Short Version : 14. Wave Motion

Wave Properties

Wave amplitude

Waveform•Pulse•Continuous wave•Wave train

Periodicity in space :Wavelength Wave number k = 2/

Periodicity in time :Period T Frequency = 2/T

Longitudinal & Transverse Waves

Longitudinal wavesTransverse waves

Water waves

LongitudinalTransverse

mixed

1-D Vibration

Water Waves

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

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

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

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

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.

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

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

Beats

Interference in 2-D

Water waves from two sources with separation

Nodal lines:amplitude 0

path difference = ½ n

Destructive Constructive

Interference

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 0k L 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 0k L

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 k L t kA k L t

cos sin 0k L t

Standing Waves

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 /approachapproach

v ff

u v

1 /recede

ff

u v

T = period of wave

Moving 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

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