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Optical clocks have reached such an impressive accuracy and stability that the future redefinition of the second will be probably based on an optical transition. Consequently, building time scales based on optical clocks has become a key problem. Unfortunately, optical clocks are still laboratory prototypes and are not yet capable of long times of autonomous operation. It is hence critical to understand the impact of this limited optical clock availability on the generated time scale. In this work, after describing a simple and effective optical time scale algorithm, based on the steering of a flywheel oscillator towards the optical clock, we investigate in detail the impact of the limited availability of the optical clock on the performances of the steering algorithm and of the generated time scale through numerical simulations. In particular, we simulate a time scale generated by a hydrogen maser (with a flicker floor of 5.5 × 10−16) steered towards an optical clock, by considering six different scenarios for the availability of the latter, spanning from the ideal one, i.e. continuous operation of the optical clock, to the worst one, i.e. non-uniformly distributed frequency measurements with long unavailability periods. The results prove that the steering algorithm is robust and effective despite its very simple implementation, and it is capable of very good performances in all the considered scenarios, provided that the hydrogen maser behaves nominally. Specifically, they show that a time scale with an accuracy of a few hundreds of picoseconds can be easily realized in the ideal scenario, whereas in a more realistic scenario, with one measurement per week only, the time accuracy is nonetheless of a few nanoseconds, competing with the best time scales currently realized worldwide. The performances degradation due to a non-nominal maser behaviour is also discussed.

We consider the response of a finite string to white noise and obtain the exact time-dependent spectrum. The complete exact solution is obtained, that is, both the transient and steady-state solution. To define the time-varying spectrum we ensemble average the Wigner distribution. We obtain the exact solution by transforming the differential equation for the string into the phase space differential equation of time and frequency and solve it directly. We also obtain the exact solution by an impulse response method which gives a different form of the solution. Also, we obtain the time-dependent variance of the process at each position. Limiting cases for small and large times are obtained. As a special case we obtain the results of van Lear Jr and Uhlenbeck and Lyon. A numerical example is given and the results plotted.