This morning I was thinking about the next experiment and it struck me that I am interested in the phase and last time I plotted the amplitude of one signal versus the other, but hadn't done the same thing with the two phases. It was easy enough to do and the result surprised me:

Phase in receiver 1 versus phase in receiver 2.

There is some structure here that is worth exploring. Since the starting phase of each signal is arbitrary, I can add a constant phase to each signal and plot again. This just has the effect of moving the points together up and down, and back and forth, wrapping around the edges. With a little judicious choosing of these two constants we get the following equivalent plot:

Shifted phase in receiver 1 versus phase in receiver 2.

There's lots of scatter, but overall its pretty clear there is a 45 degree line through the origin that most of the points tend to cluster around. It didn't take too much thinking to realize that this is what you would expect if there was a constant phase difference between the two signals, while the phases themselves were varying - depending on the propagation conditions from moment to moment. Taking the difference between the two values I used to shift the original phases by to get this plot, I know that the phase difference in this case is 1.37 radians. Of course it could also be 1.37 + 2 pi radians, and so forth, but let's go with the 1.37 radians for the moment.

Why should there be a constant phase difference between the two signals from the two separated antennas? It took only a few moments to realize that this could be due to the angle of arrival of the signal relative to the line drawn between the two antennas. To make this concrete I went to Google Earth and drew in that line. With the Ruler Tool I can measure the separation of the antennas (23.6 meters) and the bearing of the joining line (120 degrees). Now suppose the signal from WWV is coming from somewhere towards the west - I'll superimpose a sketch of the wavefronts for that signal. Since the signal frequency is 15 MHz, its easy to calculate that the corresponding wavelength is almost exactly 20 meters, so the peaks of the wavefront are separated by 20 meters. If I've got my scaling right, the following picture should be approximately correct:

Antenna and signal geometry. Yellow circles are the two antennas with the ruler showing the separation and bearing. Magenta lines are wavefronts of signal coming from the west.

We need to calculate the angle "a" in the above picture from the geometry. That's simple enough with the given numbers - in this case the angle is about 11 degrees, which gives the wavefront bearing either 109 degrees or 289 degrees (there's no way to tell which way the wavefront is going).

The final step is to calculate the bearing from my location (42 25 36.0N, 71 09 47.6W) to the WWV 15 MHz antenna, which is kindly supplied at the NIST web page as (40 40 45.0N, 105 02 24.5W). A quick spherical geometry calculation of the great circle bearing from my location gives 277 degrees, which is 12 degrees different from the number that I calculated above. That's not too bad, though I'm not quite sure what sort of error bars to put on the calculation.

Tangled Spaces