Lecture 1 Sound

Hearing

Sound Intensity

Sound Level

Assistant Prof. Matthias Möbius

mobiusm@tcd.ie

Sound Waves

Gas, liquid or solid is mechanically disturbed

• Sound waves are produced

Speed of sound in a substance depends on

•physical properties

•e.g. (density, temperature)

When sound encounters a boundary between

substances,

some sound energy is

transmitted and some reflected

Reflection makes ultrasound imaging possible

rarefaction compression

De

nsity o

f A

ir

Compressed air >>> increased pressure

Rarefied air >>> reduced pressure

organised vibrations of air molecules>> sound

A plucked string will vibrate at its natural

frequency and alternately compresses and

rarefies the air alongside it.

Sound

Sound Waves (Longitudinal waves)

direction

Sound

Sound waves-(variation in air pressure)

can cause objects to oscillate

Example: ear drum is forced to vibrate in

response to the air pressure variation

Depending on:

intensity of the sound

frequency of vibration

movement of the ear drum will

stimulate nerve cells and the sound will be

perceived.

Speed of sound (v) in materials

Material Speed (ms-1)

Air 344

Helium 965

Water 1450

Blood 1570

Body

Tissue

1570

Copper 3750

Iron 5000

Glass 5000

•Greater in solids because molecules interact

more strongly with each other

•Greater in rigid materials

Sound Waves

In general

Depends on

•Phase of the material

•Characteristics of the material

(elasticity, density & temperature)

Helium has a lower

density than air.

Resonant frequencies of

vocal cavity increase.

Spectral distribution of

sounds shift to higher

frequencies

-timbre of sound changes

solids liquids gasesv v v

Sound Waves

Speed of sound (v)

Ev

Bv

kTv

m

Gas

Liquid

Solid bar E Young’s Modulus

density

p

v

c

c

Cp specific heat constant pressure

Cv specific heat constant volume

m molecular mass

k Boltzmann’s constant

T temperature (Kelvin)

B bulk modulus

Depends

on elasticity and density

kTV

m

Calculate the speed of sound in air at 20 oC

=1.4. Boltzmann’s constant =1.38x10-23J/K

Avg. mass of “air molecule” = 47.97x10-27kg

23

27

1.4(1.38 10 / )[(20 273.15) ]

47.97 10

J K KV

kg

1343.6V ms

Sound

The speed of sound in water is 4.2 times

the speed of sound in air. A whistle on land

produces a sound wave with frequency f0. When

this sound wave enters water, its frequency is:

a) 4.2f0

b) f0

c) f0/4.2

d) Not enough information given

Speed of sound

• Frequency (f) of a wave is independent

of the medium through which the wave

travels.

–It is determined by the frequency of the

oscillator that is the source of the waves.

Diffraction

Sound

Light waves:

•Wavelengths « dimensions of everyday objects

•Little diffraction occurs

•Relatively sharp shadows occur

Sound waves:

• Wavelength ≥ size of everyday objects

•diffraction occurs

Example

Sound

source

134434.4

1

v mscm

f KHz

Longer the wavelength compared

to size of opening or object the

greater the diffraction

8 1

14

3 10500

6 10

v msnm

f Hz

The motion of the fluid disturbs hair cells within

the Cochlea, which transmit nerve impulses to

the brain corresponding to the sound heard.

Hearing

Ear can detect very low intensity sounds

Ear canal

hammer

ear drum

stirrup

anvil Cochlea

Outer ear Middle Inner ear

Oval window

sound

Hearing

Sound wave enters the ear.

Forces exerted on eardrum due to air pressure

variations cause it to vibrate.

three small bones (hammer, anvil, and stirrup) in the

middle ear amplify & transmit forces to fluid filled

inner ear through the oval window (very small

area compared with eardrum) result pressure x 30

Other amplification characteristics ??

Ear can detect extremely low intensity sounds

Audible sound waves carries very little energy

Power output: Talk ≈10-5 W

Talk 24 hours a day non stop for 114 years

≈106 hours

Total energy output is ≈10-5 w x106 hrs =10 Wh

All waves carry energy

Hearing

Equivalent to quantity of energy consumed

by a 100W bulb in 6 minutes

Waves (energy) spread out from source

Intensity (I) of a wave is defined as

•Energy (E) carried per unit time per unit area (A)

/E tI

A E

Pt

therefore P

IA

Power (P)

Unit of intensity Watt per square metre (Wm-2)

Intensity

Sunlight intensity at Earth ≈103 Wm-2

Sound Waves

Hearing

If we listen to two sounds (I1 and I2)

and I2 seems twice as loud as I1

Human perception

Measure intensities

I2 is approximately 6 to 10 times I1

Convenient scale to measure loudness is

the logarithm of the intensity

Human ear can detect extremely low intensities

≈10-12 Wm-2

Maximum intensity without ear damage

≈1 Wm-2

Large range 1012 logarithmic units useful

Intensity

Ear response to sound

• logarithmic

• not linear

Decibel scale for intensity

Sound (Intensity) level in decibels (b)

10

0

10logI

Ib

where (threshold of hearing

at 1000Hz)

12 2

0 10I Wm

decibel (b) is a relative sound level measurement

Perceived loudness is roughly Logarithmic

Threshold of discomfort = 1 Wm-2

Above this, pain is experienced, and there is

potential for long term damage

Hearing

Logarithm is the inverse of exponentiation:

Note that logarithms can have different bases.

The most common ones are:

log10, log2, ln (natural logarithm with base e)

log(a b) = log(a) + log(b)

log(a/b) = log(a) - log(b)

log(ab) = b log(a)

log(1) = 0 for all bases

Convert between different bases:

logx(A) = logy(a) / logy(x)

Logarithm

10x =120

log10 (10x) = log10 (120)

x log10 (10) = log10 (120)

x=log10 (120)

Hearing

Sensitivity of ear Can detect sound intensity of ≈10-12Wm-2

Corresponds to pressure variation of ≈ 3x10-5 Pa

(Atm. Pressure ≈ 101,325 Pa)

Random fluctuation due to thermal motion

of molecules ≈ 5x10-6 Pa

Sensitivity:

essentially due to mechanical layout

•Area ratio: ear drum to oval window ≈ 30

•hammer, anvil and stirrup amplification ≈2

•canal resonance at 3kHz pressure increase ≈2

•Total pressure amplification ≈ 30x2x2 = 120

2( )Intensity pressure

Intensity increases by factor of 1202=14,400

Brain: discriminatory role

Filters unwanted noise

Suppression: non-awareness of background noise

ear is not equally sensitive at all frequencies

Sound

level

(dB)

Intensity

(Wm-2)

Sounds

0 1x10-12 Threshold of hearing

10 1x10-11

20 1x10-10

30 1x10-9 Quiet room

40 1x10-8 computer

50 1x10-7

60 1x10-6 Normal conversation

70 1x10-5 Busy traffic

80 1x10-4 Loud radio

90 1x10-3

100 1x10-2

110 1x10-1

120 1 Rock concert, Threshold

of pain

140 1x102 Jet airplane at 30m

160 1x104 Bursting eardrums

Sound levels and Intensities

Computer 10 times louder than quiet room

Does not seem so because of the logarithmic

response of the ear

Vibration

amplitude.

air molecules

1.1x10-11m

1mm

Sound levels and Intensities

Damage Threshold

5 hours/week at > 89dB

damage after 5 years

> 100dB deemed hazardous

D a n g e r H e a r i n g l o s s

10 minutes at 120dB

Temporarily changes your threshold of hearing

from 0dB to 30dB

(a) Calculate the sound level in dB of a sound

intensity 10-8Wm-2

(b) Calculate the intensity in Wm-2 of a sound

level of 80 dB

(a) 10

0

10logI

Ib

8 2

10 12 2

1010log

10

Wm

Wmb

(b) 10

0

80 10logI

I

Sound Waves

8 12 2 4 2

4 2

10 10 10

10

I Wm Wm

I Wm

10

0

8 logI

I

8

0

10I

I

4

1010log 10 10 4 40db b

Ability to hear is not only a function of

intensity but also frequency

Intensity hearing range: 10-12Wm-2 →1Wm-2

Frequency range: 20 Hz → 20 kHz

Hearing ability

Loudness is a method of describing the acoustic

pressure (or the intensity) of a given sound

Dogs:

up to 40 kHz

Dolphins:

up to 250 kHz.

Bats:

up to 120 kHz

Humans:

Hearing

Infrasonic < 20 Hz 20 kHz < ultrasonic

Elephants:

down to 1Hz

Pigeons:

down to 0.1 Hz

Hearing ability as a function of intensity and

frequency. The blue solid line is the pure tone

threshold curve, below which the subject does

not hear.

Ear most sensitive at 3000 Hz

Pain threshold almost frequency independent

Hearing

20 100 1k 10k 20k Hz

frequency

Sound

Level

dB

120

100

80

60

40

20

0

Intensity

W/m2

100

10-2

10-4

10-6

10-8

10-10

10-12

Pain threshold

Hearing threshold

Human Hearing Ability

Why two ears

Time difference of sound arriving at both ears

used to locate the source of the sound

Hearing

Sounds from different directions arrive at each

of our ears at slightly different times and with

slightly different intensities.

Main advantage

Other advantages

•easier to understand speech in noisy background

• help judge loudness

Example:

crossing a road

direction of the car

approximately how close it is

Inverse Square Law

Distance

powerIntensity

area

1 2

14

PI

r 2 2

24

PI

r

2

1 2

2

2 1

I r

I r

Consider imaginary spheres

r2

r1

Isotropic

source

Sound intensity is reduced by moving away

from source By how much?

As the person gets further away, the sphere that

intersects with them gets larger and larger

Fraction = Area of person 4 π r12

Fraction = Area of person 4 π r22

Sound intensity

Inverse square law

2

1 2

2

2 1

I r

I r

Variation of Sound Intensity with

distance from a point source

Intensity I1 at a distance r1 from source

Intensity I2 at a distance r2 from source

Intensity is inversely proportional to the

square of the distance from the source.

NOTE: Sound level (dB) is not inversely

proportional to distance squared !

Sound intensity

A person near a source of loud noise wants to

decrease their exposure to it by a factor of 10.

How far away do they have to move?

2

1 2

2

2 1

I r

I r

2

1 2

21 1

10

I r

I r

2

2

2

1

10r

r 2

1

10 3.16r

r

They have to move 3.16 times further away

The intensity falls off as 1/r2 (where r is the

distance) so moving 4 times as far away will

decrease the exposure by a factor of 16.

Examples

Sound intensity

Waves

A bat can detect sound frequencies up to

120,000 Hz. What is the wavelength of sound

in the air at this frequency?

v f

13344

2.87 10120,000

v msmetres

f Hz

=.287cms

v

f

High frequency—short wavelength

Wave only disturbed by objects with dimensions

similar to or greater than the wavelength

Smaller objects have little effect

Bats use ultrasound for navigation

Can distinguish between insect and falling leaf

Example

Waves

Resonance

Most objects have a natural frequency:

Determined by

• size

• shape

•composition

If an object is subjected to an intense wave

oscillating at object’s natural frequency

a large response (Resonance) occurs

Resonance occurs if

frequency of the driving force equals

natural frequency of the system

Example: child being pushed on a swing.

Swing is kept in motion at its natural frequency

by a series of appropriately timed pushes.

Difficult to get it to swing at any other frequency

12

lT

f g

Simple pendulum

Only one natural frequency

Waves

Resonance: examples

Roman foot soldiers were instructed to break step

when marching over a bridge

•Prevented possible resonance response and

bridge damage

Opera singers with powerful voices

can set glasses into audible vibration

If frequency of note is the same as the natural

frequency of the glass, the glass may vibrate

with a large amplitude and may break

Air passages of the

mouth,

larynx

Nasal cavity

together form an acoustic resonator.

Voiced sound depend on

•resonant frequencies of the total system

------depends on system’s volume and shape

Resonance: examples

Electrical Resonance:

Example: Tuning in radio station

Adjust resonant frequency of the electrical circuit

to the broadcast frequency of the radio station

To “pick up” signal

Half-closed pipe Resonance (e.g. ear canal):

/f soundsonanceRe

)L4/(f sound1 Fundamental mode:

Traveling waves transfer energy from one

place to another

Sound Waves

Examples

• foghorns have a low frequency

•Elephants communicate over long distances

(up to 4 km), frequencies as low as 14 Hz

Sound energy dissipates to thermal energy

when sound travels in air.

Higher frequency sounds dissipate more quickly,

because more energy transferred to the medium;

so lower frequency sounds travel further.

Travel distance is a function of frequency

Lecture 2 Sound

Beats

Doppler Effect

Ultrasound

Applications

Waves

Beats If the two waves interfering have slightly

different frequencies (wavelengths), beats occur.

In step (in phase) In step (in phase)

Out of step (out of phase)

Superposition Simple case: Addition of two waves with

same frequency and amplitude

Wave 1

Wave 2

resultant

Waves

Beats If the two waves interfering have slightly

different frequencies (wavelengths), beats occur.

fb = f1-f2

Waves get in and out of step as time progresses

Result-

• constructive and destructive interference occurs

alternately

•Amplitude changes periodically at the beat

frequency Beat frequency

Absolute value: beat frequency always positive

Wave 1

Wave 2

Resultant

envelope

Waves

Beats fb = f1-f2

No beats occur

If frequency difference = zero

Wave 1

Wave 2

resultant

Waves

Beats

Sound waves Beats perceived as a modulated sound:

loudness varies periodically at the beat frequency

Application

Accurate determination of frequency

Piano tuning

Adjust tension in wire and listen for beats

between it and a tuning fork of known frequency

The two frequencies are equal when the beats

cease.

Easier to determine than when listening to

individual sounds of nearly equal frequencies

f1 = 264Hz

f2 = 266 Hz

Beat frequency 2Hz

Beats can happen with any type of waves

Example

Change in perceived frequency depending on

the relative motion of the source and listener.

Occurs with all types of waves – most notable

•sound waves,

•light waves.

Doppler Effect

stationary

moving→

Example:

Perceived pitch (or frequency) of a moving

source such as a fire engine siren changes as it

goes past

Christian Doppler 1803-1853

Austrian Physicist, Mathematician

Longer

Lower f

Shorter

higher f

Sound Waves

Frequency of sound emitted does not change

Waves

Doppler effect is observed because the distance

between the source of sound and the observer

is changing.

source always emits the same frequency.

Source moving towards the observer

•sound waves reaching observer perceived to

be at a more frequent rate (higher frequency)

sound waves compressed into shorter distance

Source moving away from the observer,

•sound waves reaching observer perceived to

be at a less frequent rate (lower frequency)

Sound waves expanded into longer distance

Waves

Observed frequency for a moving source

+ sign: source moving away from observer

- sign: source moving towards observer

Stationary source, moving observer

wave observerobserver source

wave

v vf f

v

- sign: observer moving away from source

+sign: observer moving towards source

f = Frequency

v = Speed

waveobserver source

wave source

vf f

v v

Waves

Example moving observer

A stationary siren has a frequency of 1000 Hz. What

frequency will be heard by drivers of cars moving at 15ms1?

a) away from the siren?

b) toward the siren?

(a) w o

o s

w

v vf f

v

w oo s

w

v vf f

v

1 1

1

344 151000 956

344o

ms msf Hz Hz

ms

(b)

1 1

1

344 151000 1044

344o

ms msf Hz Hz

ms

wave observerobserver source

wave

v vf f

v

Example: Moving Source

A Garda car with a 1000 Hz siren is moving at

20 ms-1. What frequency is heard by a

stationary listener when the police car is:

a) Moving away from

b) approaching the listener

(a) waveobserver source

wave source

vf f

v v

1

1 1

3441000 1062

344 20observer

msf Hz Hz

ms ms

(b)

1

1 1

3441000 945

344 20observer

msf Hz Hz

ms ms

waveobserver source

wave source

vf f

v v

If you were to replace the Garda car with 2 stationary

sirens emitting at the two frequencies as perceived in (a)

and (b), what would be the beat frequency between them?

Beat frequency beat a bf f f

945 1062beatf Hz Hz

117beatf Hz

Waves

Doppler effect can be used to measure speed

of the source

Police radar uses radio waves:

measures Doppler shift to determine speed of car

•compares frequency of reflected wave from car

with that emitted from radar

Radar: RAdio Detecting And Ranging

Doppler RADAR

•Weather

•Rainstorms, tornadoes

•Wind sheer at airports Swirling air & water droplets

RADAR

Wave source

Sound Waves

Reflection of waves (echoes)

•Caused by solid object

•Change in nature of medium

- Underwater navigation and observation

Sound waves applications

SONAR (sound navigation and ranging)

• Measuring the travel time of sound waves

in the ocean can help monitor sea

temperatures and global changes

Frequency greater than range of human hearing

Sound with frequencies above 20 kHz

Ultrasound

Applications

•Navigation

•Diagnostics

•Surgery

•Therapeutic

•Cleaning

Normally 1 →20MHz

Typical prey: moths (dimensions cms)

Bats use ultrasonic echolocation methods to

detect their presence.

Bats can determine distance, speed and direction

of their prey

(using reflection time and Doppler effect)

why do bats use ultrasound?

13344

6.88 10 0.750

v msm cm

f kHz

Ultrasound- Shorter wavelength

•Reflection, not diffraction occurs at moth.

Submarines, dolphins and bats use ultrasound

for navigation 30-100kHz

13344

344 10 34.41

v msm cm

f kHz

Audible

Ultrasound

Ultrasound

Ultrasound

Medical applications

Reflections of ultrasound pulses from patients

occur at interfaces between different tissues

of different density

Ultrasound probe passed over region of interest

Reflection time provides depth information

Image constructed from echo

and position information

Good contrast: reflection from boundaries

between materials of nearly the same density

Ultrasound Imaging

Ultrasound & Doppler effect

can be used to measure

• Blood flow speed in arteries and veins,

measure arterial occlusion

•Echocardiogram , examination of the heart

• measure blood flow in and out

•fetal heart beats

• pulsation of artery walls

Medical applications

Stroke: early warning

Monitor blood speed in carotid artery in neck

Ultrasound

Red blood cell

Ultrasonic Doppler flow meter Transmitter Receiver

Ultrasound

•intensity kept low (≈10-2 Wm-2) to avoid tissue

damage

Ultrasound scanning during pregnancy

Medical ultrasound without harmful effects

Surgery

Ultrasonic scalpel (55kHz)

Precise cutting and coagulation

•Tumour removal

•Tonsillectomies

Imaging

Why use ultrasound---not audible sound

Smallest detail observable ≈ one wavelength

Ultrasound

Compromise between spatial resolution of image

and penetration depth

•Frequency is selected based on the depth

of the tissue to be treated.

Example: deep heat therapy (low frequency)

115700.5

3000

v msm

f Hz

115701

150

mscm

kHz

Audible sound wavelength in tissue

Ultrasound wavelength in tissue

In tissue, higher frequencies are attenuated more

1MHz:

penetration depth ≈ 6cm

3MHz: superficial conditions (eg. Tennis elbow etc)

Example

Ultrasound speed =1500m/s in tissue.

Using an ultrasound frequency of 2MHz,

calculate (a) smallest detail visible

(b) time for reflected wave to return to probe

from a depth of 5cm

v f 6

1500 /

2 10

v m s

f Hz

(a)

(b)

time for reflected wave to return to probe

5

1

2 0.056.6 10 sec

1500

s mt

v ms

Ultrasound

l = 0.75mm

Ultrasound

•Destructive effects

•Intense ultrasound produces large

density and pressure changes

• Results

− Large stresses

−Heat is produced in most materials

− microscopic vapour bubbles formed and

implode releasing energy (cavitation)

Non-invasive removal of kidney stones

Dental applications

Consists of a ultrasound probe with a small

tip. The ultrasound in combination with water

flow effective in plaque and tartar removal

ultrasonic scalar

Other uses in medicine

Ultrasound

Auto-focus cameras

computes time taken (and hence distance

of subject) for the reflected ultrasonic sound

wave to reach the camera lens position and

then sets focus accordingly.

Component surface cleaning

Component placed in fluid in ultrasonic bath

Ultrasound creates a periodic compression

and expansion in the fluid.

Bubbles formed, grow, and implosively collapse

Results in Acoustic cavitation

localised heating (>1000K)

and high pressures (>100 atmospheres)

Result: effective surface cleaning

Sound

Moving source, approaching listener

When speed of source approaches the speed of

sound, waves ahead of source come close

together.

Supersonic speed

waveobserver source

wave source

vf f

v v

observerf approaches infinity

Nearly infinite number of wave crests reach

observer in very short time

is known as a shock wave

source wavev vWave front produced when

sonic boom

At supersonic speeds the waves overlap and

there are many points of constructive interference,

shock wave results

Sound

Supersonic speed

0sv

sv v

supersonic

subsonic

sv v

sv v

Mach 1

Waves ahead of source

come closer together

Waves pile up at front

Waves overlap:

Shock wave,

Sonic boom.

Stationary source v = 0

Circles represent wave fronts

emitted from sound source

Speed of sound in air =vs

Sound

Supersonic speed

sin sv t

vt

1

sins

vM

v

sin 1 since No shock unless

1M

vt

sv t

Sound wave travels a distance vst

Source travels distance vt

In time interval t

Tangent lines lie on surface of cone

Ratio is called Mach number M s

v

v

sv v

Circles represent wave fronts

emitted from sound source

object speedM

speed of sound

Bow

Waves

sin wwv t

vt

Speed of boat v

> Water wave speed Vww

What is the speed of ultrasound with a wavelength of

0.25 mm and a frequency of 6 MHz? How does this

compare with the speed of sound in air?

Question

v f

6 3 3 16 10 0.25 10 1.5 10v Hz m ms

3 1

1

1.5 104.4

344

ms

ms

Compare with speed of sound in air

Lightening strikes 10 km away.

(a) How long after the strike will you see the light?

(b) How long after the strike will you hear the sound?

c = 3*108 m/s, s = 10 km, t = ? (a)

(b) v = 344 m/s, s = 10 km, t = ?

s = vt t = s/v = (10,000 m)/(344 m/s) = 29 s

s = vt t = s/v

t = (10,000 m)/(3*108 m/s) = 3.3*10-5 s

If you hear the sound 3 seconds after you see

the lightening how far away is the strike?

s = vt =(344 m/s)(3 s) = 1002 m

Question

(a) What is the sound level in decibels of a sound with an

intensity of 0.0200W/m2?

(b) If you had 3 such sounds what would the sound level

be?

10

0

10logI

Ib

2 2

10 12 2

2 1010log

10

Wm

Wmb

10

1010log 2 10b 10 1010 log 2 10 log 10b

10 0.3 10b 103dBb

Question

(a)

(b) 2 2

10 12 2

3 2 1010log

10

Wm

Wmb

10

1010log 6 10b

10 1010 log 6 10 log 10b

10 0.78 10b

107.8dBb

Not equal to 3x103 dB !

Sound levels are logarithms of intensity