14. Wave Motion

1. Waves & their Properties

2. Wave Math

3. Waves on a String

4. Sound Waves

5. Interference

6. Reflection & Refraction

7. Standing Waves

8. The Doppler Effect & Shock Waves

Ocean waves travel thousands of kilometers

across the open sea before breaking on shore.

How much water moves with the waves?

Other kinds of waves:

• Sound

• Light

• Radio

• Ultrasound

• Microwave

• Earthquake / Tsunami

Wave:

Traveling disturbance that

transport energy but not matter.

None

14.1. Waves & their Properties

Mechanical waves: mechanical disturbances in material medium.

E.g., air, water, violin string, Earth’s interior, ….

Electromagnetic waves: EM disturbances anywhere (including vacuum)

E.g., Visible, infrared, & ultraviolet light, radio waves, X ray, …

Longitudinal & Transverse Waves

Longitudinal wavesTransverse waves

Water waves

LongitudinalTransverse

mixed

1-D Vibration

Water Waves

Wave Amplitude

Wave amplitude = maximum value of the disturbance.

( w.r.t. undisturbed state )

Water wave: max height above undisturbed level.

Sound wave: max excess pressure.

Wave in coupled springs: max displacement from equilibrium position.

Wave Shape

Waveform = shape of waves.

Pulse = isolated disturbance.

Continuous wave

= ongoing periodic disturbance.

Wave train

= periodic disturbance of finite duration.

Wavelength, Period, & Frequency

A continuous wave is periodic in both time & space.

Wavelength : distance over which the wave pattern repeats. ( length of 1 cycle )

Period T : duration over which the wave pattern repeats. ( time for 1 cycle )

Frequency f : number of wave cycles per unit time. ( f = 1 / T )

Wave Speed

Speed of wave depends only on the medium.

Sound in air 340 m/s 1220 km/h. in water 1450 m/s in granite 5000 m/s

Small ripples on water 20 cm/s.

Earthquake 5 km/s.

vT

fWave speed

GOT IT? 14.1.

A boat bobs up & down on a water wave, moving a vertical distance of 2 m in 1 s.

A wave crest moves a horizontal distance of 10 m in 2 s.

Is the wave speed

(a) 2 m/s, or

(b) 5 m/s ?

Explain.

( Speed of disturbance )

14.2. Wave Math

At t = 0, ,0y x f x

At t , y(0) is displaced to the right by v t.

,y x t f x v t

For a wave moving to the left : ,y x t f x v t

For a SHW (sinusoidal):

,0 cosy x A k x2

k

= wave number

SHW moving to the right :

, cosy x t A k x t 2

T

k x t = phase

vT k

= wave speed

k x v t

pk @ x = 0 pk @ x = v t

Waves

Example 14.1. Surfing

A surfer paddles to where the waves are sinusoidal with crests 14 m apart.

He bobs a vertical distance 3.6 m from trough to crest, which takes 1.5 s.

Find the wave speed, & describe the wave.

, cosy x t A k x t

13.6

2A m 1.8m

14 m 2 1.5T s 3.0 s

120.449k m

12

2.09 sT

4.7 /v m sT

, 1.8 cos 0.449 2.09y x t x t

GOT IT? 14.2.

Figure shows two waves propagating with the same speed.

Which has the greater

(a) amplitude, (b) wavelength, (c) period, (d) wave number, (e) frequency ?

U LL U U

v = / T

The Wave Equation

1-D waves in many media can be described by the partial differential equation

,y x t f x v t

2 2

2 2 2

y y

x v t

Wave Equation

whose solutions are of the form

v = velocity of wave.

E.g., •water wave ( y = wave height )•sound wave ( y = pressure )•…

, cosy x t A k x t vk

( towards x )

14.3. Waves on a String

= mass per unit length [ kg/m ]

A pulse travels to the right.

In the frame moving with the pulse, the entire string

moves to the left.

Top of pulse is in circular motion with speed v & radius

R.Centripedal accel:

2

ˆm v

mR

a y

Tension force F is cancelled out in the x direction:

2 sinyF F 2F ( small segment )

2

2m v

FR

22 R v

R

Fv

2F v

Example 14.2. Rock Climbing

A 43-m-long rope of mass 5.0 kg joins two climbers.

One climber strikes the rope, and 1.4 s later, the 2nd one feels the effect.

What’s the rope’s tension?

m

L L

vt

110 N2

m LF

t

2

5.0 43

1.4

kg m

s

2F v

Wave Power

SHO :

Segment of length x at fixed x : 2 21

2E x A

2 21

2

xP A

t

2 21

2v A

v = phase velocity of wave

2 21

2E m A

Wave Intensity

Wave front = surface of constant phase.

Plane wave : planar wave front.

Spherical wave : spherical wave front.

Intensity = power per unit area direction of propagation [ W / m2 ]

Plane wave : I const

Spherical wave :24

PI

r

Example 14.3. Reading Light

A book 1.9 m from a 75-watt light bulb is barely readable.

How far from a 40-W bulb the book should be to provide the same intensity at the page.

24

PI

r 75 40

2 275 40

P P

r r

4040 75

75

Pr r

P 40

1.975

Wm

W 1.4 m

GOT IT? 14.3.

The intensity of light from the more distant one of two identical stars is only 1% that

of the closer one. Is the more distant star

(a) twice

(b) 100 times

(c) 10 times

(d) 10 times

as far away.

14.4. Sound Waves

Sound waves = longitudinal mechanical waves through matter.

Speed of sound in air :P

v

P = background pressure.

= mass density.

= 7/5 for air & diatomic gases.

= 5/3 for monatomic gases, e.g.,

He.

P, = max , x = 0

P, = min , x = 0

P, = eqm , |x| = max

Sound & the Human Ear

Audible freq:20 Hz ~ 20 kHz

Bats: 100 kHz

Ultrasound: 10 MHz

db = 0 :Hearing Threshold @ 1k Hz

Decibels

Sound intensity level :

100

10 logI

I

12 20 10 /I W m Threshold of hearing at 1

kHz.

[ ] = decibel (dB)/10

0 10I I

22 1 10

1

10 logI

I

2 1 / 102

1

10I

I

2 110I I2 1 10 dB

3/102 110I I2 1 3 dB 12 I

Nonlinear behavior: Above 40dB, the ear percieves = 10 dB as a doubling of loudness.

Example 14.4. TV

A TV blasts at 75 dB.

If it’s then turned down to 60 dB, by what factor has the power dropped ?

60 75 / 1010

22 1 10

1

10 logI

I

2

101

10 logP

P

24

PI

r

2 1 / 102

1

10P

P 3 / 210 0.0321

30

1

10 10

10 db drop ½ in loudness

15 db drop between ½ & ¼ in loudness

14.5. Interference

constructive interference

destructive interference

Principle of superposition: tot = 1 + 2 .

Interference

Fourier Analysis

Fourier analysis:

Periodic wave = sum of SHWs.

E note from electric guitar

0

1sin

2 1n

square wave A n tn

Fourier Series

Dispersion

Non-dispersive medium

Dispersive medium

Dispersion:wave speed is wavelength (or freq) dependent

Surface wave on deep water:

2

gv

long wavelength waves reaches shore 1st.

Dispersion of square wave pulses determines max

length of wires or optical fibres in computer networks.

Dispersion

Conceptual Example 14.1. Storm Brewing

It’s a lovely, sunny day at the coast,but large waves, their crests far apart, are crashing on the beach.

How do these waves tell of a storm at sea that may affect you later?

crests far apart long wavelength

v = ( g / 2 ) large

storm that generates the waves are not far behind

Note: tsunamis generate shallow-water waves that do not obey2

gv

Making the Connection

A storm develops 600 km offshore & starts moving towards you at 40 km/h.

Large waves with crests 250 m apart are your 1st hint of the storm.

How long after you observe these waves will the storm hit?

Time for storm to reach you = 600

1540 /

kmh

km h

Speed of wave =2

g

2250 9.8 /

2

m m s

19.7 /m s 71.0 /km h

Time for wave to reach you = 600

8.4571.0 /

kmh

km h

The storm is 15 8.45 = 6.55 h 6.6 h away.

Beats

Beats: interference between 2 waves of nearly equal freq.

1 2cos cosy t A t A t

1 2 1 2

1 12 cos cos

2 2A t t

Freq of envelope = 1 2 .

smaller freq diff longer period between beats

Applications:

Synchronize airplane engines (beat freq 0).

Tune musical instruments.

High precision measurements (EM waves).

ConstructiveDestructive

Interference in 2-D

Water waves from two sources with separation

Nodal lines:amplitude 0

path difference = ½ n

Destructive Constructive

Interference

Example 14.5. Calm Water

Ocean waves pass through two small openings, 20 m apart, in a breakwater.

75 m from the breakwater & midway between the openings, water is rough.

33 m parallel to the breakwater away, the water is calm.

What’s the wavelength of the waves?

2 275 33 10AP m m m

2AP BP

86.5m

2 275 33 10BP m m m 78.4 m

2 86.5 78.4m m 16 m

GOT IT? 14.4.

Light shines through two small holes onto a screen in a dark room.

The holes spacing is comparable to the wavelength of the light.

Looking at the screen, will you see

(a) two bright spots

(b) a pattern of light & dark patches?

Explain.

14.6. Reflection & Refraction

Fixed end

Free end

Partial Reflection

A = 0;reflected wave inverted

A = max;reflected wave not inverted

light + heavy ropes

Rope

Partial reflection + oblique incidence

refraction

Partial reflection + normal incidence

Application: Probing the Earth

P wave = longitudinal

S wave = transverse

S wave shadow

liquid outer core

P wave partial reflection

solid inner core

Explosive thumps

oil / gas deposits

14.7. Standing Waves

String with both ends fixed:

2L n

, cos cosy x t A k x t B k x t

Superposition of right- travelling & reflected waves:

, 2 sin siny x t A k x t

1 1cos cos 2 sin sin

2 2A

standing wave

sin 0kL 1,2,3,n

Allowed waves = modes or harmonics

n = mode numbern = 1 fundamental moden > 1 overtones

y = 0 node y = max antinode

2L n

0, 0y t B = A

Standing Waves

1 end fixed node,

1 end free antinode.

2 14

L n

cos 0kL

1,2,3,n

22 1

2L n

, cos cosy x t A k x t B k x t

0x L

dy

dx

B A

sin sin 0kA kL t kA kL t

cos sin 0kL t

Standing Waves

Standing Wave Resonance

vf

v = const fundamental mode ~ lowest freq

overtones ~ multiples of fund. freq

Skyscraper ~ string with 1 free end & 1 fixed end.

Tacoma bridge: resonance of torsional standing waves.

Other Standing Waves:

• Water waves in confined spaces (waves in lake).

• EM waves in cavity (microwave oven).

• Sound wave in the sun.

• Electrons in atom.

Musical Instruments

Standing waves on a violin, imaged using holographic interference of laser light waves.

Standing waves in wind instruments:

(a)open at one end L = (2n1) / 4

(b) open at both ends L = n / 2

Example 14.6. Double Bassoon

Double bassoon is the lowest pitched instrument in most orchestra.

It’s “folded” to achieve an effective open-ended column of 5.5 m long.

What is the fundamental freq, assuming sound speed is 343 m/s.

vf

343 /

2 5.5

m s

m

31Hz ~ B0

/ 2

GOT IT? 14.5.

A string 1 m long is clamped tight at one end & free to slide up & down at the other.

Which of the following are possible wavelengths for standing waves on it:

4/5 m, 1 m, 4/3 m, 3/2 m, 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m ?

2 14

L n

14.8. The Doppler Effect & Shock Waves

Point source at rest in medium radiates uniformly in all directions.

When source moves, wave crests bunch up in the direction of motion ( ).

Wave speed v is a property of the medium & hence independent of source motion.

vf

f Doppler effectApproaching source:

.

t = T

u T

t = 2T 2 uT = uT

t = 0

approach u T

u = speed of source

uv

1u

v

recede u T 1u

v 1 /recede

ff

u v

T = period of wave

Moving Source

1 /approachapproach

v ff

u v

Application of the Doppler effect:

• Ultrasound: measures blood flow & fetal heartbeat.

• High freq radio wave: speeding detector.

• Starlight: stellar motion.

• Light from galaxies: expanding universe.

Example 14.7. Wrong Note

A car speeds down the highway with its stereo blasting.

An observer with perfect pitch stands by the roadside, & as the car approach,

notices that a musical note that should be G ( f = 392 Hz ) sounds like A ( 440 Hz ).

How fast is the car moving?

392343 / 1

440

Hzm s

Hz

37.4 /m s

1app

ff

uv

1app

fu v

f

134 /km h

Moving Observers

An observer moving towards a point source at rest in medium sees a faster moving wave.

Since is unchanged, observed f increases.

1toward

uf f

v

1away

uf f

v

Prob. 76

For u/v << 1:

1app

ff

uv

1u

fv

towardf

Waves from a stationary source that reflect from a moving object undergo 2 Doppler effects.

1.A f toward shift at the object.

2.A f approach shift when received at source.

Doppler Effect for Light

Doppler shift for EM waves is the same whether the source or the observer moves.

1app

u

c

correct to 1st order in u/c

1app

uf

c

Shock Waves

1app

u

v

0app if u v Shock wave: u > v

Mach number = u / v

Mach angle = sin1(v/u)

E.g.,

Bow wave of boat.

Sonic booms.

Solar wind at ionosphere

Shock wave front

Source, 1 period ago

Moving Source