14. Wave Motion

1. Waves & their Properties2. Wave Math3. Waves on a String4. Sound Waves5. Interference6. Reflection & Refraction7. Standing Waves8. 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 x2k

= wave number

SHW moving to the right :

, cosy x t A k x t 2T

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

1 3.62

A m 1.8m

14 m 2 1.5T s 3.0 s

12 0.449k m

12 2.09 s

T

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 yx 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 vmR

a y

Tension force F is cancelled out in the x direction:

2 sinyF F 2F ( small segment )

2

2 m vFR

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?

mL

Lvt

110 N2

m LFt

2

5.0 43

1.4

kg m

s

2F v

Wave Power

SHO :

Segment of length x at fixed x : 2 212

E x A

2 212

xP At

2 21

2v A

v = phase velocity of wave

2 212

E 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

PIr

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.

24PIr

75 402 2

75 40

P Pr r

4040 75

75

Pr r

P 401.9

75WmW

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 :Pv

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 log II

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

kHz.

[ ] = decibel (dB)/10

0 10I I

22 1 10

1

10 log II

2 1 / 102

1

10II

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 log II

210

1

10 log PP

24PIr

2 1 / 102

1

10PP

3 / 210 0.032130

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

1 sin2 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:

2gv

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 obey2gv

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 15

40 /km h

km h

Speed of wave =2g

2250 9.8 /2

m m s

19.7 /m s 71.0 /km h

Time for wave to reach you = 600 8.45

71.0 /km hkm 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 21 12 cos cos2 2

A 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 sin2 2

A

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

2 L 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

2 2 12

L n

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

0x L

dydx

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 sm

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 1 u

v

recede u T 1 uv

1 /recede

ffu v

T = period of wave

Moving Source

1 /approachapproach

v ffu 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 / 1440

Hzm sHz

37.4 /m s

1app

ff uv

1app

fu vf

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.

1towarduf fv

1awayuf fv

Prob. 76

For u/v << 1:

1app

ffuv

1 ufv

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.

1appuc

correct to 1st order in u/c

1appufc

Shock Waves

1appuv

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