Question

a)The sum of two plane waves (which could correspond to probability amplitude functions) u₁(z,t) and u₂(z,t), both traveling in the positive z direction with slightly different frequencies and propagation constants and with equal amplitudes. The time and space variation of the two plane waves 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 intensity envelope 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 with electric 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 phase velocity for the sum of the two fields.

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

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 a function of z; and b) for a fixed value of z as a function of time.

Answer 1

The em wave equation can be written as;

E_x = E₀cos(wt – kz + φ₀)

According to the question;

Phase = 0, wt = 2*pi;

So the above equation is reduced to;

E_x = E₀cos(2π – kz )

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

Then the characteristics of E_x with respect to z (1 to 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 = E₀cos(wt + kz + φ₀)

Adding two waves with phase difference 180⁰.

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

        = 2 E₀ cos(φ/2)sin(kz-wt+ φ/2)

If φ = 180⁰; then E_result = 0; which means two wave interfere destructively and cancel each other

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

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