Chapter 15and16

MECHANICAL WAVES

Chapter 15Chapter 15 2nd semester, AY 2008-20092nd semester, AY 2008-2009

Chapter 15Chapter 15 What’s in store for us? 2What’s in store for us? 2

• Types of waves

• Wave properties

• Superposition of waves

• Standing waves

Sound waves • Properties of sound waves

• Doppler effect

Chapter 15Chapter 15 Something in common 3Something in common 3

Chapter 15Chapter 15 What is a wave? 4What is a wave? 4

A wave is a traveling disturbance that transports energy but not matter.

Examples:• Sound waves • Mexican wave or La Ola • Water waves• Light

Chapter 15Chapter 15 Types of waves 5Types of waves 5

Transverse• the disturbance in a medium is perpendicular to the wave’s

propagation direction.

Examples:• String waves• Water waves• Light

Chapter 15Chapter 15 Types of waves 6Types of waves 6

Longitudinal• The medium’s displacement is along the wave’s direction of

propagation.

Examples:• Sound waves• Slinky

Chapter 15Chapter 15 Describing waves 7Describing waves 7

Crest – the highest point of the wave above the origin

Trough – the lowest point of the wave beneath the origin

Wavelength (λ)– the distance between two neighboring crests/troughs

Period (T) – time it takes to complete one cycle

Frequency (f) – number of cycles in one sec

Chapter 15Chapter 15 Describing waves mathematically 8Describing waves mathematically 8

A wave varies in space space and timetime.

Say a transverse wave,

)] )/2cos[(),( txAtxy

Direction of vibration or oscillation

Amplitude

Wavelength (spatial) Direction of

propagation Frequency or Period

(temporal)


Spatial property)] )/2cos[(),( txAtxy

)] )/2cos[()0,( xAxy At time t =0: time t =0:

Spatial part

+x

y

A

λ


Temporal property

At position x =0: position x =0: temporal part

t

y

A

-A

T


] )/2(cos[)cos(),0( tTAtAty

Tf

1


Wavenumber (k)

Wave speed (ν)


/2k Unit: m-1

The wave moves one wavelength in one period T so its speed is:

fT

Unit: m/s


Example: The wave function for a harmonic wave on a string is given by:

(a)In what direction does this wave travel and with what speed?

(b)Find λ, f and T of this wave

(c)What is its maximum displacement?

)]5.3 )2.2sin[(03.0),( 11 tsxmmtxy

rightsmv ,/59.1

mk

86.22

Hzf 557.0

2

sfT 80.1/1

mA 03.0


Problem set 13.1: By rocking a boat, a boy produces surface water waves on a previously quiet lake. He observes that the boat performs 12 oscillations in 12 seconds, each oscillation producing a wave crest 15 cm above the undisturbed surface of the lake. He further observes that a given wave crest reaches shore, 12 m away in 6.0 s.

(a)Find the period, speed, wavelength and the amplitude of this wave.

(b)Set-up the wave function.

Chapter 15Chapter 15 Other waveforms 14Other waveforms 14

v

v There are also “pulsespulses” caused by a brief disturbanceof the medium. (waves on a string)

v And “pulse trainspulse trains” which aresomewhere in between. (standing waves)

Other than “continuous wavescontinuous waves”

(described by your sin and cos)

Chapter 15Chapter 15 Waves on a string 15Waves on a string 15

• Consider a pulse propagating along a string:

• What type of wave is it?

• What determines its speed?

http://www.acoustics.salford.ac.uk/feschools/waves/string.htm


We find that the speed of a transverse wave on a medium is:

F

v

Important:

Increasing the tension (F) increases the speed.

Increasing the string mass density () decreases the speed.

Moreover, the speed Moreover, the speed depends only on the nature of the depends only on the nature of the mediummedium and and notnot on amplitude and frequency of the pulse. on amplitude and frequency of the pulse.

F = tension on string

μ = linear mass density

(mass/unit length)


Example: One end of a nylon rope is tied to a stationary support at the top of a vertical mine shaft 80.0 m-deep. The rope is stretched taut by a box of mineral samples with mass 20.0 kg attached at the lower end. The mass of the rope is 2.00 kg. The geologist at the bottom of the mine signals to his colleague at the top by jerking the rope sideways.

(a)What is the speed of the transverse wave on the rope?

(b)If the generated wave has a frequency 2.00 Hz, how many cycles of the wave are there in the rope’s length?


Problem set 13.2: What is the fastest transverse wave that can be sent along a steel wire? The maximum tensile stress to which steel wires should be subject is 7.0 x 108 N/m2. The density of steel is 7800 kg/m3. Show that your answer does not depend on the diameter of the wire.


Energy

• Think about grabbing the left side of the string and pulling it up and down in the y direction.

• You are doing work since F.dr > 0 as your hand moves up and down.

• A wave propagates because each part of the medium transfers its motion to an adjacent region.

because work is done = energy transferred!


Energy

• For SHM (simple sine and cosine waves):

E = ½ k A2 with k = m2

• Within one wavelength:

E = ½m) 2 A2 = ½ ( 2A2

E K U A


Power

• In one period: Pavg = E/T = ½ ( 2A2 / T

where v = / T

• So: Pavg = ½ 2A2 v

with v = (F/)½

dEP

dt

E K U

Square Square dependence on dependence on the amplitude and the amplitude and frequency of the frequency of the wavewave


Exercise:

initial

final

A wave propagates on a string. If the amplitude and the wavelength are doubled, by what factor will the average power carried by the wave change ? (Pfinal / Pinit )

Recall Pavg = ½ 2A2 v

Chapter 15Chapter 15 Waves in 3D 23Waves in 3D 23

Intensity Time average rate at which energy is transported by the wave per unit area

By COE,

24 r

PI

21

22

2

1

r

r

I

I

INVERSE SQUARE LAW INVERSE SQUARE LAW FOR INTENSITYFOR INTENSITY

Chapter 15Chapter 15 Wave-matter interaction 24Wave-matter interaction 24

When a wave encounters matter, any of these can occur:

Transmission

Reflection

Interference and superposition of waves

• With a free end, the string is free to move vertically• The pulse is reflected, but with no phase change.


Reflection (free end)

• When the pulse reaches a fixed support, the pulse moves back (reflected) along the string in the opposite direction

• The reflected pulse is inverted.


Reflection (fixed end)


Transmission

Part of the energy in the incident pulse isReflected (inverted)

Transmitted (not inverted)

2 = m2/L(massive)

1 = m1/L(lighter)


Transmission

Part of the pulse is reflected (not inverted) and part is transmitted (not inverted)

2 = m2/L(lighter)

1 = m1/L(massive)


InterferenceConsider two harmonic waves A and B (same and

amplitudes, differing only in phase traveling to the right.

What does C(x,t) = A(x,t) + B(x,t) look like?

A(x,t)=A cos(kx–t)

B(x,t)=B cos(kx–t+)

Chapter 15Chapter 15 Superposition of waves 30Superposition of waves 30

When the two waves meet, you can show:

C = 2A cos(/2) cos(kx– t+2)

Amplitude Phase shift

Chapter 15Chapter 15 Superposition of waves 31Superposition of waves 31

C = 2A cos(/2) cos(kx– t+2)

In phase

Constructive interference

Out-of-phase

Destructive interference

Chapter 15Chapter 15 Standing waves 32Standing waves 32


Now, consider two harmonic waves A (to the right) and B (to the left) with same amplitudes and same .

A(x,t)=A cos(kx – t) B(x,t)=B cos(kx+t)

nodes antinodes(string permanently at rest)


A(x,t)=A cos(kx – t) B(x,t)=B cos(kx+t)

C(x,t) = A(x,t) + B(x,t) = 2A sin(kx) cos (t)

C(x, t) = 0 when ,...)2,1,0(, nnkx

NODES,...)2,1,0(,2

nn

x

C(x, t) = maximum when ,...)2,1,0(,2

1

nnkx

ANTINODES,...)2,1,0(,22

1

nnx

Chapter 15Chapter 15 Conditions for standing waves 35Conditions for standing waves 35

so when do standing waves occur?

n

Ln

2

,...3,2,1n

L

nvfn 2

Resonant frequencies

Chapter 15Chapter 15 Harmonic series 36Harmonic series 36

n = 1, L = /2 fundamental freq.

n = 2, L = 2/2 = Second harmonic, 1st overtone

n = 3, L = 3/2

Third harmonic, 2nd overtone

n = 4, L = 4/2 = 2Fourth harmonic, 3rd overtone

f1

f2 = 2f1

f3 = 3f1

f4 = 4f1

Chapter 15Chapter 15 Combination of harmonics 37Combination of harmonics 37

A combination wave composed of the 1st harmonic and the third harmonic.

What makes instruments unique is the combination of harmonics produced by the different instruments.

Flutes produce primarily the 1st harmonic They have a very pure tone

Oboes produce a broad range of harmonics and sound very different

Chapter 15Chapter 15 Combination of harmonics 38Combination of harmonics 38


Example: A string tied to a speaker is stretched by a block of mass m. The length of the string in horizontal position is 1.2m, the linear density of the string is 1.6 g/m, and the speaker is set to have a frequency of 120 Hz. What mass allows the speaker to set up the fourth harmonic on the string?m


Problem set 13.3:A string is stretched between fixed supports separated by 75.0 cm. It is observed to have resonant frequencies of 420 Hz and 315 Hz, and no other resonant frequencies between these two.

(a)What is the lowest frequency for this string?(b)What is the wave speed for this string?

Chapter 15Chapter 15 Sound 41Sound 41

• Seismic waves to probe the Earth’s crust for oil

• Ships carry sonar to detect underwater obstacles

• Submarines detect other submarines by listening to the acoustic signature of propellers

• Ultrasound images of fetus• Allow us to communicate

through words and music

Applications

Chapter 15Chapter 15 Sound waves 42Sound waves 42

Sound is a longitudinal wave that can travel thru gas, liquid or solid.

Displacement of air molecules due to propagation of sound waves:

)] )/2cos[(),( max txstxs


• The speed of sound waves in a medium depends on the compressibility and the density of the medium.

In fluids (liquids or gases):

Bv

Yv

In solid rod:

Medium Speed (m/s)

Air 343

Helium 972

Water 1500

Steel (solid) 5600

Speed of sound


Speed of sound• The speed of sound also depends on the temperature of

the medium• This is particularly important with gases• For air, the relationship between the speed and

temperature is

C273

T1 m/s) (331 v c

The 331 m/s is the speed at 0o CTC is the air temperature in Centigrade


Intensity of sound• The amplitude of pressure wave depends on

– Frequency of harmonic sound wave– Speed of sound v and density of medium of medium – Displacement amplitude smax of element of medium

Intensity of a sound wave is

Again, proportional to (amplitude)2

Range of tolerable smax: 10-5 m to 10-11 m

v2I

2max

P

maxmax v sP


Sound level• The range of intensities detectible by the human ear is

very large• It is convenient to use a logarithmic scale to determine the

intensity level,

0

10 log 10 II

Units: in decibels (dB) reference intensity

unknown intensity

I0 = 10-12 W/m2 (lower limit of human range of hearing)


Sound level (dB)

Threshold hearing 0Rustle of leaves 10Whisper 20Office, classroom 50Normal conversation (at 1m) 60Rock group 110Threshold of pain 120Jet engine (at 30m) 130

Sound level


Loudness and frequency

Chapter 15Chapter 15 Sound level 49Sound level 49

Example:In 1976, the Who set a record for the loudest concert: the sound level 46 m in front of the speaker systems was β2 = 120 dB. What is the ratio of the intensity of the Who at that spot to the intensity of another band performing at sound level β1 = 92 dB?

6301

2 I

I

Chapter 15Chapter 15 Doppler effect 50Doppler effect 50

A change in the observed frequency when the source (S) or the detector (D) moves relative to the medium.

sourceobserver fv

t

vtf

/

Both stationary:

DS

)(/)( DD

observer

vv

t

tvvtf

sourceD

observer v

vvff



D moves towards the source:

DS

)(/)( DD

observer

vv

t

tvvtf

D moves away from source:

S is stationary.



DS

D is stationary.

sourceobserver vv

vff

s



Both moving:

DS

sourceD

observer vv

vvff

s


Example:An ambulance emitting a whine at 1600 Hz overtakes and passes a cyclist pedaling a bike at 8.00 m/s. After being passed, the cyclist hears a frequency of 1590 Hz. How fast is the ambulance moving?


Example:Suppose a horseshoe bat flies toward a moth at speed vb = 9.0 m/s, while the moth flies toward the bat with speed vm = 8.0 m/s. From its nostrils, the bat emits ultrasonic waves of frequency fbe that reflect from the moth back to the bat with frequency fbd. The bat adjusts the emitted frequency fbe until th returned frequency fbd is 83 kHz, at which the bat’s hearing is best.

(a) What is fm, the f heard and reflected by the moth?

(b) What is fbe, the f emitted by the bat?


Exercise:A stationary motion detector sends sound waves of 0.150 MHz toward a truck approaching at a speed of 45.0 m/s. What is the frequency of the waves reflected back to the detector?

ccatalanccatalan 2nd semester, AY 2008-20092nd semester, AY 2008-2009

-End of Physics 71-

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Chapter 15and16