Waves traveling through water don't look much like a sine wave. They are clearly periodic though so, I wonder, how can we model water waves with trigonometry?

In the simple model here, each particle moves in uniform circular motion. They are in sync but the radius of that circular motion decreases exponentially with depth and the phase is proportional to horizontal position. The exact coordinates of the particle moving around the center $(x_0,y_0)$ are $$ \begin{align} x(t) &= x_0 + A e^{\alpha y_0} \cos(\alpha x_0 - \beta t) \\ y(t) &= y_0 + A e^{\alpha y_0} \sin(\alpha x_0 - \beta t). \end{align} $$ Note that $A$ represents the radius of the circular motion at the surface, $\alpha$ is the reciprocal of the wave length, and $\beta$ is the reciprocal of the period. Horizontal mouse position on this page affects $\alpha$, vertical mouse position affects $A$, and you can click anywhere on the page to highlight two particles to emphasize their circular motion.