Chap. 15: Wave - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/...If you double the wavelength of a wave on a string, what happens to the wave speed v and the

Chap. 15: Wave

1

I. Outlook

II. What is wave?

III.Shape & Examples

IV. Equation of motion – Wave equations

V. Examples

Wave Motion

Outlook Translational and Rotational Motions

with Several physics quantities

Energy (E)

Momentum (p)

Angular momentum (L)

With Conservation laws

Conservation of energy

Conservation of linear momentum

Conservation of angular momentum

Wave Motion

2

Wave Motion

What is Wave? Various Types of Waves

3

Mechanical vibration

Spring system

String fixed at both ends

Sound (vibration of air density)

Water wave

Electromagnetic vibration PHYS208

Light

Wave Motion

What is Wave? Vibration

4

Wave Motion

What is Wave? Interference

5

Wave Motion

What is Wave? Oscillations

6

By Marcus Vicente Sauseda From Physics Festival 2013

Wave Motion

Wave Motion

7

Pendulum

http://www.youtube.com/watch?v=yVkdfJ9PkRQ

Transverse and Longitudinal Wave Motions

http://www.acs.psu.edu/drussell/demos/waves/wavemotion.htm

http://www.youtube.com/watch?v=yVkdfJ9PkRQ

http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html




Wave Motion

Goal of My Lectures

Try to address a single question:

• HOW can we describe the wave motion?

Let’s start with SHM!

8

Wave Motion

x

x = R0 cos q

where q = w t

R0

x(t) = R0 cos (w t)

q

Starts here

Kin. Equation of SHM

x(t) = R0 cos (w t + f )

f

“phase angle”

Review: Circular Motion to SHM Simple Harmonic Oscillator (SHO)

9

Wave Motion

Pulse on Rope y

x

[Q] How can you describe the shape of the rope?

[A]

10

2

s in )(

xAxy

Figure 15.4

and TIME dependence…

22

cos ),(

tvx

k

Atxy

w

co s )( tAty w

2

cos )(

xAxy

22

sin ),(

tvx

k

Atxy

w

11

Figure 15.10a

12

Figure 15.10b

wave speed

v = /T or = v T

13

Wave Motion

We will study this equation.

22

sin ),(

tvx

k

Atxy

w

wave speed v = /T

or = v T

14

Wave Motion

x

A

– A

2

s in )0,(

xAtxy

Motion of wave

y

Shapes

15

Wave Motion

x

y

2

s in )0,(

xAtxy

2

cos

2

2

sin

2

2

sin

4

22 sin )

4(

xA

xA

vTxA

TvxA

Tt,xy

w

16

A

– A

wave speed v = /T

or = v T

Mathematical Description of a Wave

– The wave function, y(x,t), gives a

mathematical description of a

wave. In this function, y is the

displacement of a particle at time t

and position x.

– The wave function for a sinusoidal

wave moving in the +x-direction is

y(x,t) = Acos(kx – wt), where k =

2π/ is called the wave number.

– Figure 15.8 at the right illustrates a

sinusoidal wave.

17


– The wave function, y(x,t), gives a

mathematical description of a

wave. In this function, y is the

displacement of a particle at time t

and position x.


wave moving in the +x-direction is

y(x,t) = Acos(kx – wt), where k =


– Figure 15.8 at the right illustrates a

sinusoidal wave.

18

The wave function for a sinusoidal wave

moving in the +x-direction is

y(x, t) = A cos(k x – w t),

where k = 2π/ , w = 2f, = v T …



wave moving in the x-direction is

y(x,t) = Acos(kx + wt), where k =


19


moving in the x-direction is

y(x, t) = A cos(k x + w t),

where k = 2π/ , w = 2f, = v T …

Wave Motion

A transverse traveling wave (amplitude A, wave length , and

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

y = A sin(2x/ + f).

Here f is a constant phase factor.

(a) What will be the equation for a wave traveling to the left

along the x axis as a function of x and t?

[Hint] y(x, t) = ? (find …)

(b) What is its maximum acceleration of particles on the

cord?

[Hint] ay(x, t) = ?

Example 1

20




where k = 2π/ , w = 2f, = v T …

Wave Motion



y = A sin(2x/ + f).




[Hint] y(x, t) = ? (find w)


cord?

[Hint] ay(x, t) = ?

Example 1

21




where k = 2π/ , w = 2f, = v T …

Wave Motion



y = A sin(2x/ + f).




[A] y(x, t) = A sin(2x/ + w t + f), where w = 2f


cord?

[Hint] ay(x, t) max ay?

Example 1

22

Wave Motion



y = A sin(2x/ + f).






cord?

[A] ay(x, t) = –Aw2 sin(2x/ + w t + f) aymax= …

Example 1

23

Wave Motion



y = A sin(2x/ + f).






cord?

[A] ay(x, t) = –Aw2 sin(2x/ + w t + f) aymax=Aw2

Example 1

24

Wave Motion

Example 2 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 meters and t in seconds. For this wave,

determine:

(a) the amplitude,

(b) wavelength, frequency, velocity (magnitude and direction),

(c) maximum and minimum speeds of particles of the cord, and

(d) maximum acceleration (magnitude) of the particles.

[A] …

25




where k = 2π/ , w = 2f, = v T …

Wave Motion

A transverse wave pulse travels to the right along a string with

speed v = 2.0 m/s. At t = 0, the shape of the pulse is given by the

function y = 0.45 cos(3.0x + 1.2) where y and x are in meters and t

in seconds. For this wave, determine:

(a) the wavelength, frequency, and amplitude,

(b) maximum and minimum speeds of particles of the string, and

(c) maximum and minimum accelerations (magnitudes) of the

particles.

[A] …

Example 3

26




where k = 2π/ , w = 2f, = v T …

If you double the wavelength of a wave on a string,

what happens to the wave speed v and the wave

frequency f?

A. v is doubled and f is doubled.

B. v is doubled and f is unchanged.

C. v is unchanged and f is halved.

D. v is unchanged and f is doubled.

E. v is halved and f is unchanged.

Q15.1




where k = 2π/ , w = 2f, = v T = v / f …

If you double the wavelength of a wave on a string,

what happens to the wave speed v and the wave

frequency f?

A. v is doubled and f is doubled.

B. v is doubled and f is unchanged.

C. v is unchanged and f is halved.

D. v is unchanged and f is doubled.

E. v is halved and f is unchanged.

A15.1

Which of the following wave functions describe a wave

that moves in the –x-direction?

A. y(x,t) = A sin (–kx – wt)

B. y(x,t) = A sin (kx + wt)

C. y(x,t) = A cos (kx + wt)

D. both B. and C.

E. all of A., B., and C.

Q15.2

Which of the following wave functions describe a wave

that moves in the –x-direction?

A. y(x,t) = A sin (–kx – wt)

B. y(x,t) = A sin (kx + wt)

C. y(x,t) = A cos (kx + wt)

D. both B. and C.

E. all of A., B., and C.

A15.2

Wave Motion

[Q] The figure shows the wave shape of a

sinusoidal wave traveling to the right at two

instants of time. Find the mathematical

representation of the wave?

[A] …

Example 5

31




where k = 2π/ , w = 2f, = v T …

txkAt,xy

k,.A

v,.A

tkxAt,xy

tkxAt,xy

v,k

tkxAt,xy

cos )(

, cm 53

s 3 /cm 4 cm, 6 cm 53

cos )(

right the to Moveing

cos )(

90

2

2 where

sin )(

w

w

w

w

f

w

fw

Wave Motion

Example 5

4

= 6 cm

32

v = 4 cm / 3.0 s




where k = 2π/ , w = 2f, = v T …

cos

cos

sin sin

f = ?

txkAt,xy

k,.A

v,.A

tkxAt,xy

tkxAt,xy

v,k

tkxAt,xy

cos )(

, cm 53

s 3 /cm 4 cm, 6 cm 53

cos )(


cos )(

90

2

2 where

sin )(

w

w

w

w

f

w

fw

Wave Motion

4

= 6 cm

Example 5

33

cos

cos

sin sin

+ or ?

v = 4 cm / 3.0 s

txkAt,xy

k,.A

v,.A

tkxAt,xy

tkxAt,xy

v,k

tkxAt,xy

cos )(

, cm 53

s 3 /cm 4 cm, 6 cm 53

cos )(


cos )(

90

2

2 where

sin )(

w

w

w

w

f

w

fw

Wave Motion

4

= 6 cm

Example 5

34

v = 4 cm / 3.0 s

A = ? k = ? w = ?

Wave Motion

4

= 6 cm

Example 5

35

txkAt,xy

k,.A

v,.A

tkxAt,xy

tkxAt,xy

v,k

tkxAt,xy

cos )(

, cm 53

s 3 /cm 4 cm, 6 cm 53

cos )(


cos )(

90

2

2 where

sin )(

w

w

w

w

f

w

fw

Wave Motion

Math & Physics

– Equation of Motions –

?) (2

sin )(

d

d ) cos( )(

)( 2

1 )(

2

2

2

2

00

tvxAt,xy

xt

xtAtx

m/Fatatvxtx

ww

36

Wave Motion

Math

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

2

sin

[RH]

sin

[LH]

2

2 where

cos

) (

2 sin )(

x

y

kt

y

tkxAkx

y

tkxAt

y

v,k

,tkxAt

y

x

yv

t

ytvxAt,xy

w

w

ww

w

ww

Differential Equation

Solution of D.Eq.

2nd derivative?

37

Wave Motion

Math

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

2

sin

[RH]

sin

[LH]

2

2 where

cos

) (

2 sin )(

x

y

kt

y

tkxAkx

y

tkxAt

y

v,k

,tkxAt

y

x

yv

t

ytvxAt,xy

w

w

ww

w

ww


Solution of D.Eq.

Where is Physics?

38

Wave Motion

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

2

sin

[RH]

sin

[LH]

2

2 where

cos

) (

2 sin )(

x

y

kt

y

tkxAkx

y

tkxAt

y

v,k

,tkxAt

y

x

yv

t

ytvxAt,xy

w

w

ww

w

ww

Math & Physics – Equation of Motions –

ay = –w2 y (You have seen this!)

39

2

2

2

vk

w

40

Wave Motion

41

Interferences

Boundary Conditions

– When a wave reflects from a fixed end, the pulse inverts as it reflects. See Fig. 15.19(a) at the right.

– When a wave reflects from a free end, the pulse reflects without inverting. See Fig. 15.19(b) at the right.

42

Dancing with Interferences – Waves traveling in opposite directions on a taut string

interfere with each other.

– The result is a standing wave pattern that does not move on the string.

– Destructive Interference occurs where the wave displacements cancel, and Constructive Interference occurs where the displacements add.

– At the nodes (N) no motion occurs, and at the antinodes (A) the amplitude of the motion is greatest.

– Figure 15.23 on the next slide shows photographs of several standing wave patterns.

43

Standing Wave

Wave Motion

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

44

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

Figure 15.23

45

Figure 15.26

F

Lf

2

11

n

Ln

2

nn vT

nn

f

vvT

n

L

2

vL

nfn

2

Fv

(p.484-p.485)

Example 15.3

46

Figure 15.26

F

Lf

2

11

n

Ln

2

nn vT

nn

f

vvT

n

L

2

vL

nfn

2

Fv

(p.484-p.485)

Example 15.3

47

Figure 15.26

F

Lf

2

11

n

Ln

2

nn vT

nn

f

vvT

n

L

2

vL

nfn

2

Example 15.3

48

Fv

(p.484-p.485)

“Dancing” Fire

Wave Motion

49

Wave Motion

50

“Sound Magic”

http://www.youtube.com/watch?v=s9GBf8y0lY0l

Informed by Taylor Campsey



51

1

52

2

53

3

54

4

interference

http://weloveteaching.com/0sci208/lectures/w

aves/interference.html http://www.gwoptics.org/ebook/interferome

ters.php

55

4

http://en.wikipedia.org/wiki/Acti

ve_noise_control

http://94j51c5-15-

1337.wikispaces.com/Active+Nois

e+Cancellation

Active Noise Cancellation

56

5 Example 15.3 Eq. 15.19 P15.72 Eq. 15.6 Sec. 15.3 and 15.4

Keep Checking Website

Wave Motion

57

BACKUP

Wave Motion

58

Wave Motion

Pulse on Rope “Periodic Motion” y

x


[A]

59

Wave Motion

Pulse on Rope


[A] Well, I use w (because it is a periodic motion)!

[Q] Anything else?

“Periodic Motion” y

x

60

Wave Motion

Pulse on Rope

[Q] Anything else?

[A] The rope can be treated as a group of particles,

each being a SHO in y axis.

Rope = Group of particles

“Continuum of SHOs” y

x

y(t) = ? 61

Wave Motion

Pulse on Rope

[Q] Anything else?

[A] The rope can be treated as a group of particles,

each being a SHO in y axis.



x

y(t) = A cos (w t + f )

A

-A

62

Wave Motion

Pulse on Rope

[Q] Can we describe y for all SHO’s?

[A] Hmmm … Not so easy … Let’s go slowly.

Let’s consider a shape of rope at t = 0.



x

A

-A

63

Wave Motion

Shape y

x

A

– A

2

s in )(

xAxy

Wave number is

“number of waves

in unit length”:

k = 2/

So, how many waves

in unit of 2?

0 1 m


[A]

64

Wave Motion

Shape y

x

A

– A

2

s in )(

xAxy

Wave number is

“number of waves

in unit length”:

k = 2/

So, how many waves

in unit of 2?: 1.5

0 1 m


[A] Use T, (or k), and A

[Q] Anything else? 65

Wave Motion

Shape y

x

A

– A

and TIME dependence…

2

s in )(

xAxy

Wave number is

“number of waves

in unit length”:

k = 2/

0 1 m

66

Wave Motion

x

A

– A

2

s in )0,(

xAtxy

Motion of wave

y

Shapes

67

Wave Motion

x

A

– A

2

s in )0,(

xAtxy

1

2 3

4

¼ of one cycle ¼ T

Motion of wave by /4

Motion of SHO

y

wave speed v = (/4)/(T/4)

v = /T or = v T

Shapes

68

Wave Motion

x

y

A

– A

2

s in )0,(

xAtxy

[y] SHM angular frequency

[x] Motion with a constant velocity

Two Components

Bottom line is … 69

Wave Motion

x

y

A

– A

2

s in )0,(

xAtxy

22

sin ),(

tvx

k

Atxy

w

42

2

xx

44

2

4

22

Tvx

Tvx

The shape of the rope (wave)

moves to +x direction.

70

Wave Motion

x

y

A

– A

2

s in )0,(

xAtxy

2

2

2

2

2

4

22

4

xcosA

xsinA

vTxsinA

TvxsinA)

Tt,x(y

w

71

Wave Motion

Wave Motion

I. Outlook

II. What is wave?

III.Kinematics & Examples

IV. Equation of motion – Wave equations

V. More Examples

Sections 1,2,4,5,

72

Wave Motion

x

y

A

– A

2

s in )0,(

xAtxy

[y] SHM angular frequency

[x] Wave propagation:

moving with a constant velocity

22

sin ),(

tvxAtxy

w

Visualization

73

Wave Motion

Math & Physics

– Equation of Motions –

?) (2

sin )(

d

d ) cos( )(

)( 2

1 )(

2

2

2

2

00

tvxAt,xy

xt

xtAtx

m/Fatatvxtx

ww

74

Wave Motion

Finding Wave Equations

- Transverse Wave on Rope -

Consider a segment (mass m)

of the rope under FT.

y

x

75

Wave Motion

Finding Wave Equation

Consider a segment (mass m)


Look at the vertical (y) motion.

1

2

FT

FT

m

y

x

76

Wave Motion

1

2

FT

FT

m

Finding Wave Equation y

x

m

Fa

y

y

77

Wave Motion

1

2

FT

FT

m

x

x

y

F

t

y

x

F

t

y

m

F

t

y

m

F

t

y

m

FF

t

y

m

FF

t

y

m

Fa

yy

y

y

qq

qq

SLOPE) of e(differenc

)SLOPE(SLOPE

)(

sinsin

T

2

2

T

2

2

12T

2

2

12T

2

2

1T2T

2

2

12

2

2

:velocity Wave

:Equation Wave

T

2

2

T

2

2

Fv

x

yF

t

y

Finding Wave Equation y

x

78

Wave Motion

Finding Wave Equation

1

2

FT

FT

m

:velocity Wave

:Equation Wave

T

2

2

T

2

2

Fv

x

yF

t

y

y

x Consider a segment (mass m)


Look at the vertical (y) motion.

79

Wave Motion

[27] Determine if the function y = A sin (k x – w t)

is a solution of the wave equation.

[A] …

Example 4

80

Wave Motion

Math

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

2

sin

[RH]

sin

[LH]

2

2 where

cos

) (

2 sin )(

x

y

kt

y

tkxAkx

y

tkxAt

y

v,k

,tkxAt

y

x

yv

t

ytvxAt,xy

w

w

ww

w

ww


Solution of D.Eq.

Where is Physics?

81

Wave Motion

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

2

sin

[RH]

sin

[LH]

2

2 where

cos

) (

2 sin )(

x

y

kt

y

tkxAkx

y

tkxAt

y

v,k

,tkxAt

y

x

yv

t

ytvxAt,xy

w

w

ww

w

ww


ay = –w2 y (You have seen this!)

82

Wave Motion

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

2

sin

[RH]

sin

[LH]

2

2 where

cos

) (

2 sin )(

x

y

kt

y

tkxAkx

y

tkxAt

y

v,k

,tkxAt

y

x

yv

t

ytvxAt,xy

w

w

ww

w

ww


ay = –w2 y (SHM)

kx – w t = /2 @crest

k x – w t = 0

x / t (= v) = w / k 83

Chap. 15: Wave - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/...If you double the wavelength of a wave on a string, what happens to the wave speed v and the

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