So, we know where the intensity is maximum and where it's zero. How does the intensity change in between these points?

Pick a random point on the screen. Let's say the light arriving from the first source has an electric field which varies as

E₁ = E_o sin(ωt).

The electric field from the second source is almost the same, just with a phase shift:

E₂ = E_o sin(ωt + φ).

Thus the total electric field at our point is:

E = E₁ + E₂ = E_o [ sin(ωt) + sin(ωt + φ) ]

Use the trig. identity sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]

where A = ωt + φ and B = ωt

This gives E = 2 E_o cos(φ/2) sin(ωt + φ/2)

The intensity of the wave is proportional to E².

Therefore I α 4 E_o² cos²(φ/2) sin²(ωt + φ/2)

Averaging over time using the fact that the average value of sin²(θ) = 1/2 gives:

I_av α 2 E_o² cos²(φ/2) or I_av = I_max cos²(φ/2)

The phase difference depends on the path length difference. When the path length difference is one wavelength, for instance, what is the phase difference between the waves?

When δ = λ the phase φ = 2π.

This gives φ/2π = δ/λ, so:

φ = 2pd/λ = 2πd sin(θ)/λ

Thus our expression for the average intensity is:

I_av = I_max cos² (φ /2) = I_max cos² (πd sin(θ)/λ)

For small angles I_av = I_max cos² (πd y/λL)

where y is the distance along the screen measured from the center.