Huygen's principle: interference of many point sources

Christiaan Huygens was able to explain wave phenomena like interference and diffraction in terms point sources. A point source in two dimensions emits circular waves of the form,

$\large \frac{A}{\sqrt{r}} \cos\left(\frac{2\pi r}{\lambda}-\frac{2\pi t}{T} +\phi \right).$

Here $r$ is the distance from the point source, $t$ is the time, $\lambda$ is the wavelength, $T$ is the period, and $\phi$ is the phase. The amplitude of the waves decreases with distance like $1/\sqrt{r}$. Put $N=1$ into the form below and it will plot the waves generated but a point source at the left side of the image.

In three dimensions, the only difference is that the amplitude of the waves falls like $A/r$.

$\large \frac{A}{r} \cos\left(\frac{2\pi r}{\lambda}-\frac{2\pi t}{T} +\phi \right)$

The form below lets you place $N$ point sources in a vertical line at the left edge of the image. They all emit surface waves with the same amplitude, frequency, and phase. These waves interfere with each other and the amplitude is the sum of the individual waves.

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

Here $|\vec{r}-\vec{r}_i|$ is the distance from source $i$. The wavenumber $k$ is related to the wavelength $\lambda$ by $k=\frac{2\pi}{\lambda}$ and the angular frequency $\omega$ is related to the period $T$ by $\omega=\frac{2\pi}{T}$. The point sources are equally spaced in an interval $a$ which is indicated by the small white spots. Try making the wavelength larger and smaller than $a$. If the point sources are closely spaced compared to the wavelength, the diffraction pattern will approximate the single slit diffraction pattern.

Red is positive and blue is negative. Black regions have no wave amplitude.