency of the applied
• force is close to one of normal
• modes of the system, resonance
• occurs.

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Problem 10
Why does a clarinet play a lower note than a
flute when both instruments are about the same
length ?
A flute is an open-open tube. A clarinet
is open at one end and closed at the other end by
the players lips and reed.
open
open
open
closed
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A
A
A
Solution 10
FLUTE is open at both ends ? particle
displacement antinodes at the open ends
Particle displacement variations
Fundamental (1st harmonic)
2nd harmonic (1st overone)
All harmonics can be excited
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Solution 10
A
N
N
CLARINET is open at one and closed at the other ?
pressure node at open end antinode at closed end
pressure variations
A
N
Fundamental (1st harmonic)
3nd harmonic (1st overone)
All odd harmonics can be excited
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• Problem 11
• Resonance

The sound waves generated by the fork are
reinforced when the length of the air column
corresponds to one of the resonant frequencies of
the tube. Suppose the smallest value of L for
which a peak occurs in the sound intensity is
90.0 mm.

• Find the frequency of the
• tuning fork.

Lsmalles t 9.00 cm
(b) Find the wavelength and the next two water
levels giving resonance.
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Solution 11 (alternative)
Tube closed at one end and open at the other ?
odd harmonics can be excited
The frequency of the tuning fork is fixed and
hence the wavelength also has a fixed value. The
length of the tube is variable.
A
pressure variation
N
N
N
A
N
L1 ?/4
L3 3?/4

• 1st resonance
• L1 90.0 mm 90.010-3 m v 343 m.s-1 f1 ?
  Hz
• 4 L 0.360 m v f ? f1 v / ?
  (343)/(9010-3) Hz 953 Hz
• 2nd and 3rd resonances
• L3 3?/4 (3)(0.36)/(4) m 0.270 m
• L5 5?/4 (5)(0.36)/(4) m 0.450 m

L5 5?/4
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Problem 12
Why does a chimney moan ?
Chimney acts like an organ pipe open at both ends
Pressure node
Speed of sound in air v 340 m.s-1 Length of
chimney L 3.00 m L ? / 2 ? 2 L v f
? f v / ? 340 / (2)(3) Hz f 56 Hz
low moan
N
A
N
Pressure node
Fundamental mode of vibration
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Natural frequencies of a trumpet closed at mouth
and open at the flared end
Fundamental f1 60 Hz
1st overtone 163 Hz
2nd overtone 247 Hz
NOT a harmonic sequence for the natural
frequencies of vibration
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Simulation of the human voice tract natural
frequencies