1 chapter 16 waves-i. 2 mechanical wave sound (f: 20hz ~ 20khz) water wave, ultrasound (f: 1mhz ~...
TRANSCRIPT
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
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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
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如何描述波函數 (Wave Function)
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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.
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如何描述 Harmonic Wave ( 諧波 ) 函數
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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
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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
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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
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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:
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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.
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Elastic potential energy
If the disturbance is small,
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2
22
= vF k
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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:
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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
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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.
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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
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16.9: Interference of Waves
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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
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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
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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.