Intensity of the interference pattern of many point sources

Surface waves generated by $N$ point sources have the form,

$\large z = \sum\limits_{j=1}^N \frac{A}{\sqrt{|\vec{r}-\vec{r}_j|}}\cos (k|\vec{r}-\vec{r}_j|-\omega t )$ cm

Here $|\vec{r}-\vec{r}_j|$ is the distance from source $j$. The wavenumber $k$ is related to the wavelenth $\lambda$ by $k=\frac{2\pi}{\lambda}$ and the angular frequency $\omega$ is related to the period $T$ by $\omega=\frac{2\pi}{T}$. Consider the case where the point sources are equally spaced in an interval $a$ which is indicated by the small blue spots on the left.

If you focus on one point, it executes simple harmonic motion with a certain amplitude. Simple harmonic motion is the real part of circular motion in the complex plane. Written in complex form, the oscillations are,

$ \large \sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j|-\omega t +\phi_j)} = \left( \sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j| +\phi_j)}\right)e^{-i\omega t}.$

The right side of this equation describes a phasor that moves in a circle in the complex plane with an amplitude $A$,

$\large A= \sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j| +\phi_j)}.$

The intensity is the square of the amplitude,

$\large I \propto A^*A = \left(\sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} \cos(k|\vec{r}-\vec{r}_j|+\phi_j)\right)^2+\left(\sum \limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} \sin(k|\vec{r}-\vec{r}_j|+\phi_j)\right)^2.$

The form below plots the logarithm of the intensity of the waves as a function of the position for $A_j=1$ and $\phi_j=0$.

There is a red point on the intensity pattern. The coordinates of this point are $P_x$ and $P_y$. To the left of the intensity pattern is a representation in the complex plane of the harmonic oscillations at the red point. The blue phasors represent the harmonic motion caused by the waves traveling from the sources. The red phasor is the sum of the blue phasors. The real part of the red phasor is the motion observed at the red point.