ABSTRACT
The electromagnetic waves are transverse propagating in space, where the distance between two consecutive crests at a given time is called the wavelength, as illustrated in Figure 4.2. If an observer is fixed in space, two consecutive crests pass through that point, separated in time by the period T. The frequency ν is the number of crests passing in a unit time, which is given by the inverse of the period as follows:
ν = 1
T . (4.1)
If the crest waves propagate with a speed v, the frequency and the wavelength are related by
ln = v. (4.2)
Two additional parameters are defined to specify the wavelength and frequency of the electromagnetic wave. These are the angular frequency, ω, measured in radians per unit time
w p= 2 ν (4.3)
and the wave number k
k = 2p
l . (4.4)
Thus, Equation 4.2 can be written as
w k
v= . (4.5)
Let us assume that the wave propagates, starting at the origin (x = 0) at the time t = 0, along the x-axis, in the positive direction. The instantaneous value of the electric field E(x, t) at a given point x in space and at a given time t can be expressed by
E x,t A kx t( ) cos( ).= - +w f0 (4.6)
This may be called the “instantaneous amplitude,” but more frequently, the “disturbance.” The maximum value of the disturbance is A, and it is called the “amplitude.” The phase of the wave at the origin, at the time t = 0, is ϕ0. Thus, the phase ϕ of the wave at the point x and the time t is given by kx − ωt + ϕ0. Given the amplitude A and a certain phase, the disturbance can be calculated with this expression. The two parameters defining the disturbance are the amplitude and the phase. So, it is desirable to express the disturbance by a 2D vector, where the two components are defined by the amplitude and the phase. Thus, waves can be added as vectors and at the end we have both the final amplitude and the final phase. This representation is quite simple in terms of complex numbers, as follows:
E x,t kx t i kx t
x y i x y
( ) cos sin
cos ( , ) sin ( , )
= - +( ) + - +( )
= +
w f w f
f f
= +R x t iI x t( , ) ( , ).
