y_{1}(r_{1}, t) = A (\frac{1}{r_{1}} ) sin(kr_{1} − ωt)

y_{2}(r_{2}, t) = A (\frac{1}{r_{2}} ) sin(kr_{2} − ωt)

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

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

sin(α) + sin(β) = 2 sin (\frac{α + β}{2} ) cos (\frac{α − β}{2} )

y_{sum} ≃ \frac{2A}{r_{avg}} \{sin (\frac{kr_{1} − ωt + kr_{2} − ωt}{2}) cos (\frac{kr_{1} − ωt − kr_{2} + ωt}{2})\}

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

cos (\frac{k[r_{1} − r_{2}]}{2}) = 0

\frac{k[r_{1} − r_{2}]}{2} = (2N − 1) \frac{π}{2} , N = 1, 2, 3, ...

k = \frac{2π}{λ}

sum = zero when |r_{1} − r_{2}| = \frac{(2N − 1)}{2} λ

cos (\frac{k[r_{1} − r_{2}]}{2}) = 1

sum = max when |r_{1} − r_{2}| = N λ

x = d sin(θ)

zeroes x = \frac{2N − 1}{2} λ = d sin(θ)

if d << L , sin(θ) = \frac{y}{r_{avg}} ≃ \frac{y}{L}

zeroes \frac{y d}{L} = \frac{2N − 1}{2} λ