a)The sum of two plane waves (

a)The sum of two plane waves (which could correspond toprobability amplitude functions) u1(z,t) andu2(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:

u1(z,t) = Y0cos(wt – kz)

u2(z,t) = Y0cos ([w + Dw]t – [k +Dk]z)

We found that the sum of these two waves produce an intensityenvelope in time and space given by: u1(z,t) +u2(z,t) = 2Y0cos(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:

v1(z,t) = Y0sin(wt – kz)

v2(z,t) = Y0sin([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

v1(z,t) = sin(wt – kz) and v2(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;

Ex = E0cos(wt – kz + φ0)

According to the question;

Phase = 0, wt = 2*pi;

So the above equation is reduced to;

Ex = E0cos(2π – kz )

Assuming E0 = 1 and k (wave number ) = 1;

Then the characteristics of Ex 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;

Ex = E0cos(wt + kz + φ0)

Adding two waves with phase difference 1800.

Eresult = E0cos(wt – kz ) +E0cos(wt – kz + φ)

        = 2 E0cos(φ/2)sin(kz-wt+ φ/2)

If φ = 1800; then Eresult = 0; which meanstwo wave interfere destructively and cancel each other

If φ = 2π rad; then Eresult = – 2 E0sin(kz-wt+ π) = 2 E0 sin(kz-wt)

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


 
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