Despite all of the research that I did for that thesis and in the studies I did both before and afterward, it has never been clear to me just how far WLW was heard on a regular basis when they were 500,000 watts. I've heard that the daytime signal was so great that the station could be clearly received many hundreds of miles away. I've even heard that it could also be received in downtown Los Angeles. I do recall being told that even if a radio station increases its power ten times over what it was (50,000 watts to 500,000 watts) doesn't mean it can be heard ten times farther. Of course, suffice to say during that 1934-1939 period, WLW covered quite a few miles. It might be noted that even after the power was restored to 50,000 watts, WLW's 700 k.c. frequency was on a clear channel.
To double the coverage, a four time increase in power is needed. So WLW's 50 kw coverage would be doubled with 200 kw, and then increased by another roughly 50% by going to 500 kw.
But there are other factors.
Back in the 30's there was far less man-made noise. And those old tube radios were actually very good. So a much weaker signal could be heard farther away than in the more recent years. That goes for both day and night reception.
The AM band was not very crowded... about a fifth of the number of stations there are today. So there was far less local interference from other stations. At night, stations like WSM with "just" 50 kw got mail from all of the 48 states... just not as much from Washington as from Georgia, but mail came from all over the nation.
And, whether today or back then, wide coverage stations really don't have "perfect circle" coverage. Difference in ground conductivity favored certain directions, and cut into the signal in others. WLW apparently did very well in Indiana and Illinois, but the lower conductivity of Kentucky and West Virgina and western PA reduced the effective distances to the east and south.
At night, conductivity is not the relevant factor for skywave. On a clear frequency, with no interference on 690 or 710, the signal was viable for great distance. But around Toronto, New York, Seattle and Los Angeles, those adjacent channels likely made reception difficult.
WLWs own daytime coverage map showed a "good signal" up into southern Michigan, down to the TN border and comparable distance to the west and east.
Important to remember is that in the Golden Age of radio, nights were prime time... unlike today when many stations automate nights and can't sell much advertising there. That is when the big network shows were on and where the money was made. And at night, WLW was truly "the nation's station".
For those unfamiliar, here is a booklet about the station and its 500,000 watt transmitter:
https://www.americanradiohistory.com/Archive-Station-Albums/WLW_Brochure-1935-LR.pdf
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