CCh

ap. 1

2: W

ave

1

•To describe m

echanical waves. •

To study superposition, standing waves and sound. •

To present sound as a standing longitudinal wave. •

To see that waves will interfere (add constructively and destructively).

•To study sound intensity and beats.

•To solve for frequency shifts (the D

oppler Effect). •

To examine applications of acoustics and m

usical tones.

(1)Characterization of “wave” �

1, 2, 3, 4 (2)

Wave phenom

ena �

5 – Reflection �

6 & 7 – Standing wave �

8 - Interference (3)

Something in real life

�9 – Sound & H

earing �

10 – Sound intensity �

11 – Beats �

12 – Doppler effect

onoof

“wavve”

HHow

to stu

dy …

2

•To describe m

echanical waves.

•To study superposition, standing waves and sound.

•To present sound as a standing longitudinal wave.

•To see that waves will interfere (add constructively and destructively).

•To study sound intensity and beats.

•To solve for frequency shifts (the D

oppler Effect). •

To examine applications of

acoustics and musical tones.

[2015C] 16 �

13 problems

EE1

. Visu

alizin

g Wave F

orm

3

A, �,�k,��

� T

AAn

ato

my o

f E1

4

A, �,�k,��

� T

55

The graph show

n below closely approxim

ates the displacement x of

a tuning fork as a function of time t as it is playing a single note.

What are (a) the am

plitude, (b) period, (c) frequency, and (d) angular frequency of this fork’s m

otion?

E1

. Visu

alizin

g Wave F

orm

(II)

66

An

ato

my o

f E1

(II)

HW

…

EE2

. Visu

alizin

g Sta

ndin

g Wave

7

A string is attached at one end to an electric oscillator set to vibrate

at 100 Hz. The other end is draped over a pulley and has a m

ass M

attached to it. If the string has a mass of 3 gram

s, a length of 3 meters, and there are 2 m

eters of string between theoscillator and

the pulley, what should M be so that two antinodes appear? Take g

to be

10 m/s

2. Ignore

the tension

created by

the extra

hanging string.

88

HW

…

An

ato

my o

f E2

EE3

9

AAn

ato

my o

f E3

10

Later …

FFirst Q

uestio

n: W

hat is W

ave?

Oscillations &

Vibrations

11

By Marcus Vicente Sauseda

From Physics Festival 2013

Try to address a single question: HOW

can we describe the wave m

otion? - Let’s start with Circular Motion �

SH

M �

Wave !

where �� = � t

y(t) = R sin ��

= R sin (�

t)

Kin. Equation of SHM

Step

1: C

ircula

r Motio

n to

SH

M

12

T = 2� / �

arc length s = R ��

s / T = R ��/ T

v = R �

�

Uniform

Circular Motion � = angular speed

SHM

� = angular frequency � T

x

R

��

y

y(t) = A sin(� t)

where �� = � t

y(t) = R cos ��

= R cos (�

t)

Kin. Equation of SHM

Step

1: C

ircula

r Motio

n to

SH

M (II)

13

arc length s = R ��

s / T = R ��/ T

v = R �

�

Uniform

Circular Motion � = angular speed

SHM

� = angular frequency � T

R

��A

T = 2� / �

x

y T

Wave on

Rope ! y

t

y(t) = A cos(� t)

SStep

2: W

ave F

orm

on

Rope

y

x 14

1

��

SStep

2: W

ave F

orm

on

Rope

y

x 15

2

��

SStep

2: W

ave F

orm

on

Rope

y

x 16

3

��

SStep

2: W

ave F

orm

on

Rope

y

x 17

4

��

SStep

2: W

ave F

orm

on

Rope

y

x 18

5

��

SStep

2: W

ave F

orm

on

Rope

y

x 19

6

��

SStep

2: W

ave F

orm

on

Rope

y

x 20

7

��

SStep

2: W

ave F

orm

on

Rope

y

x 21

[Q] H

ow can you describe the shape of the rope? [A

] A

, ��

��

vw

ave A

��

and TIM

E dependence…

22

WWave F

orm

on

Rope (T

ensio

n)

y

x 23

vwave = ��/ T�

vwave = ��(� /2�)�

vwave = �� f�tension

The wave function, y(x,t), gives a mathem

atical description of a wave. In this function, y is the displacem

ent of a particle at time t

and position x.

WWaves o

n a

lon

g rope u

nder ten

sion

(a)The velocity of the wave

will depend on the type and size of rope as well as the tension

(b)Wave length if f = 20 H

z

24

Example 12.2

� = linear mass density

225

A, �,�k,��

� T

Look

back

at A

nato

my o

f E1

[Q] D

irection?

Math

ematica

l Descrip

tion

of a

Wave 22

6

The w

ave function for a sinusoidal wave m

oving in the +x-direction is

y(x, t) = A sin(� t – k x), w

here k = 2π/ � , � = 2�f, ��= v T …

The w

ave function for a sinusoidal wave m

oving in the x-direction is

y(x, t) = A sin( � t + k x), w

here k = 2π/� , � = 2�f, ��= v T …

P.12-10 P.12-12

Which of the follow

ing wave functions describe a w

ave that m

oves in the –x-direction?

A. y(x,t) = A sin (–kx – ��t)

B. y(x,t) = A sin (kx + �t)

C. y(x,t) = A cos (kx + �t)

D. both B

and C.

E. all of A

, B, and C

.

Q12.2

27

Which of the follow

ing wave functions describe a w

ave that m

oves in the –x-direction?

A. y(x,t) = A sin (–kx – ��t) = A sin ( – (kx + �t) )

B. y(x,t) = A sin (kx + �t)

C. y(x,t) = A cos (kx + �t)

D. both B

and C

E. all of A

, B, and C

A12.2

28

SSu

mm

ary o

f Wave F

orm

Fea

tures

29

Wave M

otion

330

Wave M

otion

A transverse traveling w

ave (amplitude A

, wave length ��,�and

frequency f) on a cord at t = 0 is represented by

y = A sin(2�x/�).

What w

ill be the equation for a wave traveling to the left along

the x axis as a function of x and t? [H

int] y(x, t) = ?

Exa

mple 3

31

The w

ave function for a sinusoidal wave

moving in the +x-direction is

y(x, t) = A sin(� t – k x),

where k = 2π/� , � = 2�f, ��= v T …

Wave M

otion

A transverse traveling w

ave (amplitude A

, wave length ��,�and

frequency f) on a cord at t = 0 is represented by

y = A sin(2�x/�).

What w

ill be the equation for a wave traveling to the left along

the x axis as a function of x and t? [A

] y(x, t) = A sin(2�x/� + �

�t ), where � = 2�f

Exa

mple 3

(Work

Book

)

32

The w

ave function for a sinusoidal wave

moving in the +x-direction is

y(x, t) = A sin(� t – k x),

where k = 2π/� , � = 2�f, ��= v T …

Wave M

otion

EExa

mple 4

A

transverse traveling wave on a cord is represented by

y(x, t) = 0.48 sin(0.56x + 84t)

where y and x are in m

eters and t in seconds. For this wave,

determine:

(a)the am

plitude, (b)

wavelength, frequency, velocity (m

agnitude and direction), (c)

maxim

um and m

inimum

speeds of particles of the cord, and (d)

maxim

um acceleration (m

agnitude) of the particles. [A

] …

33

The w

ave function for a sinusoidal wave

moving in the +x-direction is

y(x, t) = A sin(� t – k x),

where k = 2π/� , � = 2�f, ��= v T …

Wave M

otion

EExa

mple 4

(Work

Book

) A

transverse traveling wave on a cord is represented by

y(x, t) = 0.48 sin(0.56x + 84t)

where y and x are in m

eters and t in seconds. For this wave,

determine:

(a)the am

plitude, (b)

wavelength, frequency, velocity (m

agnitude and direction), (c)

maxim

um and m

inimum

speeds of particles of the cord, and (d)

maxim

um acceleration (m

agnitude) of the particles. [A

] A = 0.48, …

34

The w

ave function for a sinusoidal wave

moving in the +x-direction is

y(x, t) = A sin(� t – k x),

where k = 2π/� , � = 2�f, ��= v T …

Wave M

otion

A transverse w

ave pulse travels to the right along a string with

speed v = 2.0 m/s. A

t t = 0, the shape of the pulse is given by the function y = 0.45 cos(3.0x) w

here y and x are in meters and t in

seconds. For this wave, determ

ine: (a)

the wavelength, frequency, and am

plitude, (b)

maxim

um and m

inimum

speeds of particles of the string, and (c)

maxim

um and m

inimum

accelerations (magnitudes) of the

particles. [A

] …

EExa

mple 5

35

Wave M

otion

A transverse w

ave pulse travels to the right along a string with

speed v = 2.0 m/s. A

t t = 0, the shape of the pulse is given by the function y = 0.45 cos(3.0x) w

here y and x are in meters and t in

seconds. For this wave, determ

ine: (a)

the wavelength, frequency, and am

plitude, (b)

maxim

um and m

inimum

speeds of particles of the string, and (c)

maxim

um and m

inimum

accelerations (magnitudes) of the

particles. [A

] …

Wave function is:

y(x, t) = 0.45 cos(3.0x 6.0t)

Exa

mple 5

(Work

Book

)

36

The w

ave function for a sinusoidal wave

moving in the +x-direction is

y(x, t) = A sin(� t – k x),

where k = 2π/� , � = 2�f, ��= v T …

Wave M

otion

Interference

37

Seco

nd Q

uestio

n: W

hat is W

ave?

Bou

ndary C

on

ditio

ns

–W

hen a wave reflects from

a fixed end, the pulse inverts as it reflects.

–W

hen a wave reflects from

a free end, the pulse reflects without inverting..

338

http://ww

w.physicsclassroom

.com

/mm

edia/waves/fix.cfm

http://w

ww

.physicsclassroom.co

m/m

media/w

aves/free.cfm

339

Wave M

otion

AAm

azin

g Wave M

otio

n

40

Pendulum

http://ww

w.youtube.com

/watch?v=yV

kdfJ9PkRQ

Transverse and Longitudinal W

ave Motions

http://ww

w.acs.psu.edu/drussell/dem

os/waves/w

avemotion.htm

�M

echanical vibration �

Spring system

�Pendulum

system

�String fixed at both ends

�Sound (vibration of air density)

�W

ater wave �

Electromagnetic vibration �

PHYS202

�Light

Wave M

otion

441

Interferen

ces

442

Active N

oise C

an

cellatio

n

http://weloveteaching.com

/0sci208/lectures/waves/interference.htm

l http://w

ww

.gwoptics.org/ebook/interferom

eters.php

http://en.wikipedia.org/w

iki/Active_noise_co

ntrol http://94j51c5-15-1337.w

ikispaces.com/A

ctive+Noise+C

ancellation

43

HHave yo

u n

oticed

?

43

444333

Exam

ple 12.6

444

Try T

hem

30

31

32

Dan

cing w

ith In

terferences

–W

aves traveling in opposite directions on a taut string interfere with each other.

–The result is a standing wave pattern that does not m

ove on the string.

–Destructive Interference occurs where the wave displacem

ents cancel, and Constructive Interference occurs where the displacem

ents add.

–A

t the nodes (N) no m

otion occurs, and at the antinodes (A

) the amplitude of the m

otion is greatest. –

Figure on the next slide shows photographs of several standing wave patterns.

445

446

447

AAn

ato

my o

f Sta

ndin

g Wave

http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/doku.php?id=phy123:lab_8 48

12 anti-nodes

SSta

ndin

g Waves o

n a

string

(a)F

T if � = 40.0 g/m and f1 =

20.0 Hz?

(b)f2 and w

avelength of second harm

onic? (c)

f3 and wavelength of second

overtone (or 3rd harm

onic)?

f3 = 3 f1

49

f2 = 2 f1

““Dan

cing” F

ire

50

Roy Montalvo

Wave M

otion

551

“Sound Magic”

http://ww

w.youtube.com

/watch?v=s9G

Bf8y0lY

0l

Informed by Taylor C

ampsey

552

In the primitive m

usical instrument shown in the figure, the

tension in the vibrating part of the wire depends on the weight of the block. If the wire has a frequency f, what will the frequency be if you replace the block with one four tim

es m

ore massive?

(a)2 f

(b)4 f (c)

2 f ©

2016 Pearson Education, Inc.

QQu

ick Q

uiz

53

LLook

back

at E

2

54

A string is attached at one end to an electric oscillator set to vibrate

at 100 Hz. The other end is draped over a pulley and has a m

ass M

attached to it. If the string has a mass of 3 gram

s, a length of 3 meters, and there are 2 m

eters of string between theoscillator and

the pulley, what should M be so that two antinodes appear? Take g

to be

10 m/s

2. Ignore

the tension

created by

the extra

hanging string.

AAm

azin

g Hea

ring

55

�20 – 200,000H

z is the approximate

range of human hearing. Below that

is infrasonic and above …. ultrasonic.

�N

ote: there are variations between anim

al species and effects on any hearing due to pressure changes.

Superhuman H

earing – Can hear far more sounds with far m

ore detail and at far

greater distances

than norm

ally hum

anly possible,

including sounds

on frequencies undetectable

by humans such as dog whistles. Superm

an is able to m

entally screen out most of these sounds to be able to function norm

ally, even in a noisy environm

ent, and can focus in on specific things, like aperson's voice or

heartbeat, even if they are in another part of the city. He can som

etimes be

shown to hear sounds on other planets, which makes it likely that his hearing is

fundamentally different from

that of a human, as sound cannot travel through

the vacuum of space. Like hum

ans and most anim

als, he is skilled at automatically

noticing his own name out of the jum

ble of several overheard conversations, m

aking him adept at quickly responding to calls of distress all over the city.

http://en.wikipedia.org/wiki/Powers_and_abilities_of_Superman (N

ov 2014)

Amplify faint or distant sounds; 100 Yard sound pick up; 40 hour battery life

A proven parabolic dish m

icrophone also known as an omnidirectional m

icrophone that am

plifies noises when pointed at a given direction. The Bionic Ear and Booster Set is a 12" parabolic dish with a record output jack to capture and record faint and distant sounds up to 100 yards away. 5 plus years in the spy m

arket and the Bionic Ear and Booster Set is still one of the most infam

ously used spy gadgets. Specifications: o

Frequency response of 100-10KHz

oGain am

p cut off at 90dB o

Am

plifies sounds by 40 dB o

Independent volume controls on headset

oO

utput jack to record sounds o

Line in jack impedance of 10V m

aximum

o

46dB Mic sensitivity

o40 hour battery life; 9V battery (not included)

556

Bionic Ear & Booster Set (Parabolic Dish)

A Bionic Sound A

mplifier M

icrophone that Picks Sound From

100 Yards Away

HHave yo

u n

oticed

?

57

This is useful for P.12-40, but not given in the form

ula sheet below, because this can be obtained from

the above equation. HH

ave yo

u n

oticed

?

58

40

�Shifts in observed frequency can be caused by m

otion of the source, the listener, or both. Exam

ples 12.10-12.13.

TTh

e Doppler E

ffect –– Eq.1

2.1

9

59

�Shifts in observed frequency can be caused by m

otion of the source, the listener, or both. Exam

ples 12.10-12.13.

HHave yo

u n

oticed

?

60

661

Th

e Doppler E

ffect [I]

L

S

662

Th

e Doppler E

ffect [II]

L

S

�Find the observed frequency by each of two listeners. Exam

ples 12.10-12.13.

EExa

mple 1

63

L2

L1

fS = 300 Hz

v = 340 m

/s

fS = 300 Hz

v = 340 m

/s

EExa

mple 1

(Work

Book

)

64

L2

L1

�Find the observed frequency by each of two listeners. Exam

ples 12.10-12.13.

vL = 0 m

/s

vL = 0 m

/s

EExa

mple 2

65

EExa

mple 2

(Work

Book

)

L

SA

S

B

�Shifts in observed frequency can be caused by m

otion of the source, the listener, or both. Exam

ples 12.10-12.13 and P.12-53,54,60

66

EExa

mple 3

67

EExa

mple 3

(Work

Book

)

L

S

68

L

S

EE3

69

AAn

ato

my o

f E3

70

UUn

dersta

ndin

g Pro

blem

71

UUn

dersta

ndin

g Pro

blem

72

12.42: "By w

hat factor must the sound intensity be increased to

increase the sound intensity level by 12.5 dB?”

�I2 /I1 = ? = 10^{

2 /10 – 1 /10 } = 10^{ (

2 – 1 ) / 10}

�12.5 dB

= 2 –

1

UUn

dersta

ndin

g Pro

blem

73

UUn

dersta

ndin

g Pro

blem

s

74

775

Un

dersta

ndin

g Ch

apter 1

2

776

Blank Page

Exam

ple 12.3 779