StructureFunctionOfTheSineFunction

The structure function $\gamma(h)$ is defined as:

$$\begin{align*} \gamma(h) &= \text{Var}(f(t) - f(t + h)) \\ &= \left< ( f(t + h) - f(t) )^2 \right> \\ &= \frac{1}{2\pi} \int_{0}^{2\pi} ( f(t + h) - f(t) )^2 dt \end{align*}$$

For $f(t) = \sin(t)$, the difference $f(t + h) - f(t)$ is: $$\sin(t + h) - \sin(t) = 2 \cos\left(t + \frac{h}{2}\right) \sin\left(\frac{h}{2}\right)$$

Squaring this gives: $$\left( \sin(t + h) - \sin(t) \right)^2 = 4 \cos^2\left(t + \frac{h}{2}\right) \sin^2\left(\frac{h}{2}\right)$$

Taking the expectation value over $t$ and the integral form yield:

$$\begin{align*} \gamma(h) &= \left< 4 \cos^2\left(t + \frac{h}{2}\right) \sin^2\left(\frac{h}{2}\right) \right> \\ &= \frac{1}{2\pi} \int_{0}^{2\pi} 4 \cos^2\left(t + \frac{h}{2}\right) \sin^2\left(\frac{h}{2}\right) dt \\ &= 2 \sin^2\left(\frac{h}{2}\right) \end{align*}$$