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Interference

The superposition of two or more waves resulting in a new wave pattern, demonstrating wave properties like superposition and constructive/destructive interference.
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The statement of the theorem

Let E1(r,t)\mathbf{E}_1(\mathbf{r}, t) and E2(r,t)\mathbf{E}_2(\mathbf{r}, t) be two monochromatic electromagnetic fields propagating in a linear, isotropic medium, satisfying the homogeneous wave equation: ∇2Ej−1v2∂2∂t2Ej=0\nabla^2 \mathbf{E}_j - \frac{1}{v^2} \frac{\partial^2}{\partial t^2} \mathbf{E}_j = 0 for j=1,2j=1, 2. Define the resultant field Etotal(r,t)\mathbf{E}_{total}(\mathbf{r}, t) via the principle of superposition: Etotal(r,t)=E1(r,t)+E2(r,t)\mathbf{E}_{total}(\mathbf{r}, t) = \mathbf{E}_1(\mathbf{r}, t) + \mathbf{E}_2(\mathbf{r}, t). Assume the fields can be represented by their complex amplitudes: Ej(r,t)=E0j(r)ei(kj⋅r−ωt)\mathbf{E}_j(\mathbf{r}, t) = \mathbf{E}_{0j}(\mathbf{r}) e^{i (\mathbf{k}_j \cdot \mathbf{r} - \omega t)}. The intensity II (proportional to the time-averaged Poynting vector magnitude) of the resultant field is given by: I(r)=12μ0c∣Etotal(r,t)∣2=I1(r)+I2(r)+2I1(r)I2(r)cos⁡(Δϕ(r))I(\mathbf{r}) = \frac{1}{2\mu_0 c} \left| \mathbf{E}_{total}(\mathbf{r}, t) \right|^2 = I_1(\mathbf{r}) + I_2(\mathbf{r}) + 2\sqrt{I_1(\mathbf{r}) I_2(\mathbf{r})} \cos(\Delta \phi(\mathbf{r})) where Ij(r)=12Re(E0j⋅E0j∗)I_j(\mathbf{r}) = \frac{1}{2} \text{Re}(\mathbf{E}_{0j} \cdot \mathbf{E}_{0j}^*) is the intensity of the individual waves, and Δϕ(r)=k1⋅r−k2⋅r+ϕ01−ϕ02\Delta \phi(\mathbf{r}) = \mathbf{k}_{1} \cdot \mathbf{r} - \mathbf{k}_{2} \cdot \mathbf{r} + \phi_{01} - \phi_{02} is the phase difference, determining the constructive (Δϕ=2πn\Delta \phi = 2\pi n) or destructive (Δϕ=π(2n+1)\Delta \phi = \pi (2n+1)) interference conditions.
Source: Wikipedia