Are there rigorous mathematical definitions for these waves? My friend linked this .gif to me tonight, and asked me if I knew of any equations that might model these bottom two waves (the blue and green waves). Unfortunately, I am not far enough in my education to recognize if any such model exists. Are these waves modeled after some equation, or is this just some piece of eye candy?


So, for the fist one obviously, it's sine function.
$$
h(t) = r \sin \omega t
$$
As for the second one,
$$
h(t) = \frac a2 \cdot \left \{ \begin{array}{lcc}
\tan \left (-\frac \pi 4 + \omega t \right )& \text{if} & 0 \le t \le \frac T4 \\
1 & \text{if} & \frac T4 \le t \le \frac T2 \\
\tan \left ( \frac {3\pi}4 + \omega t \right ) & \text{if} & \frac T2 \le t \le \frac {3T}4 \\
-1 & \text{if} & \frac {3T}4 \le t \le T
\end{array}\right .
$$
For the last one,
$$
h(t) = a \cdot \left \{ \begin{array}{lcc}
\frac {\sqrt 3}2 \tan \left ( -\frac \pi 6 + \omega t\right) & \text{if} & 0 \le t \le \frac T6 \\
\frac 12 + \frac 12 \left [\frac 12 + \frac {\sqrt 3}2 \tan \left( -\frac \pi 6 + \omega t - \frac \pi 3\right) \right ] & \text{if} & \frac T6 \le t \le \frac T3 \\
1 - \frac 12 \left [\frac 12 + \frac {\sqrt 3}2 \tan \left( -\frac \pi 6 + \omega t - \frac {2\pi} 3\right) \right ] & \text{if} & \frac T3 \le t \le \frac T2 \\
\frac {\sqrt 3}2 \tan \left( \pi - \omega t + \frac \pi 6 \right) & \text{if} & \frac T2 \le t \le \frac {2T}3 \\
-\frac 12 - \frac 12 \left [ \frac 12 + \frac {\sqrt 3}2 \tan \left ( -\frac \pi 6 + \omega t - \frac {4 \pi}3 \right )\right ] & \text{if} & \frac {2T}3 \le t \le \frac {5T}6 \\
-1 + \frac 12 \left [ \frac 12 + \frac {\sqrt 3}2 \tan \left ( -\frac \pi 6 + \omega t - \frac {5 \pi}3\right )\right ] & \text{if} & \frac {5T}6 \le t \le T
\end{array}\right .
$$
PS: For last two cases $a$ is a square or hexagon side. $T = \frac {2\pi}\omega$
Initial time angles are chosen according to this

