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The daytime range of an AM station does not increase as much as the square root of the power ratio. This is because of absorption of the groundwave by the earth. The square root rule would only apply if the earth were a good conductor. In urban areas, the earth is usually a poor conductor, and the groundwave absorption is high. The high end of the AM broadcast band has much more groundwave absorption than the low end. Most Part 15 AM operators operate near the high end of the band to maximize antenna efficiency.
Prof. Bill Everitt, author of “Communication Engineering,” McGraw-Hill, 1937, gave a rough rule of thumb that the range of an AM station increases as the fourth root (square root done twice) of the ratio of the power increase. Dr. Everitt was Professor of E.E. at U. of Illinois, Urbana, when I was a student in the Sixties. The rule gives a range increase of only 1.78 for a tenfold power increase. To get a range increase of 3.16, a power ratio of 100 is needed. This is ten watts! One of the contributors to this topic, Jim B, expresses dissatisfaction with increasing power from 100 mW to 1 W. Clearly, a power increase of this amount does not bring happiness.
We are talking about a 100mW range of 1 to 3 miles, which is not very likely unless the antenna is much longer than the 3 meters allowed by the rules. For an antenna that is legal, the range may be so short that there is hardly any groundwave absorption at all. Even in that case, the square root rule does not apply. This is because, at a short distance from the antenna, the near field intensity decreases rapidly with distance.