# a)The sum of two plane waves (

a)The sum of two plane waves (which could correspond toprobability amplitude functions) u_{1}(z,t) andu_{2}(z,t), both traveling in the positive z direction withslightly different frequencies and propagation constants and withequal amplitudes. The time and space variation of the two planewaves were given by:

u_{1}(z,t) = Y_{0}cos(wt – kz)

u_{2}(z,t) = Y_{0}cos ([w + Dw]t – [k +Dk]z)

We found that the sum of these two waves produce an intensityenvelope in time and space given by: u_{1}(z,t) +u_{2}(z,t) = 2Y_{0}cos(Dwt/2 – Dk z/2) cos([w +Dw/2]t – [k + Dk/2]z)

The velocity of the envelope is called the group velocity.

For this problem, consider two slightly different waves withelectric fields of the form:

v_{1}(z,t) = Y_{0}sin(wt – kz)

v_{2}(z,t) = Y_{0}sin([w + Dw]t – [k + Dk]z)

Derive analytically an expression for the group and phasevelocity for the sum of the two fields.

b) Choose appropriate values of w, Dw, k and Dk and plot the sumof

v_{1}(z,t) = sin(wt – kz) and v_{2}(z,t) =sin([w + Dw]t – [k + Dk]z) for a) a fixed value of time as afunction of z; and b) for a fixed value of z as a function oftime.

Answer:

The em wave equation can be written as;

E_{x} = E_{0}cos(wt – kz + φ_{0})

According to the question;

Phase = 0, wt = 2*pi;

So the above equation is reduced to;

E_{x} = E_{0}cos(2π – kz )

Assuming E_{0} = 1 and k (wave number ) = 1;

Then the characteristics of E_{x} with respect to z (1to 10) is shown in the figure below;

2) when the wave will be travelling in the negative z direction;the equation of the wave propagation can be written as;

E_{x} = E_{0}cos(wt + kz + φ_{0})

Adding two waves with phase difference 180^{0}.

E_{result} = E_{0}cos(wt – kz ) +E_{0}cos(wt – kz + φ)

= 2 E_{0}cos(φ/2)sin(kz-wt+ φ/2)

If φ = 180^{0}; then E_{result} = 0; which meanstwo wave interfere destructively and cancel each other

If φ = 2π rad; then E_{result} = – 2 E_{0}sin(kz-wt+ π) = 2 E_{0} sin(kz-wt)

So they interfere constructively and the amplitude of theresultant signal gets twice than original one.