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Update 1: I've done some digging and I think this is related to signal coherence, namely, that I'm seeing a coherence time of ~3 σ, which is consistent with the definition where Ct=1/Δv where Δv is the bandwidth of a signal. I've approximated the bandwith of my ensemble of signals being ~3 σ which encapsulates 99% of possible period values.

I have a question that appears to be related to the beat frequencies of multiple oscillatory signals. Suppose I have $n$ number of sinusoidal waves, each of which follow $ sin(2 \pi \frac1\tau t+\phi)$ and imagine that we add a variability to the frequency, such that $\tau$~$N(\tau,\sigma^2)$.

At time $t=0$, all $n$ waves will be in-phase, following a divergence to out-phase and, depending on the number of waves and their difference in frequencies, a return to in-phase.

I've ran a couple of simulation in an attempt to gain some insight into how the number of oscillatory signals and the magnitude of noise in $\tau$ affects the capacity for the a population of signals to eventually reach a state in which they are in-phase.

I visualized the degree of "in-phase"ness of a population of signals by summing the signals' amplitudes (highest absolute value of the sum should indicate phase matching).

The sum of 2 signals with $\tau$~$N(\pi,.07\pi)$ enter image description here The sum of 100 signals with $\tau$~$N(\pi,.07\pi)$ enter image description here The sum of 500 signals with $\tau$~$N(\pi,.07\pi)$ enter image description here

I also varied $\tau's$ noise as:

The sum of 100 signals with $\tau$~$N(\pi,.14\pi)$ enter image description here The sum of 100 signals with $\tau$~$N(\pi,.3\pi)$ enter image description here

So my question is has anyone encountered any theory related to the apparent phenomenon where there seems to be an exponential convergence to a state of asynchronous that is dependent on the number of signals being considered? Also, the magnitude of the noise of the frequencies seems to dampen the approach to asynchrony.

I saw hints of an answer in this post which is why I've added the quantum mechanics tag as the knowledge of dealing with adding waves of probabilities is probably useful. Any help would be greatly appreciated.

Synthetic
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  • I've done some digging and I think this is related to signal coherence, namely, that I'm seeing a coherence time of ~3 $\sigma$, which is consistent with the definition where $C_t = 1/\Delta v$ where $\Delta v$ is the bandwidth of a signal. I've approximated the bandwith of my ensemble of signals being ~3 $\sigma$ which encapsulates 99% of possible period values. Anyone know of a good resource in deriving coherence time and how it relates to the number of signals being considered? – Synthetic Jul 01 '19 at 19:24

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