(Acoustics)

Topic 02

Mechanical Waves

Updated 2012April24

Dr. Bill PezzagliaPhysics CSUEB

1Outline

A. Wave Phenomena

B. Energy in Waves

C. Wave Propagation

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A. Wave Phenomena

1. Types (modes) of Waves

2. Structure of a wave

3. Wavespeed

3 1. Types of Waves(a) Longitudinal (“p” waves) travel through gas, liquid and

solids. Particle motion in direction of wave

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(b) Transverse (“s” waves) travel only in solids. Particle motion perpendicular to direction of wave.

(c) Surface Waves• Ocean Waves are “Cycloid” Waves. Particles travel in

circles.

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• Rayleigh Waves (“R” wave) are similar, but exist in solids (more damaging in earthquakes than “p” or “s” waves)

(d) Light Waves • Light is a wave of electric and magnetic phenomena that can travel

through empty space (no medium!)

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(e) Gravity Waves?• Not discovered yet, but we are looking for them.

They would distort space as they travel by!

7 f Seismographs 8

• 100 AD: Chinese (Zhang Heng) have primitive device

• 1880 John Milne invents modern seismograph

• P (primary) wavesarrive before “S”(secondary) waves

(shear waves)

g Two Basic Wave Types (Andrija Mohorovičić 1909)

S waves can only travel in solids, and are slowerP can travel in solid, liquid and gas!

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(pressure waves)

(shear waves)

h Liquid Core

Richard Dixon Oldham (July 31, 1858 – July 15, 1936) was a British geologist who, in 1906, argued that the Earth must have a molten interior as S waves were not able to travel through liquids nor through the Earth's interior.“S Shadow Zone”

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2. The Structure of a Wave

a) Nodes : no displacement

b) Antinode : maximum displacement

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Wave parameters

(c) Wavelength (measured in meters)(d) Amplitude (loudness) is the maximum wave

displacement

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2e Wavefunction

Description of a (transverse) wave of wavelength λλλλ and frequency “ f”, with amplitude “ A”, as a function of distance “ x” from origin and time “ t”, moving in direction of positive x.

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−= tfx

Atxyλ

π2cos),(

To describe moving to the left, change the minus sign to plus.

2f. Sound is displacement of pressure 14

• Rarefaction : the “trough” of the wave where pressure (density) is low• Compression : the “peak” of the wave where pressure (density) is high

−+= tfx

PPtxP atm λπ2cos),( 0

3. The Speed of Wave

(a) Measurements of Speed of Sound

(b) Speed of wave in media

(c) Dispersion

15a.i Speed of sound is finite

• Leonardo da Vinci(1452-1519)

• A bell far away, will be heard to resonate in response to a bell ringing, after a delay in time.

• 1500 Did he measure the speed of sound? (some references say yes).

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a.ii Speed of sound depends upon medium!

• (1640) classic experiment on the sound radiation by a ticking watch in a partially evacuated glass vessel provided evidence that air is necessary, either for the production or transmission of sound.

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Robert Boyle (1627-1691 AD)

a.iii Measurement of Speed of Sound

• William Derham (1657-1735)First to accurately measure speed of sound (in air)(341 meters/second)

• Newton used his value in the Principia (1686), although it was 16% higher than the value Newton theoretically calculated.

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a.iv Speed of Sound in Water (1826)

Ion Lake Geneva, Switzerland, Jean-Daniel Colladen, a physicist, and Charles-Francois Sturm, a mathematician, measured speed to be 5x faster than in air.

In their experiment, the underwater bell was struck simultaneously with ignition of gunpowder on the first boat. The sound of the bell and flash from the gunpowder were observed 10 miles away on the second boat. The time between the gunpowder flash and the sound reaching the second boat was used to calculate the speed of sound in water.

Speed did NOT depend upon frequency!

19a.vi. What Wavespeed does NOT depend upon

• Gassendi: demonstrated speed of sound is independent of pitch by comparing measurements from cannon and rifle (no “dispersion”)

• Speed of sound (in air) does not depend on wavelength or frequency

• Speed of sound (in air) does not depend upon amplitude (loudness)

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Pierre Gassendi(1592-1655)

3b. Speed of Wave in Media• Generally speed of wave “ v”: is

frequency ×××× wavelength ( λλλλ)

• For waves on a string the wavespeed only depends upon tension “ T” and mass per unit length “ µµµµ”

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λfv =

µT

v =

b.ii Speed of Waves in media depends on density

• Beno Gutenberg 1914 (worked with Richter)• determines travel times for waves.• Measures Temp and pressure inside of earth• Measures size of core from P Wave Shadow Zone.

P wave velocities: in granite: 6 km/secin basalt: 7 km/secin peridotite: 8 km/sec

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b.iii. Velocities of P and S waves in different layers of the Earth 23 b.iv. Theory of Speed of Sound

• Speed of sound depends upon the properties of the medium throughwhich it travels:

• “B” is coefficient of stiffness (springiness, or “Bulk Modulus”) measured in “Pascals”(the unit of pressure, Newton/meter2)

• “ρ” density (kilograms per cubic meter)

• “c” is wavespeed in meters/second

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ρB

c =

b.v. Speed of Sound in Gas, Liquid and Solid 25

Item B (Pa) ρ (kg/m3) V (m/s)

Air 1.42×105 1.184 346

Water 2.2×109 997 1485

Ice 8.8×109 910 3500

Iron 12×1010 7690 5900

Note generally speed in solid > speed in liquid > speed in gas

b.vi. Speed in Gas

For a gas, formula can be rewritten:

• “T” Temperature (degrees Kelvin)

• “m” molecular mass (kg/mole)– Heavier gas slower (CO2)– Lighter gas faster (Helium)

• “γ” Fudge Factor (7/5 for diatomic molecules)• “R” gas constant 8.314 Joules/°K Mole

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m

RTc

γ=

3c. Dispersion

Speed of Ocean waves is quite complex, depending upon acceleration of gravity “g”, the depth “d” of water, and the wavelength “λ”.

In shallow water (d<< λ), the wavespeed becomes independent of the wavelength (no dispersion), but slows with rise of ocean floor, hence waves “break”.

In deep water (d>> λ) the wavespeed depends upon wavelength (i.e. “dispersion”). Long wavelengths travel faster.

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=λπ

πλ dg

c2

tanh2

dgc =

λπλ

25.12

== gc

Theory of water waves: G. Airy (1841)

c.ii. Ernst Chladni (1756—1827)

• First measurement of speed of sound in solids (up to 40x faster than in air!)

• Measures speed of sound in different gases(slower in heavier gases)

• 1787 “Chladni Plate” shows vibration of sound using sand on a plate.

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c.iii Wavespeed in Chladni Plate

• Determine wavelength from pattern in sand. Calculate wavespeed v=fλ.

• Transverse (s) waves in metal plates (thickness “h”) and rods is dispersive

• Speed is proportional to root of frequency or inversely proportional to wavelength.

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12

hK

Bc

KcKcfv

=

=

==

ρ

λππ

B. Energy in Waves30

1. Impedance2. Kinetic Energy3. Decibels

1. Wave speed vs Displacement Speeda) Wavespeed: is how fast the nodes (or antinodes) are moving. In air, all

sounds move the same speed whether loud, soft, high or low.b) Particle (Displacement) Speed: is how fast the particles in the medium are

moving, which can be quite different! This is related to the intensity of the sound (soft sound they move slowly and not very far)

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c) In a “standing wave”, the particles are moving but the wave is not moving at all!

1b. Amplitude and Particle Velocity

The maximum speed of the molecules in a sound wave is related to the maximum displacement (amplitude “A”) by the frequency “ f ” :

Afv π2=

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1c. Impedance & Acoustic Velocity

• Impedance is the relation between the maximum wave pressure “P”and the maximum oscillating speed “v” of the molecules.

• The “Specific” Acoustic Impedance “z” can be calculated from the medium’s density “ρ” and coefficient of stiffness “B”

• Units: “rayl” (Pascal•sec/meter)z=420 rayls for air at room temp

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cBz ρρ ==

zvP =

2. Definition of Energy

a) Kinetic Energy of molecule (where “v” is the “particle speed”, NOT the wavespeed “c”):

b) Energy Density (energy per unit volume) where ρ is mass density of gas in kg/m3.

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221 mvKE =

221 vu ρ=

2.c Total Energy

c) Rewrite equation

• substitute ρ=z/c• substitute v=P/z

– “z” is the impedance,– “c” is the wavespeed– “P” the amplitude of the pressure

wave.

• For a “loud” P=1 Pascal wave, energy density is only 3x10-6 J/m3

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cz

Pu

2

2

=

2d. Power & Intensity

• Power is rate of using energy. Units=Watts=Joule/sec

• Intensity is the power per unit area delivered by wave:Intensity measured in Watts/square meter[e.g. sunlight is 1400 Watts/meter2 ]The ear can hear as small as 10-12 Watts/m2

• Intensity “I” is proportional to thesquare of the pressure amplitudeMinimum ear can hear is 2x10-5 Pascals

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z

PucI

2

2

==

3a. Decibels: Fechner’s Law

• 1860 Fechner’s Law

• As stimuli are increased by multiplication, sensations increase by addition (Sensation grows as the logarithm of the stimulus)

• Example: A 10x bigger intensity sound is “heard”as only 2x bigger by the ear

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Gustav Theodor Fechner(1801-1887)

3b. Which sounds half as loud as first?• Reference: http://www.phys.unsw.edu.au/jw/dB.html

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3c. Decibel Scale

• The decibel is a logarithmic scale

• A multiplicative factor of 10x in intensity is +10 db

• 0 db is threshold of hearing• 1 db is just noticeable difference• 15 db is a whisper• 60 db is talking• 120 db is maximum safe level• 150 db is jet engine (ear damage)• 180 db stun grenade

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==================Power Ratio dB___________________0.5 -31 02 +35 +710 1020 1350 17100 201000 3010000 40==================

= −

21210 10

10m

W atts

IntensityLogdB

C. Propagation of Waves

1. Inverse Square Law(a) Alkindus (al-Kindi 801-873), Based upon optics of Euclid, knew that light rays are scattered in a cone with the light source as apex, hence PROBABLY knew that the intensity of light drops off in proportion to the increase in the surface area (i.e. square of the distance)

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C1b. Inverse Square Law 41 C1c. Inverse Square Law 42

•Apparent Luminosity drops off inversely proportional to squared distance.

•Sun at Jupiter (5x further away than earth) would appear 1/25 as bright.

•Gravity follows this•So does sound!•Intensity “I” as function of luminosity L and distance “r”:

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4)(

r

LrI

π=

C.2 Inverse Square law and dB

• Factor in 10 in distance is 1/100 the intensity of sound

• Corresponds to drop of 20 dB.

• Below, assume a 120 dB sound at 1 meter distance

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Distance Intensity dB

1 1 120

10 0.01 100

100 0.0001 80

1000 0.000,001 60

10,000 10-8 40

100,000 10-10 20

1,000,000 10-12 0

C.2b SPL: Sound Pressure Level

• Intensity I (Watts/m2) is the energy density “u” times the wavespeed c.

• Hence Intensity can be written:

• So, a ratio of intensities is proportional to ratio of squares of the pressures in a media

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ucI =

z

PI

2

2

=2

00

=

P

P

I

I

C.2c Sound Pressure and Distance

• Intensity drops with square of distance (total power L0)

• Intensity is proportional to square of amplitude (i.e. pressure amplitude)

• Hence, pressure amplitude drops proportional to inverse of distance

• Amplitude of displacement will then also be inversely proportional to distance

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20

4)(

r

LrI

π=

z

PI

2

2

=

rrP

1)( ∝

rrA

1)( ∝

C2d Monopole Radiating 46

3. Doppler Effect 1842 Christian Doppler shows “detected” frequency fd depends

upon:

• fs frequency of source• v relative speed that detector is moving away from source• c velocity of sound in medium• So if moving 10% speed of sound towards you, the frequency

will be increased 10%

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−=c

vff sd 1

3b. More on Doppler

+−=

s

dsd vc

vcff

Actually, one must distinguish between who is moving relative to the medium (air).

• fs frequency of source• c velocity of sound in medium• vs speed of source away from detector relative to air• vd speed of detector away from source relative to air• fd frequency detected

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If source is traveling at speed of sound, get infinite frequency (sonic boom!)

3c. Shock Waves If travel faster than speed of sound,

the waves form a cone of angle θ

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NumberMach sin ==c

vθ

?? Equation not work if v>c ??

3d. Doppler effect for Light

• For light, there is no media. The effect only depends upon the relative speed “v”between source and detector (moving away from each other) and speed of light “c”.

• A yellow star (λs=580 nm) moving towards you at 8.6% the speed of light would appear green (λd=520 nm)

• A yellow star moving away from you at 8.6% the speed of light would appear red (λd=630 nm)

• In the near future we will discuss atomic spectra (the dark lines). Their shift in position gives very precise measurements of astronomical speeds.

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−+=

+−=

vc

vc

vc

vcff

sd

sd

λλ

Videos/Animations• Acoustic Levitation: http://www.youtube.com/watch?v=94KzmB2bI7s• Ruben’s Tube Demo:

– http://www.youtube.com/watch?v=kQ6jYR0-svE– http://www.youtube.com/watch?v=JD90K6KfQes– http://www.youtube.com/watch?v=EhnbhOoPIBc– Mythbusters: http://www.youtube.com/watch?v=ynqzeIYA7Iw

• Dispersion of sound waves– Part 1: http://www.youtube.com/watch?v=HtUGYKjG09g– Part 2: http://www.youtube.com/watch?v=rm8cMV95gWY

• Echo Tube (shows dispersion of sound)– http://www.youtube.com/v/H9LocUJCR5c– http://www.youtube.com/watch?v=H9LocUJCR5c– http://www.youtube.com/watch?v=IYRoWutoyxM

51References

• References

– Donald E. Hall, Musical Acoustics (3rd edition) [Brooks/Cole 2002]Rossing, Moore & Wheeler, “The Science of Sound”(3rd ed, Addison Wesley 2002)

– Fletcher & Rossing, “The Physics of Musical instruments”(2nd ed, Springer 1998)

– Olson, “Music, Physics & Engineering” (2nd ed, Dover 1967)

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