Wave Motion & EM Waves
(II)
Chih-Chieh Kang
Electrooptical Eng.Dept. STUT
email:kangc@mail.stut.edu.tw
Sinusoidal Traveling Waves
 z, t 
2
 2

 A cos
t
z  i 

T

 A cost  kz  i 
phase : t  kz  i
For simplicity ,
initial phase i  0, 
 z, t 
2
 2
 A cos
t

T
 A cost  kz

z

Sinusoidal Waves

Snapshot of a traveling
sinusoidal wave (at a
fixed time, t = 0), and
0=0

Vertical displacement of
the traveling wave:

 2 
 z, t  0   z   A cos  z 
 
Wavelength  ：the distance between two successive crests or

Amplitude A：one half the wave height or the distance from

troughs.
either the crest or the trough to the equilibrium points
Phase  = 2z/
Sinusoidal Traveling Waves

A wave does not change its
shape as it travels through
space. For a traveling
sinusoidal wave moving at a
speed v, the wave function 
at some later time t :
 2

 z, t   A cos  z  vt 




2vt 
 2
 A cos  z 

 



0 

 2 
at t  0  z, t  0  A cos
z
  
Phase Lead & Phase Lag
 2

 z   A cos  z  0 


(Ulaby)
Sinusoidal Traveling Waves
0 

2vt

 2
vt  
For the time a wave traveling a distance of one
wavelength is called period T
T



v
The frequency of a sinusoidal wave f
1
f 
T

v  f
Sinusoidal Traveling Waves

The angular wave number (or propagation number)
of a sinusoidal wave k
2
k

 Wave function
 2
z  vt   A cos 2 z  2v t 
 


 
2v 
2 


 A cos kz 
t   A cos kz 
t
 
T 


 z, t   A cos 
Harmonic Traveling Waves

For a traveling sinusoidal wave
(at a fixed point z = 0)
2t
 z  0, t   A cos
T
 A cos 2ft  A cos t
angular frequency=2/T=2f
2

 wave function  z, t   A cos kz 
T


t   A coskz  t 

Speed of a Wave

For a traveling wave, its waveform
retains the same phase
  kz  t  constant
d kz  t 
dz
 k  t  0,
dt

dt
dz

v
dt
k
Phase velocity v : the velocity of the
waveform as it moves across the
medium
v

k


T
 f
Mathematical Description of a
Wave

Waves are solutions to the wave equation：
 2
1  2
 2 2
2
z
v t
1-D waves
：wave function, v：phase velocity
- Where does wave equation come from?
- What do solutions look like?
- How much energy do they carry?
Wave Equation for a String



Each small piece of string obeys Newton’s Law：
Small displacement, so sin   tan   y
x
Net force is proportional to curvature：
y
 y  x  x  y  x  

Fy

T


T
x



2
x
x 

x
2
Wave Equation for a String

Newton’s 2nd Law…
 Fy  m a


y
x

2


T

2
t
y
2
y


x


m
2
2
x
t
2
y
>>(mass density   m / x
…leads to the wave equation with

- wave function=transverse
displacement  x  t   y
2
2


1



- phase velocity
2
2
2
 x c t
T <-restoring force
c
 <-inertia
T

2
y
2
Solutions of 1-D Wave
Equation
 z ,t   C1 
f z  vt   C 2 g z  vt 





forward wave
backward wave
Consider 1 z ,t   f z  vt  f  ,
 / z  1
 1  1   1


z
 z

  z  vt
 / t  v
 1  1 


 v 1
t
 t

2
 2 1    1     1    1    1     1 

 
    v

 
  
2
2
t  t  t 
 
z
z  z  z    t
2
   1  
   1    2 1
2  1


 v
v




2
    t
 2
    z 
Solutions of 1-D Wave
Equation
1  2 1  2 1  2 1

 2
2
2
2
v t

z
 1 z ,t   f z  vt is a solution
the same reason, 2 z, t   g z  vt is a solution
 z ,t   C11 z ,t   C2 2 z ,t   C1 f z  vt  C2 g z  vt
is a solution too
any linear combination of solutions is also a solution
: superposition
Description of Traveling
Waves

waves traveling in the +z
direction
 z, t   f z  vt  cosz  vt
no change in shape
 z  vt, t    z,0
point P moving with time
  z  0, t  0   A cos 0  A
  z   , t  0
   z  5 / 4, t  T / 4 
   z  3 / 2, t  T / 2 
1-D Harmonic Traveling
Waves
 z, t   f z  vt 
  z 
  z , t   f   v  t  
  v 
  z 

 
/v = 2f / f = 2/= k : Angular
 z, t   
F t  kz

wave number
 z, t   F   t  
v
froward wave

1-D time-harmonic traveling waves propagating in
the +z direction
 z, t   A cost  kz
 z, t   A sin t  kz
1-D Harmonic Traveling
Waves

Complex representation of harmonic traveling waves
propagating in +z direction
ˆ z, t   Aˆ ei t kz   Aei t kz  
 A cost  kz  i   iA sin t  kz  i 
i
Aˆ  Aˆ eii  Aeii ,
if i  0
i : initial phase

 z, t   ReAei t kz    A cost  kz 

i t  kz 
 A sin t  kz
 z, t   ImAe
ˆ z, t   Aei t kz 

ˆ z, t  0  Aeikz

 If  z, t   Acost  kz    instead of  z, t   Acost  kz is a solution of wave equations
i
References


F. T. Ulaby, Fundamentals of Applied
Electromagnetics, Prentice Hall.
J. D. Cutnell, and K. W. Johnson, Physics, Wiley.