Telecom/Physics of travelling wave

1. One-Dimensional Travelling Wave
 The amplitude/value of a travelling wave is a
function of both distance and time.
 Consider a moving pulse y = f(x) where y is the
transverse displacement/disturbance of a
pulse/wave from equilibrium.
 If the pulse is moved to the right, then x
becomes x-a, where a is a positive number.
 Thus y=f(x-a)
 If v is the relative velocity of the wave in the
+x direction, then a=vt and y=f(x-vt)
f(x)
f(x-3)
f(x-2)
f(x-1)
x
0 1 2 3

2. One-dimensional Travelling Wave
So f(x-vt) represents a rightward, or forward,
propagating/ travelling wave.
Similarly, f(x+vt) represents a leftward or backward,
propagating wave.
The function f depends on the nature/shape of the
signal.
For example, f can be:
oy= A sin(k[x-vt]) i.e travelling periodic function in +x direction
oy= ek(x+vt)
i.e travelling -x direction

3. Harmonic Wave Function
 Harmonic waves are smooth patterns that repeat endlessly.
 They involve the sine and cosine functions:
• y= A sin(k[x±vt]) or y= A cos(k[x±vt])
 If we consider spatial variations of a wave at a fixed time t then
the repetitive spatial unit of the wave is the wavelength, .
 Thus,y= A sin(k[(x+ )+vt]) = A sin[k(x+ vt)+ 2]
• Recall that sin x = sin (x + 2), thus k  = 2
• k =
 k is called the propagation constant.

4. Harmonic Wave Function
 Alternatively, if we consider time variations of a wave at a
fixed point x along the wave, then the repetitive temporal
unit of the wave is the period, T.
 Thus,y= A sin(k[x + v(t+T)]) = A sin[k(x+ vt)+ 2]
y= A sin(kx + kvt + kvT) = A sin[kx+ kvt+ 2]
Therefore, k = and
•

5. One Dimensional Wave Equation
 A wave equation is a real partial differential equation in which the
second derivative with respect to position x is proportional to the
second derivative with respect to time t.
 Hence, the wave equation has the general form;
where is the velocity of the wave

6. General Solution to the Wave Equation
 The wave equation has the simple solution:
 In order to show that f(x±vt) is a general solution to the wave
equation,
• let f(x±vt) = f(u) , i.e. u = x±vt
( , ) ( v )
f x t f x t
 
where f (x,t) can be any twice-differentiable function.

7. General Solution to the Wave Equation
and
Using the chain rule:
and
Thus Þ and Þ
Substituting into the wave equation:
1
u
x



v
u
t



f f u
x u x
  

  
f f u
t u t
  

  
f f
x u
 

 
v
f f
t u
 

 
2 2
2
2 2
v
f f
t u
 

 
2 2
2 2
f f
x u
 

 
2 2 2 2
2
2 2 2 2 2 2
1 1
v 0
v v
f f f f
x t u u
 
   
   
 
   
 

8. Maxwell’s Equations
Maxwell's Equations are a set of 4 equations that
describe the world of electromagnetics.
The equations explain how electric and magnetic
fields are generated by charges, currents, and
changes of the fields.

9. Maxwell’s Equations

10. Maxwell’s Equations
 In free space, the volume charge density (ρ) = 0 and
conduction current density (J1) = 0 (since  = 0 ).
 The differential forms of Faraday and Ampere’s laws become:
and
where , c is the speed of light in free space= 3*108
m/s.
and are permeability and permittivity of free space
respectively.
o o
E
B
t
 

 

B
E
t

 


11. The Curl Operator
 If × F is, for F composed of [Fx, Fy, Fz]:
∇
where i, j, and k are the unit vectors
for
the x-, y-, and z-axes,
respectively.
then

12. Moving Fields
 If the electric and magnetic fields are constrained to the y
and z directions, respectively, and they are both functions of
x and t only, then the result will be a linearly polarized plane
wave travelling in the x direction at the speed of light c.

13. Moving Fields
 This means that the spatial variation of the electric field gives
rise to a time-varying magnetic field and vice versa.
Using Faraday’s Law,
B
E
t

 

But
Faraday’s Law becomes …….. Equation 1

14. Moving Fields
 This means that the spatial variation of the magnetic field
gives rise to a time-varying electric field and vice versa.
Using Ampere-Maxwell’s Law ,
But
Ampere-Maxwell’ Law becomes
o o
E
B
t
 

 

…….. Equation 2

15. Electromagnetic Wave Equation
 Taking the partial derivative of equation 1 with respect to x
and combining with the results from equation 2:
 Therefore,
 Similarly,

16. Electromagnetic Wave Equation
 In one dimension, the electric field part becomes:
, recall that
• Thus which is a wave equation
• As shown earlier example solution to the wave equation
could be :
2 2
2 2
y y
o o
E E
x t
 
 

 
𝜇𝑜 𝜀𝑜=
1
𝑐
2

17. Exercise
1. Show that the expression, is a solution to a wave
equation.
2. A plane harmonic infrared wave travelling through a
transparent medium is given by
Assuming that all values are in SI units, determine:
i. Refractive index of the transparent medium
ii. Wavelength of the disturbance in vacuum
iii. Refractive index