Read the equation
A pure sine has zero average over a whole number of periods and zero limiting time average over arbitrarily long intervals. The averaging interval and convention matter for a finite sample. Multiplying the sine by a changing, correlated amplitude can give a product with a nonzero average.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| golden | Frequency multiplier φ |
| t | Time |
How it works
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Assumptions & limits
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Original context
Read the surrounding source passage
\begin{equation}
\alpha(E^* - C^*) + \beta \cdot L^* \cdot \sin(\golden t) \cdot \golden = 0
\end{equation}
Time-averaging eliminates the oscillatory term (since $\langle \sin(\golden t) \rangle = 0$), yielding $E^* = C^*$.
Similarly, $dE/dt = 0$ at equilibrium gives:
\begin{equation}
-\gamma C^* + I^* \cdot (1 + \cos(\golden t \cdot \fib{n})) = 0
\end{equation}
The cosine term averages to zero for large Fibonacci numbers, giving $C^* = I^*/\gamma$.Exact original notation
\langle \sin(\golden t) \rangle = 0