#  自由振盪、阻滯振盪、共振、非簡諧振盪與混沌擺

(1)

t dt D

x dx dt

x d

t A

dt kx b dx

dt x m d

t dt A

b dx kx

F

d

cos cos cos

2 2

2

2 2

x

x x

sin

) 1

( 2

(2)

## 4 波動與聲波

A sand scorpion

How does a

horseshoe bat detect a moth in

total darkness?

(3)

### 

 governed by Newton’s laws, exist only within a material medium ( 介質 )

### 

 governed by Maxwell’s laws, require no material medium to exist

 all travel at c = 299,792,458m/s

(4)

### 

transverse

 displacement wave ⊥ ( 橫波 )

### 

longitudinal

 displacement wave ∥ ( 縱波 )

(5)

Wave train pulse

(6)

t s

m d

s m

s d m

v d v

t d

l t

) /

75 (

/ ) 150

1 /

50 ( 1

(7)

m

2D

3D

(8)

m

### sin

Snapshot at t = 0 (or any other instant)

(9)

y = y(t) at a fixed position

m

m

### sin 

(10)

2 k 2 k

• Why picking the sine wave ？

• Need a videotape to show it in real time.

• Wavelength and angular wave number – wavelength (λ):

• Angular wave number(k):

) sin(

) (

sin

sinkx1 y k x1 y kx1 k

ym m m

### k

(11)

Period, angular frequency, and frequency f

1 1 1

m m m

2 , 1

2

f T

T

(12)

The wave speed

dt v dx

Traveling Wave

(13)

### A left-going wave

- k v

t kx

y t

x

y( , )  m sin( 

), =

(14)

) sin(

) ,

(x t y kx t

ym

(15)

y a

t kx

t y a u

t kx

t y u y

t kx

y t

x y

y

m y

m m

2

2 sin( )

) cos(

) sin(

) , (

### 

(16)

R

F l

2sin  (2) = 由牛頓第二定律推導

(17)

 

 

R v l v R

R a v

l m

R l F

2

2

l

,

= ) 2

(

sin 2

(18)

1 2

1

2  l2 , 4 l

2 1

1 2

2 1

2 2

1 1

2 2

/ /

=

/

=

v v

v v v v

 

(19)

2 1

1 2

2 1

2 2

1 1

2 2

/

/

=

/

=

v v v

v

v v

2

2 2 2

2 2

2 1

1 2

8 5

2

2 /

= v

l d

v

d l

v l

v

d l

v l t d

### –

(20)

) , ( )

, ( )

,

(x t y1 x t y2 x t

y  

Fourier Analysis

3 sin3 sin2 1

2 sin 1

) 1

(t t t t 

y

 

 

The Principle of Superposition for Waves

### 4.6

(21)

(22)

(23)

• 合成波

) sin(

) , (

) sin(

) , (

2 1

t kx

y t

x y

t kx

y t

x y

m m

) , ( )

, ( )

,

(x t y1 x t y2 x t

y  

) 2 (

cos1 )

2( sin1 2

sin sin

) sin(

) sin(

ym kx t ym kx t

(24)

) sin(

)

sin(kx

ty kx

t

### 

ym m

2 cos 1

2

2 ) sin( 1

2 ] cos 1 2

[ )

, (

m m

m

y y

t kx

y t

x y

Fully constructive and fully destructive interference

(25)

180

=

0 )

, (

0

=

) sin(

2 )

, (

 

 

t x y

t kx

y t

x

y m

m

m

### y  

(exactly in phase/out of phase)

(26)

### 4.8

(27)

) sin(

) , (

) sin(

) , (

2 1

t kx

y t

x y

t kx

y t

x y

m m

) , ( )

, ( )

,

(x t y1 x t y2 x t

y

) sin(

)

sin(kx t y kx t

ym

m

### 

t kx

y t

x

y( , ) [2 m sin ]cos

(28)

t kx

y t

x

y( , )  [2 m sin ]cos





, , ,

= n

n x

n kx

, , ,

= n n x

n kx

2 1 0

2 2 )

( 1

= 2)

( 1

2 1 0

2

=

(29)

(30)

### 4.9

A string of length L

n: harmonic number



n = , , ,

L n v

f v n

L 0 1 2

2

= 2

(31)

(32)

(33)

m

2D

3D

) sin(

) ,

(x t y k x k y t

ym xy

) sin(

) sin(

) , (

t r

k y

t z

k y

k x

k y

t x y

m

z y

x m

 

(34)

### 4.10

Sonar, Seismic Waves and Ultrasound

(35)

### 4.11

property

inertial

property

elastic

v

V V

B p

 /

 

 Bulk Modulus B

v B 容積彈性模數

(36)

(37)

v D v

D v t D

v D v

t d

w

w w

sin

90 sin

sin

13

sin vw

v

apparent angle

(38)

Displacement amplitude of an air element

m

(39)

m m

m

m

m m

s k v

s Bk p

t kx

Bks p

t kx

x Bks B s

p

x A

s B A

V B V

p

t kx

s t

x s

2

### 

(40)

pm 28 Pa sm 11m

pm s

pm 28 10 5 Pa m 11

m m

m m

m

s f v

s v

s k v

s Bk p

) 2

( )

(

) (

)

( 2



Hz 1000

kg/m 21

.

1 3

f

(41)

 

 

2) ( 1

) 2

2 )(

( 1

)

2 (

2 2

m L

m

m L

m

L L

### 4.13

Fully

constructive

destructive interference

(42)

(a)λP1= 0, λP2= D =1.5λ

φP1 = 0, φP2= (1.5λ/λ)2π= 3π (b) N = 6 around the circle

(43)

• 通過單位面積之平均功率

2 2

m

Intensity

The Sound Level

(44)

s s

2

### 2 

(45)

• The Decibel Scale

(46)

0 1

log )

dB 10 10 (

log10 )

dB 10 ( )

10 (

W/m 10

log

) dB 10 (

12 12 12

2 12

0 0

I

I I

(47)

### WHO –the loudest concert

The Who: 120 dB 鑽孔機 : 92

dB

I1: I2 = 630!

(48)

### 4.15

pipe

 

2 1 2 3

2 1 2 3

L

n n f v nv

L n

, , , ,

, , , , (2 open ends)

(49)

(50)

(51)

(a) 靠近耳朵

(b) 離開耳朵

(52)

s s s s t t

s t t

m m

   

 

    

1 2 1 2

1 2 1 2

2

1 2 1 2

(cos cos )

[ cos ]cos

/ ( ) / ( )

s s t t

f f f

m

beat beat

 

      

 

[ cos ]cos

( )

2

2 2 1

2 1 2 1 2

1 2

(53)

(54)

### Effect

(a) Detector moving; source stationary

(55)

f vt t

v f vt v t t

v v

f v v

v f f v v

v f f v v

v

D D

D D D

      

  

 

   

/ ( ) /

/

(56)

### detector stationary

(57)

) 1

(

/ /

v f u

v f v

v f v

f

v v

f v v f

v f v f

v v

f v

f v f

v

v T

v vT

v f v

s D

s s

s

s

(general) (low speed)

### – (b)

(58)

sin vt v t

v

s vs

Mach Cone/Shock Wave Mach Number

### –

(59)

f = 1250Hz v = 242m/s (a) f ’ 偵測器接收的頻率

s

### f

D D

(b) f ’’ 火箭偵測到反射波的頻率

(60)

(a) fm when fbd = 83kHz

(b) fbe when fbd = 83kHz

kHz

 79

 

  m

m m b

bd f

v v

v f v

f

m be m

b

be

###    75kHz

Vb = 9 m/s vm= 8 m/s

(61)

1

1

] ) 1 (

1 [ )

1

( x 1 x x 

magnitude

(62)

200 ly

(63)

s 2 9

2

5

M 10

1 . 2

m/s 10

39 .

5

nm 90

. 0

nm 7

. 500

 

r M ma mu

r

GmM

c

u

ly) /

m 10

46 . 9 )(

ly 100

( 15

r

(64)

(65)

(66)

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## References

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