波长λ与角波数

波與粒子
波的類型 機械波
governed by Newton’s laws, exist only within a material medium (介質)
電磁波
governed by Maxwell’s laws, require no material medium to exist
• Angular wave number(k):
k 2 k 2
週期(T)與角頻率 (ω)
Period(T), angular frequency(w), and frequency(f)
ym sin t1 ym sin (t1 T )
ym sin(t1 T )
2 , f 1
T
T 2
行進波的波速
Traveling Wave The wave speed v dx
dt
kx t a constant,k dx 0
dt
v 来自百度文库 f
kT
A left-going wave
y( x, t )

ym
sin(k x
cos 1 2
]
sin(kx t

1 2
)
ym

2 ym
cos 1
2
Fully constructive and fully destructive interference
繞射(Diffraction)
同相(in phase)與異相(out of phase)
(exactly in phase/out of phase)
ym

2 ym
cos 1
2
y(x,t) 2 ym sin(kx t) = 0
y(x,t) 0
= 180
駐波
• 節點/反節點 node/antinode
在邊界反射之相位變化
駐波與共振
A string of length L
= 2L f v n v n= 0,1,2,

2
dK dt


1 2
v 2 ym2
反射(Reflection)
折射(Refraction)
f v1 v2
1 2 sin1 v1 sin2 v2
干涉(Interference)
波的重疊原理
• The Principle of Superposition for Waves y(x,t) y1(x,t) y2(x,t)
n
2L
n: harmonic number
ym sin(kx t) ym sin(kx t )
sin sin 2sin 1 ( ) cos 1 ( )
2
2
完全建設性與完全破壞性(干涉)
ym sin(kx t) ym sin(kx t )
y(x, t)

[2
ym
t),
v
=
-

k
橫向速率與加速度
y(x,t) ym sin(kx t)
u


y t

ym
cos(kx t)
ay


u t

2
ym
sin(kx t)
ay 2 y
能量傳遞率
dK 1 dmu2 2
u

y t

ym
cos(kx t)
dK
波長(λ)與角波數(k)
• Why picking the sine wave？ • Need a videotape to show it in
real time. • Wavelength and angular wave
number – wavelength (λ):
ym sin kx1 ym sin k(x1 ) ym sin(kx1 k)

1 2
(dx)(ym )2
cos2 (kx
t)
能量傳遞率–II
dK

1 2
(dx)(ym
)2
c os2 (k x

t)
dK dt

1 2
v 2 ym2
cos2 (kx
t)
1/2
dK dt

1 2
v 2 ym2
cos2 (kx
t)

1 4
v 2 ym2
P
波長與頻率
一維波函數具有兩個變數:
y(x,t) ym sin(kxt)
常見之波動參數
振幅(amplitude) 相位(phase) 波長(wavelength) 角波數(angular wave number) 週期(period) 角頻率(angular frequency) 頻率(frequency)
• Fourier Analysis
y(t) 1 sin t 1 sin2t 1 sin3t

2
3
波的干涉
• 合成波
y1(x,t) ym sin(kxt) y2 (x,t) ym sin(kxt )
y(x,t) y1(x,t) y2(x,t)
all travel at c = 299,792,458m/s
物質波:matter
橫波(Transverse Wave)
Transversal wave displacement ⊥wave (橫波)
縱波(Longitudinal Wave)
longitudinal wave displacement ∥wave (縱波)