Wave Motion

Wave Types

• Longitudinal– Motion parallel to

energy transport

• Transverse– Motion perpendicular

to energy transport

Properties of Waves

• Wavelength– , distance between crests

• Frequency– f, # oscillations per second

• Speed– How quickly the disturbance moves

EquilibriumA

fv

Tf

1

CDR Radio• What is the wavelength of CDR Radio?

For FM stations the call numbersis the frequency of the station in megahertz

f = 90.3×106 Hz

Velocity for radio waves is the sameas the speed of light.

v = 3.0×108 m/s

fv

Time Dependence• A moving function has both time and

position dependence.

y = f(x-v·t) (travel to the right)

y = f(x+v·t) (travel to the left)

• Ex. Haar Wavelet

15.0,1

5.00,1)(

x

xxf

0 m 1 m 2 m-1 m-2 m

T = 2s T = 0s

Wave Number• Time dependent wave

• Wave number

• k is to as is to T

• Another form

2

k

tvxAy 2cos

T

2

txkAy cos

• Wave velocity

• Ex. What is the velocity of a wave pulse on a 1 mm diameter Copper wire w/ tension of 230 N?

(cu = 8.92 × 103 kg/m3)

Waves on a String

T

v T = Tension = mass/length

Sound Waves• Compression (High Pressure)• Rarefaction (Low Pressure)

• Frequency Ranges– Infrasonic < 20 Hz

– Audible 20 – 20k Hz

– Ultrasonic > 20k Hz

Equil.

Comp.

Rare.

Speed of Sound

• Air– Bulk modulus (B) - How easy it is to compress

a volume of air.– At 20°C and sea level v = 343 m/s (air)

• Solids– Young’s modulus (Y) - How easy it is to

compress a solid.

Bv K

Tsmv 273)/331(

Yv

Sound Level• Intensity - Power transmitted per unit area.

• Intensity Level – Perceived intensity of sound.– Measured in decibels (dB)

– 10 times the intensity is perceived as plus 10 dB

– 2 times the intensity is perceived as plus 3 dB

• Threshold of hearing, I0 = 1012 W/m2

0

log10I

I

24 r

P

area

powerI

(Spherical Wave)

Fireworks• 100 m away from an explosion the intensity

level is 120 dB. What is the intensity level at 500 m?

The Ear• Mechanical energy is converted

into a neural signal in the cochea.– Nerve cells are triggered by the

displacement of the basilar membrane.

Superposition Principle

• When 2 or more waves are present, the resulting wave is an algebraic sum of all the waves.

• Interference

Constructive - Waves in phase

+A

-A

A

-A2A

-2A

+A

-A

A

-A2A

-2A

Destructive - Waves 180° out of phase

21 yyyT

Adding Waves At t = 0s, the crests of two waves are located at the

origin. If the first wave has an amplitude of 5 cm and wavelength of 2 m and the second wave has an amplitude of 2 cm and a wavelength of 0.5 m, what does the resulting wave look like?

Superposition

-10

-5

0

5

10

0 1 2 3 4

Position (m)

Am

pli

tud

e (c

m)

Beats• Periodic variation in intensity with two

waves close in frequency.

tfAy 11 2cos tfAy 22 2cos

From Trig: 22 coscos2coscos bababa

Therefore, ttAy ffff22

2121 2cos2cos2

ModulationEnvelope

OscillateAverage freq.

Beat Frequency• One speaker is transmitting a 10 Hz signal

and a second is transmitting a 11 Hz signal. What beat frequency is experienced?

21 fffb HzHzfb 1110

Hzfb 1

Beats

-15-10-5051015

0 1 2 3 4

Position (m)

Am

pli

tud

e (c

m)

Beats

-15-10-5051015

0 0.2 0.4 0.6 0.8 1

Position (m)

Am

pli

tud

e (c

m)

Interference• Remember the Superposition Principle

Constructive

Destructive

),2,1,0( 12 nnrr

),2,1,0( 21

12 nnrr

#1

#2

r1

r2Source

Effect

Earthquakes

• S waves – Transverse– Shear

• P waves – Longitudinal– Compression

Northridge, CA 1994

Standing Waves• Two identical waves traveling in opposite

directions. tkxAy sin1 tkxAy sin2

tkxAyyyT cossin221

• Antinodes - Max Amplitude5 3, 1,n

4

nx

• Nodes - Zero Amplitude4 2, 0,n

4

nx

Harmonic Series

• Have a node at each end of the string.

• Possible wavelengths

• Frequency

n

Ln

2

v

f

L

n=1

n=2

n=3

fundamental

1st overtone

2nd overtone

1st harmonic

2nd harmonic

3rd harmonic

Waves on a String• General Form for normal modes

– Nth harmonic

n

Ln

2

F

L

nfn 2

(n=1, 2, 3, …)

Ex. A 4.0 g wire is stretched to its full length of 1.75 meters under a tension of 400 N. What frequency is heard as it vibrates in the wind?

Double OpenEnded Pipe

• Air oscillates in and out of the pipe’s ends.

n

Ln

2

L

nvfn 2

(n=1, 2, 3, …)

Air Displacement

2nd Harmonic

3rd Harmonic

Fundamental

Single Open Ended Pipe

• Air oscillates in and out of the pipe’s ends.

n

Ln

4

L

nvfn 4

(n=1, 3, 5, …)

Air Displacement

1st Overtone

2nd Overtone

Fundamental

Organ PipeEx. A pipe organ needs to

play a low Bb (116.54 Hz). If a single open ended pipe is used, how long should the pipe be?

What is the frequency of the 2nd overtone for this pipe?

Doppler Effect• If a source is stationary wave

are emitted radially outward.

• Wavelength in direction of motion is compressed.

• Therefore, frequency heard is

S

O

vv

vvff

' (vS and vO considered positive when approaching each other)

Doppler Effect

• While standing at a corner, the siren of the police car goes from 530 Hz to 470 Hz as it passes by. How fast is the car traveling?

Stationary Source Moving Source

Shock Waves• Supersonic speeds

– Mach Number

– Angle of shock wave

sv

vsin

v – velocity of sound

vs – velocity of the supersonic object

v

vMach s#

vs

v