BIRDSALL: COHERENCE 



All this is said to formulate a multipath situation. Paths will appear 

 in pairs, with one path of each pair leaving the source at some de- 

 pression angle, and the other path of the pair leaving the source at 

 an elevation angle of almost equal magnitude. The difference in the 

 absolute value of the angles is quite small, of the order of .1 degree. 

 At the assumed long range there will be several such pairs, and the 

 reception will be the vector sum over all paths with their travel times 

 and losses; it will be a 250 Hz signal. 



Now let the source open range at 6 knots. Grossly speaking this 

 will cause a 0.5-Hz Doppler shift. Speaking more carefully, there will 

 be a 0.5*cos (angle) shift. Consider one pair of paths. If their 

 angles differ by 0.1 degree, their Doppler shifts will differ by 0.152 

 millihertz. That's not much, but therein lies the key number in this 

 model. Since the difference is so small no current receiver will 

 separate them, and their sum will appear as a single frequency, with 

 an apparent fade rate of 109 minutes. (Of course I am thinking about 

 much shorter observation times than 109 minutes.) The conclusion is 

 that the 'insignificant' differential Doppler will have almost no effect 

 on the measured frequency, but has a substantial effect on the measured 

 amplitude. 



There will be a number of ray-path pairs. For simplicity assume 

 that a hypothetical analyzer can isolate two pairs in one narrow filter 

 with relative frequencies and amplitudes as listed below: 



The amplitude, linear with pressure, will show rapid fluctuations 

 with a period of 100 seconds separating deep fades about every 13 

 minutes . A phase tracker shows the instantaneous frequency of the 

 reception fluctuating wildly at the 100-second rate, but using the 



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