11

Chapter 16Chapter 16

Waves-IWaves-I

22

Mechanical Wave Mechanical Wave Sound (f: 20Hz ~ 20KHz)Sound (f: 20Hz ~ 20KHz) Water wave,Water wave,Ultrasound (f: 1MHz ~ 10MHz)Ultrasound (f: 1MHz ~ 10MHz) Wave on a vibrating stringWave on a vibrating string

Electromagnetic WaveElectromagnetic Wave Radio wave Radio wave Micro Wave ( Micro Wave (0.10.1 ~ 30cm) ~ 30cm) Light ( Light (400400 ~700nm) ~700nm) X-ray ( X-ray (0.010.01 ~ 1nm) ~ 1nm)

Matter WaveWave

particle a of momentum:p

constant splanck':h ;

p

h

33

WavesWaves Wave CharacteristicsWave Characteristics Mathematical Expression of Traveling WaveMathematical Expression of Traveling Wave

Pulse WavePulse Wave Harmonic WaveHarmonic Wave

Speed of a transverse wave on a stringSpeed of a transverse wave on a string longitudinal (sound) in a fluid longitudinal (sound) in a fluid

Energy TransportEnergy Transport Superposition of WavesSuperposition of Waves Reflection and TransmissionReflection and Transmission

(Boundary Problem) (Boundary Problem)

44

Wave CharacteristicsWave Characteristics1)1) Definite speed (vDefinite speed (vww))

A: Amplitude A: Amplitude : Wave length: Wave length T: Period T: Period

2)2) Transport Energy (Not matter)Transport Energy (Not matter)

3)3) Particles of the medium moves Particles of the medium moves back and forthback and forth “or” “or” up and downup and down about their equilibrium point about their equilibrium point (For Mechanical Wave only)(For Mechanical Wave only)

Tff

Tvw

1 velocity Wave

55

In a transverse wave, the displacement of every such oscillating element along the wave is perpendicular to the direction of travel of the wave, as indicated in Fig. 16-1.

In a longitudinal wave the motion ofthe oscillating particles is parallel to the direction of the wave’s travel, as shown in Fig. 16-2.

16.3 Transverse and Longitudinal Waves

66

如何描述波函數 (Wave Function)

77

Traveling PulseTraveling Pulse

( )y f x ae 2bx

x0

, v y x t F x t-v

2b(x vt )y(x, t) f (x, t) ae 2b(x vt)y(x, t) f (x, t) ae

The shape of the Pulse does not change with time.The shape of the Pulse does not change with time.

88

如何描述 Harmonic Wave ( 諧波 ) 函數

99

Harmonic WaveHarmonic Wavefv

0t

xyxy

1010

2 At time 0 ; sin

After time ; ,

2sin

sin 2

2 2sin sin

Ge , sinerally n

t y x F x A x

t y x t F x vt

A x vt

x tA

T

A x t A kx tT

y x t A k

k

x t

22 f

Tk ;

2

vT

1111

16.4 Wave variables

1212

16.4 The Speed of a Traveling Wave

If point A retains its displacement as it moves, the phase giving it that displacement mustremain a constant:

1313

0

22

2

At fixed point

S.H.M

x x

d yy

dt

0xx

Wave speed vT

0

0

22

02

sin ;

cos

sin

y A k x t

dyA kx t

dt

d yA kx t

dt

Proof :

Velocity of a particle of the medium

1414

Which of the following functions Which of the following functions represent traveling waves ?represent traveling waves ?

y( , ) = A cos( - )(b) tx t e kx t

2y( , ) = Asi( n - c) x t kx t

2

Ay( , ) =

B + ( 3a)

)(

-x t

x t

1515

2

20(a) y( , ) =

3 + ( - 3 ) x t

x t

y(x,t)

x

t=0t=2

t=5

1616

y( , ) =(b) 5 cos(3 - 3 )tx t e x t

t=0 t=1 t=2y(x,t)

x

y(x,t)

t

x=0

1717

2y( , ) = 5 sin - 3 (c) x t 3x t

t=0y(x,t)

x

t=1 t=2

1818

利用牛頓定律推導波動方程式利用牛頓定律推導波動方程式

The Wave EquationThe Wave Equation

Wave Speed on a Stretched String

1919

x

y

Wave Speed on a Stretched String

2020

Θ

;

2 22 2x 2y

2y 2y

2x 2

F F F

F Ftan = sin =

F F

2 1F F F����������������������������

2y 2x

2 2x

If F F sin tan

F F

2121

x

y

x

dx

x+dx

y+dyyF=|F1|

F=|F2|

dyΘ(x+dx)

Θ(x)

F2y=FsinΘ(x+dx)

F1y=FsinΘ(x)

ay=d2y/dt2

; 2

2y 1y 2

d yF - F = Fsin (x + dx) - Fsin (x

Newton's

) = m m = d

2nd

xdt

Law

y

利用

方 向

2

2

Fsin (x + dx) - Fsin (x) d y=

dx dt

2222

2

2dx 0

sin (x + dx) - sin (x) d yF =

dx dtlim

2

2

d d yF sin (x) =

dx dt

For small vibration

y dysin (x) = tan (x) =

x dx

dx

dy

Θ(x)

2

2

d d yF =

dx

dy

dx dt

2

2

2

2 d y =Fd y

dx dt /

22

2

2

2

d d y

dx

y

dF =

t/ v

2323

The Wave EquationThe Wave Equation

2

2

2

2

2 2 22

2 2 2 2 2

y A

y Aty A

t

y Ax

y y yx t v t

si n ;

cos

2si n

同理2si n

1

kx t

kx t

kx t

k kx t

k

2 /

2

2 2 22

2

1 1

vk

f f

2424

0

22

2

At fixed point

S.H.M

x x

d yy

dt

0xx

Wave speed vT

0

0

22

02

sin ;

cos

sin

y A k x t

dyA kx t

dt

d yA kx t

dt

Proof :

Velocity of a particle of the medium

2525

16.6: Energy and Power of a Wave Traveling along a String

The average power, which is the average rate at which energy of both kinds (kinetic energy and elastic potential energy) is transmitted by the wave, is:

2626

15.2 Simple Harmonic Motion

In the figure snapshots of a simple oscillatory system is shown. A particle repeatedly moves back and forth about the point x=0.

The time taken for one complete oscillation is the period, T. In the time of one T, the system travels from x=+xm, to –xm, and then back to its original position xm.

The velocity vector arrows are scaled to indicate the magnitude of the speed of the system at different times. At x=±xm, the velocity is zero.

2727

Elastic potential energy

If the disturbance is small,

2828

2

22

= vF k

2929

3030

Energy Transmitted by Harmonic Wave on Energy Transmitted by Harmonic Wave on stringstring

0

1

1

21 1

2

;

2

T

dtT

2

2

2 2

( )

v

0(x=x )在一固定空間點 的能量

2 2m

avg time a

0

2m

2 2m

verage

avgm

y cos kx - ωt dx

y d

dE dE

dE

d

x

dxy

tty

d

3131

16.6: Energy and Power of a Wave Traveling along a String

The average power, which is the average rate at which energy of both kinds (kinetic energy and elastic potential energy) is transmitted by the wave, is:

3232

HomeworkHomework

Chapter 16 ( page 438 )Chapter 16 ( page 438 )

9 , 21, 24, 25, 29, 31, 34, 46, 58, 599 , 21, 24, 25, 29, 31, 34, 46, 58, 59

3333

16.9: The Superposition of Waves

•Overlapping waves algebraically add to produce a resultant wave (or net wave).

•Overlapping waves do not in any way alter the travel of each other.

3434

16.9: Interference of Waves

If two sinusoidal waves of the same amplitude and wavelength travel in the same direction along a stretched string, they interfere to produce a resultant sinusoidal wave traveling in that direction.

1 2 1 2 1 2

1 1sin sin 2sin ( )cos ( )

2 2

3535

16.9: Interference of Waves

3636

16.9: Interference of Waves

3737

Superposition of two Harmonic WaveSuperposition of two Harmonic Wave

difference phase

h wavelengtsame the

frequency same the

amplitude same the

)sin(),(

)sin(),(

2

1

tKxAtxy

tKxAtxy

3838)! veStandingWa(

)2

cos()2

sin(2),(

)(2

1cos)(

2

1sin2sinsin

Applying

)sin()sin(

),(),(),(

212121

21

駐波

合成

合成

tKxAtxy

tKxAtKxA

txytxytxy

3939

16.12: Standing Waves, Reflections at a Boundary

4040

例例：：

22

average time

22

2

2

1

Area

Power Intensity

)(sin4

)Amplitude(Intensity

)cos(

Amplitude

)sin(2),(

0

0),0( ; 0

AvI

KxA

tKxAtxy

txyx

駐波

4141

4242

2

33

23

22

2

2

11

21

...4,3,2,12

...4,3,2,2

02

sin

2

LL

n

LL

n

LL

n

nnL

L

L

k

2cos

2sin2

0sin

0,2

02

0sin

0,01

tkxAy

kL

tLxy

txy

合成

合成

合成

由

由

邊界條件

合成

合成

2...0

1...00

L,txy

,txy

4343

16.12: Standing Waves

4444

16.13: Standing Waves and Resonance

Fig. 16-19 Stroboscopic photographs reveal (imperfect) standing wave patterns on a string being made to oscillate by an oscillator at the left end. The patterns occur at certain frequencies of oscillation. (Richard Megna/Fundamental Photographs)

For certain frequencies, the interference produces a standing wave pattern(or oscillation mode) with nodes and large antinodes like those in Fig. 16-19.

Such a standing wave is said to be produced at resonance, and the string is said to resonate at these certain frequencies, called resonant frequencies.