x = \frac{a}{2} sin(θ)

zeroes x = \frac{1}{2} λ = \frac{a}{2} sin(θ)

sin(θ) = \frac{λ}{a}

sin(θ) ≃ \frac{y}{L}  ⇒  y = 1 \frac{Lλ}{a}

x = \frac{a}{4} sin(θ)

sin(θ) = \frac{2λ}{a}

sin(θ) ≃ \frac{y}{L}  ⇒  y = 2 \frac{Lλ}{a}

y = M \frac{Lλ}{a} , M = 1, 2, 3, ...

x = dz sin(θ)

\frac{x}{λ} = \frac{dz}{λ} sin(θ)

\frac{2πx}{λ} = \frac{2πdz}{λ} sin(θ)

A_{m} = \frac{A}{a/dz} = A (\frac{dz}{a} )

y_{1} = A_{m} \frac{1}{r_{1}} sin(kr_{1} − ωt)

y_{2} = A_{m} \frac{1}{r_{2}} sin(kr_{2} − ωt)

y_{1} = \frac{A_{m}}{r_{avg}} sin(kr_{1} − ωt)

y_{2} = \frac{A_{m}}{r_{avg}} sin(kr_{2} − ωt)

phase offset = \frac{2π sin(θ)}{λ} dz

y_{1} = \frac{A_{m}}{r_{avg}} sin (0 − ωt)

y_{2} = \frac{A_{m}}{r_{avg}} sin (\frac{2π sin(θ)}{λ}z − ωt)

sum of all waves = ∫_{z=0}^{z=a} \frac{A_{m}}{r_{avg}} sin (\frac{2π sin(θ)}{λ}z − ωt)

sum of all waves = ∫_{z=0}^{z=a} (\frac{A dz}{a} ) \frac{1}{r_{avg}} sin (\frac{2π sin(θ)}{λ}z − ωt)

sum of all waves = \frac{A}{r_{avg}} ∫_{z=0}^{z=a} \frac{1}{a} sin (\frac{2π sin(θ)}{λ}z − ωt) dz

sum = \frac{A}{r_{avg}} \{−\frac{1}{a} \frac{λ}{2π sin(θ)} cos (\frac{2π sin(θ)}{λ}a − ωt) + \frac{1}{a} \frac{λ}{2π sin(θ)} cos (0 − ωt)\}

β = \frac{2π sin(θ)a}{λ}

sum = \frac{A}{r_{avg}} \{−\frac{1}{β} cos (β − ωt) + \frac{1}{β} cos (0 − ωt)\}

cos(P) − cos(Q) = −2 sin (\frac{P + Q}{2}) sin (\frac{P − Q}{2})

sum = \frac{A}{r_{avg}} \{\frac{2}{β} sin (\frac{β − 2ωt}{2}) sin (\frac{β}{2} )\}

intensity = amplitude^{2} = \frac{A^{2}}{r_{avg}^{2}} \{[sin (\frac{β − 2ωt}{2})]^{2} (\frac{sin(\frac{β}{2} )}{\frac{β}{2}})^{2}\}

time − averaged intensity = \frac{A^{2}}{r_{avg}^{2}} (\frac{sin(\frac{β}{2} )}{\frac{β}{2}})^{2}

sin(θ) ≃ \frac{y}{L}

β = \frac{2πa}{λ} \frac{y}{L}

y_{sum} = A \frac{1}{r_{1}} sin(kr_{1} − ωt) + A \frac{1}{r_{2}} sin(kr_{2} − ωt) + A \frac{1}{r_{3}} sin(kr_{3} − ωt)

neighbor phase offset ϕ = \frac{2πd sin(θ)}{λ} (radians)

y_{sum} = (\frac{A}{r_{avg}} ) [sin(0 − ωt) + sin(ϕ − ωt) + sin(2ϕ − ωt)]

y_{sum} = (\frac{A}{r_{avg}} ) 2 sin (\frac{2ϕ − 2ωt}{2}) cos (\frac{2ϕ}{2}) + (\frac{A}{r_{avg}} ) sin(ϕ − ωt)

y_{sum} = (\frac{A}{r_{avg}} ) [2 cos(ϕ) + 1] sin(ϕ − ωt)

intensity = (amplitude)^{2} = (\frac{A}{r_{avg}} )^{2} [2 cos(ϕ) + 1]^{2}sin(ϕ − ωt)^{2}

time − averaged intensity = \frac{1}{2} (\frac{A}{r_{avg}} )^{2} [2 cos(ϕ) + 1]^{2}

sum of N slits ∝ A [\frac{sin(Nϕ/2)}{sin(ϕ/2)}]

intensity of N slits ∝ A^{2} [\frac{sin(Nϕ/2)}{sin(ϕ/2)}]^{2}
d sin(θ) = λ
d sin(θ) = λ + ϵ

10 [d sin(θ)] = 10λ + 10ϵ

y_{sum} = (\frac{A}{r_{avg}} ) [sin(0 − ωt) + sin(ϕ − ωt)]

y_{sum} = (\frac{A}{r_{avg}} ) [sin(0 − ωt) + sin(ϕ − ωt) + sin(2ϕ − ωt) + sin(3ϕ − ωt)]

y_{sum} = A [\frac{sin(Nϕ/2)}{sin(ϕ/2)}]