Chapter 20 Traveling Waves

Chapter Goal: To learn the basic properties of traveling waves.

Slide 20-2

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Slide 20-3

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Slide 20-5

• result from periodic disturbance

• same period (frequency) as source

• Longitudinal or Transverse Waves

• Characterized by

– amplitude (how far do the “bits” move from their

equilibrium positions? Amplitude of MEDIUM)

– period or frequency (how long does it take for each “bit” to

go through one cycle?)

– wavelength (over what distance does the cycle repeat in a

freeze frame?)

– wave speed (how fast is the energy transferred?) v f

1f

3Hz

5Hz

Wavelength and Frequency are Inversely related: v

f

The shorter the wavelength, the higher the frequency.

The longer the wavelength, the lower the frequency.

Spherical Waves

Wave speed: Depends on Properties of the Medium:

Temperature, Density, Elasticity, Tension, Relative Motion

v f

Transverse Wave • A traveling wave or pulse that causes the elements of

the disturbed medium to move perpendicular to the

direction of propagation is called a transverse wave

Longitudinal Wave

Pulse

Tuning Fork

Guitar String

A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave:

Sound String

Types of Waves

Wave PULSE:

• traveling disturbance

• transfers energy and momentum

• no bulk motion of the medium

• comes in two flavors

• LONGitudinal

• TRANSverse

Traveling Pulse

• For a pulse traveling to the right

– y (x, t) = f (x – vt)

• For a pulse traveling to the left

– y (x, t) = f (x + vt)

• The function y is also called the wave function: y (x, t)

• The wave function represents the y coordinate of any element located at position x at any time t

– The y coordinate is the transverse position

• If t is fixed then the wave function is called the waveform

– It defines a curve representing the actual geometric shape of the pulse at that time

Traveling Pulse

Wave Form

Space Snap Shots

2

2( , )

( 3 ) 1y x t

x t

2

2@ 0 , ( ,0)

( ) 1t s y x

x

2

2@ 1 , ( ,1)

( 3) 1t s y x

x

2

2@ 2 , ( ,2)

( 6) 1t s y x

x

Time Plot

One position

Changing in time

2

2( , )

( 3 ) 1y x t

x t

2

2@ 5, (5, )

(5 3 ) 1x y t

t

Traveling Waves

The media moves in SHM. The wave travels at constant speed.

The wave has the same frequency as the ‘shaking’ source!

Traveling Waves

• The wave represented by the

curve shown is a sinusoidal

wave

• It is the same curve as sin q

plotted against q

• This is the simplest example of a

periodic continuous wave

– It can be used to build more

complex waves

• Each element moves up and

down in simple harmonic motion

• Distinguish between the motion

of the wave and the motion of the

particles of the medium

2

( , ) siny x t A x vt

( , ) sin( )y x t A kx t

Wave Functions are Solutions to the

Wave Equation

2k

22 f

T

( , ) sin 2x t

y x t AT

2

( , ) siny x t A x vt

( , ) sin( )y x t A kx t

v fT k

( , ) sin 2 ( )x

y x t A f tv

2 2

2 2 2

1y y

x v t

Derive these:

Speed of wave depends on

properties of the MEDIUM

Speed of particle in the

Medium depends on

SOURCE: SHM

v f

2

( , ) sin( )

( , ) cos( )

( , ) sin( )

y x t A kx t

v x t A kx t

a x t A kx t

Wave Speed

v f

This gives the relationship between the wavelength

and frequency for constant wave speed.

The frequency depends on the source and the speed

depends on the properties of the medium.

The speed of sound is independent of the frequency.

When traveling from one medium to another, if the

speed changes, the wavelength changes but the

frequency (energy) remains the same.

Time Plot

Snap shot in Space.

This is an image of

one piece of a string

and how it moves as

the waves goes by in

time. The one piece

oscillates in SHM.

( , ) sin( )y x t A kx t

Space Plot

Snap shot in TIME.

Time is fixed. This is an

image of the entire string

or the medium’s

displacement from

equilibrium at one

instant. Can represent

either transverse or

longitudinal waves!!

( , ) sin( )y x t A kx t

Wave 1

Ocean waves with a crest-to-crest distance of

10.0 m can be described by the wave

function

y(x, t) = (0.800 m) sin[0.628(x – vt)]

where v = 1.20 m/s.

(a) Sketch y(x, t) at t = 0.

(b) Sketch y(x, t) at t = 2.00 s.

2

( , ) siny x t A x vt

When x = 5n, we get a node!!! x = 5, 10, 15, 20

Space Snap Shots in Time

Note how the entire wave form has shifted

2.40 m in the positive x direction in this time

interval: x= vt =(1.2m/s)(2s)=2.4m!!!

Wave Speed is Constant! Medium Accelerates!!

String

2

( , ) sin( )

( , ) cos( )

( , ) sin( )

y x t A kx t

v x t A kx t

a x t A kx t

y max = A

vy, max = A

ay, max = 2 A

COMPARE: Motion Equations for

Simple Harmonic Motion Chapter 15

x is fixed!! Cos or Sin changes phase!

22

2

( ) cos ( )

sin( t )

cos( t )

x t A t

dxv A

dt

d xa A

dt

2a x

Notice:

y = (15.0 cm) cos(0.157x – 50.3t).

At a certain instant, let point A be at the origin

and point B be the first point along the x axis

where the wave is 60.0 out of phase with point

A. What is the coordinate of point B?

22 2 ( )

vt f t t v t k x

/ 36.67

0.157x cm

k

2k x x

2

x

Wave Function

10 /v cm s

( , ) sin( )y x t A kx t

2k

22 f

T

If the wave speed is

Find:

Wave Function

4

6

10 /

1.67

A cm

cm

v cm s

f Hz

( , ) sin( )y x t A kx t

2k

22 f

T

( , ) 4sin( 10.5 )3

y x t x t

3k

2 10.5f

v f

( )

/ (linear mass density)

Fv F Tension

m L

Waves on Strings

Problem:

The displacement of a vibrating string vs position along the

string is shown. The wave speed is 10cm/s.

D) If the linear density of the string is .01kg/m, what is the

tension of the string?

v f

Problem:

The displacement of a vibrating string vs position along the

string is shown. The wave speed is 10cm/s.

D) If the linear density of the string is .01kg/m, what is the

tension of the string?

2( / )F v m L

2 5(.1 ) (.01 / ) 10F m kg m N

/

Fv

m L

Problem:

The displacement of a vibrating string vs position along the

string is shown. The wave speed is 10cm/s.

e) If the the tension doubles, how does the wave speed change?

Frequency? Wavelength?

/

Fv

m L

22

/

Fv

m L

22

/

Fv

m L

Wave speed increases by a factor of 2

Problem:

The displacement of a vibrating string vs position along the

string is shown. The wave speed is 10cm/s.

e) If the the tension doubles, how does the wave speed change?

Frequency? Wavelength?

/

Fv

m L

v f

Problem:

The displacement of a vibrating string vs position along the

string is shown. The wave speed is 10cm/s.

e) If the the tension doubles, how does the wave speed change?

Frequency? Wavelength?

/

Fv

m L

v f Frequency depends on

the SOURCE of vibration

Wavelength depends on BOTH!

Wave speed depends

on the MEDIUM

Problem:

The displacement of a vibrating string vs position along the

string is shown. The wave speed is 10cm/s.

e) If the the tension doubles, how does the wave speed change?

Frequency? Wavelength?

/

Fv

m L

2 2 22 2 v v f f

v f

Sound Generation Energy is transmitted as a pressure wave.

There is no net motion of the medium.

The medium oscillates in simple harmonic motion.

The frequency of the wave is the same as the vibrating source.

Vibrating

String

Spherically Symmetric

Sound Source (bell).

Spherical Waves

Representations of Waves

• Wave fronts are the concentric arcs

– The distance between successive

wave fronts is the wavelength

• Rays are the radial lines pointing

out from the source and

perpendicular to the wave fronts

• Far away from the source, the

wave fronts are nearly parallel

planes

• The rays are nearly parallel lines

Reflect

Ray Picture: ECHO

Echo vs Reverb

A reverberation is perceived when the reflected sound wave reaches your ear in less than 0.1 second after the original sound wave. Since the original sound wave is still held in memory, there is no time delay between the perception of the reflected sound wave and the original sound wave. The two sound waves tend to combine as one very prolonged sound wave.

Diffract We can hear around corners.

Why can’t we see around corners?

If the size of the wave (wavelength) is close in size to the

object (door way) then the wave will diffract (bend).

Refract Sound waves refract (bend) when moving between

mediums in which it travels at different speeds.

ADD Superposition Chapter 21

Fig. 18-17, p. 518

is a Pressure Wave

High Pressure: Compression

Low Pressure: Rarefactions

P = Pmax sin (kx – t)

343 m/s in Air @ 20 C

5960 m/s in Steel @ 20 C

1522 m/s in Ocean Water @ 20 C

Speed of Sound in a Vacuum?

Speed of Sound: Temperature

C(331 m/s) 1273 C

Tv

WARNING!

Some textbooks and teachers mistakenly state that the speed of sound increases with increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel. With only these three examples it indeed appears that speed is correlated to density, yet including only a few more examples would show this assumption to be incorrect. All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer ones.

• The speed of sound waves in a medium depends on

the elasticity, density and temperature: decreases with

increasing density and increases with Temperature!

• The compressibility can be expressed in terms of the

elastic modulus of the material:

Bv

elastic property

inertial propertyv

Yv

Liquid or Gas:

1-D String: Fv

Solid Rod:

Speed of Sound: Medium

All other things being

equal, sound will travel

more slowly in denser

materials, and faster in

stiffer ones.

i

Fvolume stress AB

Vvolume strainV

Bv

elastic property

inertial propertyv

Yv

Fv

Speed of Sound: Medium

All other things being equal, sound will travel more slowly in

denser materials, and faster in stiffer ones.

For instance, sound will travel faster in iron than uranium,

and faster in hydrogen than nitrogen, due to the lower density

of the first material of each set. At the same time, sound will

travel faster in iron than hydrogen, because the internal

bonds in a solid are much stronger than the gaseous bonds

between hydrogen molecules. In general, solids will have a

higher speed of sound than liquids, and liquids will have a

higher speed of sound than gases – not because of greater

density, but stronger bonds!

Toy Model

The transmission of sound can be illustrated by using a toy model consisting of an array of balls interconnected by springs. For real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the springs (stiffer springs transmit energy more quickly). In a real material, the ‘stiffness of the springs’ is called the elastic modulus, and the mass corresponds to the density. All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer ones.

elastic property

inertial propertyv

Bv

Yv

is a Longitudinal Wave

is a Pressure Wave

The power transmitted through a closed surface by a wave is

proportional to the amplitude of the wave.

Sound Power

2

W

m

owerI

Area

The intensity of a wave, the power per unit area, is the rate at

which energy is being transported by the wave through a unit

area A perpendicular to the direction of travel of the wave:

Intensity

2 2

W

4 m

PI

r

2

~Intensity Amplitude

Spherical Waves

• A spherical wave propagates radially outward from the oscillating sphere

• The energy propagates equally in all directions

• To compare intensities at two locations, the inverse square relationship can be used

2

1 2

2

2 1

I r

I r

2 2

W

4 m

av avIA r

Intensity

A point source emits sound with a power output of

100 watts. What is the intensity (in W/m2) at a

distance of 10.0 m from the source?

2 2

W

4 m

PI

r

0

10 logI

dBI

12 2

0Threshold of hearing : 10 /I W m

Decibel Index:

Whisper: 20db

Conversation: 60db

Loud Music: 120 db

Jet: 140 dB

Rocket: 250dB

At 90db, wear ear plugs!!!

Review rules of LOGS!!!

12 2

0Threshold of hearing : 10 /I W m

4 2Bursting of eardrums : 10 /I W m

6 2Normal Conversation: 10 /I W m

10 2Whisper: 10 /I W m

2

0

10

WhisperI

I

0

log 2WI

I 2 bels

10 1decibels bel 20 decibels

0 dB

20 dB

60 dB

160 dB

0

10 logI

dBI

Loudness Perception: Phons Perception of Loudness depends on Frequency & Intensity

Fig. 17-6, p. 482

OSHA Safety Standards

OSHA - Occupational Safety and Health Act - The OSHA criteria document reevaluates and reaffirms the Recommended Exposure Limit (REL) for occupational noise exposure established by the National Institute for Occupational Safety and Health (NIOSH) in 1972. The REL is 85 decibels, A-weighted, as an 8-hr time-weighted average (85 dBA as an 8-hr TWA). Exposures at or above this level are hazardous.

Whisper: 20db

Conversation: 60db

Loud Music: 120 db

Jet: 140 dB

Rocket: 250dB

At 90db, wear ear plugs!

You Try

A point source emits sound with a power output of

100 watts. What is the intensity (in W/m2) at a

distance of 10.0 m from the source in dB?

0

10 logI

dBI

2 2

W

4 m

PI

r

11

0

10 logI

dBI

If a sound is twice as intense, how much greater is the sound

level, in db?

22

0

10 logI

dBI

2 12 1

0 0

10 log 10 logI I

dB dBI I

2 12 1

0 0

10 log /I I

dBI I

2

1

10 logI

dBI

2 1 10 log2dB

3.01dB

53 dB is twice as intense as 50dB. Log Scale!!

Quiz: Increasing the intensity of sound

by a factor of 100 causes the sound

level to increase by what amount?

1. 100dB

2. 10dB

3. 20dB

4. 200dB

5. 2 dB

22

1

10 logI

dBI

2 10 log 100 10 2 20dB x

11

0

10 logI

dBI

The decibel level of a jackhammer is 130 dB relative to

the threshold of hearing. Determine the sound intensity

produced by the jackhammer.

1

0

130 10 logI

dB dBI

1

0

13 logI

I

1

0

log1310 10

I

I

13 1

0

10I

I

13

1 010I I 13 1210 10 210 /W m

is a What you Hear

The Pressure Wave sets the Ear Drum into Vibration.

The ear converts sound energy to mechanical energy to a nerve impulse which is transmitted to the brain.

Drum to Stirrup: Simple Machine

Amplification Since the pressure wave striking the large area of the eardrum is concentrated into the smaller area of the stirrup, the force of the vibrating stirrup is nearly 15 times larger than that of the eardrum. This feature enhances our ability of hear the faintest of sounds.

Resonance of the Cilia Nerves The inner surface of the cochlea is lined with over

20 000 hair-like cilia connected to nerve cells, each differing in length by minuscule amounts. Each hair cell has a natural sensitivity to a particular frequency of vibration. When the frequency of the sound wave matches the natural frequency of the nerve cell, that nerve cell will resonate with a larger amplitude of vibration which induces the cell to release an electrical impulse along the auditory nerve towards the brain.

Cochlear Cilia Nerve Damage

Normal Ear Damaged Ear

Excessive exposure to loud sound can damage your cilia.

Sonic: 20 Hz 20 kHz

INFRAsonic: 20Hz

ULTRAsonic: 20kHz

f

f

A middle C vibrates 252 times per second.

Sound Frequencies

Scientists first detected infrasound in

1883, when the eruption of the

Krakatoa volcano in Indonesia sent

inaudible sound waves careening

around the world, affecting barometric

readings. 310dB estimated

The eruption of the Fuego volcano in

Guatemala last year generated high-

amplitude infrasound, mostly below 10

hertz. The pressure readings show that

the strength of these sound waves can

reach the equivalent of 120 decibels.

Infrasonic: < 20Hz

Ultrasound:Pulverizing Tumors

5 2

~ 23

~ 10 /

f kHz

I W m

Deep Heat

3 2

~ 1

~ 10 /

f MHz

I W m

Ultrasound

Intensity of reflected sound wave (echo) is

related to change in density in target.

Ultrasound beam:

-2

7 1 detail

~ 10

MHz mm

I W

William@10 Weeks

"A Womb With a View" and

"Fetal Fotos” “Peek in the Pod”

Hi Cost Hi-Definition Ultrasound

Are there RISKS?

"We do know in animal

studies, certain levels of

ultrasound can cause

damages in growing bones,

in developing bones," said

Dr. Dan Schultz of the Food

and Drug Administration.

Animal Perception of Sound

•domestic cats •100-32,000 Hz

•domestic dogs •40-46,000 Hz

•African

elephants •16-12,000 Hz

•bats •1000-150,000

Hz

•rodents •70-150,000 Hz

Human: 20-20,00Hz

Infrasonic Contact Calls

Female African elephants use "contact calls" to communicate

with other elephants in their bands (usually a family group).

These infrasonic calls, with a frequency of about 21 Hz and a

normal duration of 4-5 seconds, carry for long distances (several

kilometers), and help elephants to determine the location of

other Elephants. Calls vary among individual elephants, so that

others respond differently to familiar calls than to unfamiliar

calls. Perhaps elephants can recognize the identity of the caller.

Echolocation: Sonic Vision

Dolphins produce high frequency (100kHz) clicks that pass through

the melon. These sound waves bounce off objects in the water and

return to the dolphin in the form of an echo. The brain receives the

sound waves in the form of nerve impulses. By this complex system

of echolocation, dolphins can determine size, shape, speed, distance,

direction, and even some of the internal structure of objects in the

water.

Dolphin

Vocalization

The LFAS system consists of a 35-

ton block of 18 huge underwater

speakers and dozens of

microphones. The speakers emit a

consistent low-frequency tone,

between 100 and 500 Hertz, at

240dB, which travels out into the

water at a depth of several hundred

meters. The low frequency permits

the sound to travel tremendous

distances, detecting objects many

hundreds of miles away by

echolocation.

At 100 mile radius from the ship the noise only drops to 160

db which causes shearing of the tissues in the air sack behind

whales' and dolphins' brain. This air sack is highly sensitive

since it is used in echolocation.

“Sound bombing" of ocean floors

to test for oil and gas for National

Security?

2004: More than 100 whales and dolphins died in two separate

beachings in 24 hours on remote Australian islands after US and

Australian navies sound bombed the ocean nearby.

Sea Quakes produce

powerful pressure

waves that rupture

the sinuses and

middle ear of whales

and dolphins.

Sound Weapons

Which is traveling at subsonic,

sonic, or supersonic speeds?

a)

b)

c)

Subsonic

Sonic

Supersonic

RADAR: RAdio Detecting And Ranging

•Cosmological Redshift: Expanding Universe

•Stellar Motions: Rotations and Radial Motions

•Solar Physics: Surface Studies and Rotations

•Gravitational Redshift: Black Holes & Lensing

•Extra-solar Planets via Doppler Wobbler

Case 1: Moving Source Stationary Observer 0Ov

Sv

Observer Reference Frame

Sv

Observer Reference Frame

Case 1: Moving Source Stationary Observer 0Ov

Sv

Observer Reference Frame

Case 1: Moving Source Stationary Observer 0Ov

Sv

Observer Reference Frame

Case 1: Moving Source Stationary Observer 0Ov

Sv

Observer Reference Frame

Case 1: Moving Source Stationary Observer 0Ov

S O ?v

Speed of a wave is determined

by the properties of the Medium!

Case 1: Moving Source Stationary Observer 0Ov

What is the speed of sound to the observer?

wavev v

S O

Speed of a wave is determined

by the properties of the Medium!

Case 1: Moving Source Stationary Observer 0Ov

What is the speed of sound to the observer?

v v

wavev v

What is wavelength and frequency to the observer?

, f f

Sv

source moves in time (period) a distance Sv

Case 1: Moving Source Stationary Observer 0Ov

Sv

emits another wavelength

Case 1: Moving Source Stationary Observer 0Ov

Sv

travels a distance and emits again...Sv

Case 1: Moving Source Stationary Observer 0Ov

Sv

and so on...

Case 1: Moving Source Stationary Observer 0Ov

Sv

bunching up the wavecrests by Sv

Case 1: Moving Source Stationary Observer 0Ov

Sv

is shortened by

= Sv

Case 1: Moving Source Stationary Observer 0Ov

= (1 )Sv

v

= (1 )Sv

v

= (1 )Sv

= ( )sv v

v

?f

Case 1: Moving Source Stationary Observer 0Ov

Use v v

= sv v

v

( )s

f f fv v

v

'f f

1

1 S

f fv

v

Sv

What if ?Sv v

If Sv v

1

(1 )S

f fv

v

1

(1 1)f

1

0

= Mach #Sv

v

When the duck speed is equal or greater than

the speed of waves in water, the waves form a bow wave.

Case 2: Observer Moving & Stationary Source

S Ov

?

?

?

v

f

Case 2: Observer Moving & Stationary Source

S Ov v v

f f

Case 2: Observer Moving & Stationary Source

S Ov

Observer Moving TOWARD (+) and

AWAY (-) from Source

ov vf f

v

ov v v

0(1 )v

f fv

In Sum, if both Source and Observer

are moving…..

sound

observer

source

o

s

O

S

v vf f

v v

v

v

v

0(1 )v

f fv

1

(1 )S

f fv

v

Only Source Moving: Only Observer Moving:

Both: The signs depend on

the relative motion.

For the velocity:

Moving away: minus

Moving toward: plus

Doppler Shift

A car approaches a stationary police car at 36 m/s. The frequency of the siren (relative to the police car) is 500 Hz. What is the frequency (in Hz) heard by an observer in the moving car as he approaches the police car? (Assume the velocity of sound in air is 343 m/s.)

a. 220

b. 448

c. 5264

d. 552

e. 383

sound

observer

source

o

s

O

S

v vf f

v v

v

v

v

?

?

?

v

f

Doppler Shift

A truck moving at 36 m/s passes a police car moving at 45 m/s in the opposite direction. If the frequency of the siren relative to the police car is 500 Hz, what is the frequency heard by an observer in the truck as the police car approaches the truck? (The speed of sound in air is 343 m/s.)

a. 396

b. 636

c. 361

d. 393

e. 617

sound

observer

source

o

s

O

S

v vf f

v v

v

v

v

?

?

?

v

f

Doppler Shift

A truck moving at 36 m/s passes a police car moving at 45 m/s in the opposite direction. If the frequency of the siren is 500 Hz relative to the police car, what is the frequency heard by an observer in the truck after the police car passes the truck? (The speed of sound in air is 343 m/s.)

a. 361

b. 636

c. 393

d. 396

e. 383 sound

observer

source

o

s

O

S

v vf f

v v

v

v

v

?

?

?

v

f

Windy Wave Speed Question

How does the wind affect the sound of a fog horn you hear

on a windy day? What changes?

a) Frequency b) wavelength c) speed d) nothing

Ultrasound Question 8. How far apart are two layers of tissue that produce echoes

having round-trip times that differ by 0.750s? What minimum

frequency must the ultrasound have to see detail this small?

The speed of sound in human tissue is 1540m/s.

6

4w1540 m s 0.750 10 s

5.78 10 m2 2

v td

v f fv

ww 6m s

m2.67 10 Hz

1540

578 10 4.

2 1 s 2 s 1 s / 2d d d v t v t v t

You Try

Calculate the intensity level in dB of a sound

wave that has an intensity of 15 10–4 W/m2.

a. 20

b. 200

c. 92

d. 9

e. 10

0

10 logI

dBI

2 2

W

4 m

PI

r

You Try

By what factor will an intensity change when the corresponding sound level increases by 3 dB?

a. 3

b. 0.5

c. 2

d. 4

e. 0.3

0

10 logI

dBI

2 2

W

4 m

PI

r