In my physics class we were asked to derive Snell's law from Huygens principle in the context of this hypothetical situation:
There are plane waves moving through the ocean in a direction parallel to our x-axis. If I draw a line $x=c$, then I can see the state of the water (its oscillation) at any point and at any time along that line, then approximately re-create the original plane wave using $n$ point sources stretched between two points on the line $x=c$. If the original waves are parallel to $x=c$ then the waves created by each point source will be in phase with each other. But if the waves are moving at an angle $\theta$ relative to the $x$-axis, then the phase $\phi$ of the waves at each point source is not the same. I needed to find $\phi$.
After working with classmates I got the answer $\phi = k n \sin(\theta)$ and was able to use this to derive Snell's law. But I don't understand why. I was able to "work it backwards" to sort of understand why that solution works, but I want to know how to get there, and that is something that my classmates could not explain to me. Can someone please help me understand how to find $\phi$ as a function of $\theta$?
My backwards working of the solution:
Let $y'$ be the distance between crests along a cross-section parallel to the $y$-axis of the plane wave and $λ \le y' < \infty$.
$$k = 2\pi / \lambda$$
$$y' = \lambda / \sin(\theta)$$
$$\phi = k n \sin(\theta) = (2\pi / \lambda) n \sin(\theta) = (2\pi / y') n$$
As $y'$ increases towards infinity, the plane waves become more and more parallel to $x=c$, and so $\phi$ gets closer and closer to zero. As $y'$ decreases towards $\lambda$, $\phi$ increases towards its maximum value (though I don't know how to express/find its maximum value). So I can see that $\phi$ is inversely proportional to $y'$, and so also to $\sin(\theta)/k$. I understand that the wave is periodic, and multiplying by $2\pi$ is used often to convert to angular units, so that makes sense. And $n$ is the number of point sources used, so that it would be involved makes sense. But beyond that I am lost. How can I get to $\phi = k n \sin(\theta)$ working it "the right way around"?
I already tried asking classmates, reading our text books, and googling/searching on this site, but everything I find is either too advanced, too simple, or not related.