**Mathematical modelisation.**

As according to the manufacturer of the clocks, the typical humidity sensitivity is:

[df/f]/% humidity = 1.E(-14)/%, the effect observed between two distant clocks (24 ns in 12 h), needs for example a differential step of variation of humidity of 55%, two times a day, during 178 days! Conclusion, humidity variations are not responsible of the persistent periodic phase shift observed.

About pressure, the manufacturer confirms that __ no__ measurable frequency change during pressure variations around 760 mm Hg has been observed.

When temperature is considered, the typical sensitivity around rom temperature is [df/f]/°C =0.25.E(-13)/°C and implies, for example, a differential step of rom temperature variation of 24°C, two times a day, during 178 days! More with a rom temperature maintained nearly constant around 20°C by the thermostats of the buildings. Conclusion, the possible temperature variations of the clocks may not be responsible of the periodic phase shift observed between distant clocks. All that, without to consider the thermal resistance of the housings of the clocks which smooths the possible temperature variations.

Finally, the typical magnetic sensitivity of [df/f]/Gauss=1.4 E(-13)/Gauss, needs, for example, differential steps of field induction variations of 4 gauss, two times a day, during 178 days. But the terrestrial magnetic induction in Belgium is only in the order of 0.2 Gauss and thus its variations are much less strong (except during a possible magnetic storm). About possible parasitic variable DC currents in the vicinity of the clocks, I have to remind that 4 Gauss needs a variation of 2000 amperes in a conductor at 1 m and thus is not the possible artifact.

Temperature, pressure, humidity and magnetic induction variations on the frequencies of the clocks have thus been completely neglected in the present experiment, whose goal was not to measure accurately the ether-wind velocity, but simply to detect it.

If the clocks are not responsible of the effect observed, the mathematical analysis of the set-up of the experiment (below) and the results lead to the conclusion that the ether-wind has been detected.

1) As the phase comparison in Paille street (one battery of the three A clocks) between the A1 and B1 clocks may be mathematically modelized (if a phase constant is not written)as:

where is the long term (178 days) fractional frequency difference between the A1 clock and B1 clock, the intrinsic instantaneous frequency instability of the A1 clock (or difference) relatively to its average and detected by the comparators connected to the A2 and A3 clocks in Paille street, the same in Marais street (the other battery of B clocks), the nominal frequency(5MHz) , t the absolute time, L[T(t)] the delay in the line fonction only of the temperature and thus of the time and the variation(fonction of the time) of this delay due to the ether wind of velocity v or due to an anisotropical change of frequency of the clocks (theoretically it is the case) .

2) And as in Marais street the modelisation is (**a phase constant has also been negleted**):

with the same second term because the two lines have the same length in the same underground places where the temperature is the same.

3) Thus, by substracting the Paille street phase shift fonction from the Marais's we obtain:

, **a phase shift which is absolutely independent of the temperature of the lines.**

4) As nearly all the days where the outputs of the comparators connected to the B and C clocks are straight lines we may be sure that the A clocks are stable and that the long term fractional frequency difference :E= 1.56.E(-12) is the fonction in the integral (because and are nil, we obtain: , the ether wind fonction which is represented below (divided by two) with an amplitude of 28 ns (peak-to-peak).

The phase constant has been adjusted to center the graphic (**thus the nil is not true**).

I have to remind here that the coaxial cable is orientated about in North-South direction from Marais to Paille Streets in Brussels.

A peak-to-peak amplitude of 24 ns has been obtained with:

.

It is more logic but a bit strange that a weaker amplitude has been obtained (-4ns) nevertheless the calculation is more complicated and not as rigorous to eliminate the possible effect of the temperature on the lines and may have produced an error.

I must say that today, I consider that the greater order velocity obtained (500 km/s) relatively to the MWB anisotropy velocity (360 km/s) may have been produced by a negative dispersion of velocity (around 5 MHz) in the coaxial cable that I have considered to be nil. To know that I will have to perform a new experiment to measure it.

It would be thus interesting to build or to chose a coaxial cable with the lower possible velocity and with the greater negative dispersion to obtain the best sensitivity for ether-wind detection.

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