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First thing you need to understand is that Japanese naval code JN-25 was already partially compromised before the events with fake water supply problem. In fact, Americans already knew there soon would've been operation at objective "AF". Rochefort suspected it was Midway, but he had to prove it. But as you can see, US had significant advantage.
Second, Japanese had no idea JN-25 was partially broken. This was their biggest weakness in whole affair. Had they known they would not have used it, especially for important messages concerning strategic operations.
Final piece of the puzzle, fake message about desalinization plant, is just icing on the cake. Japanese listening stations had to report intercepted US radio traffic...after all it was their job. Analyzing authenticity of the message was not. Those who did analyze the message, its authenticity and the source, decided to play safe and transmit message about additional desalinization equipment for objective AF.
Even if the Japanese did not re-transmit message about desalinization plant, identity of objective AF would likely be discovered trough usual traffic like weather reports (trick all sides used in WW2). For example, if Japanese transmitted that weather over objective AF is such and such, US could simply deduce possible locations by comparing their own weather reports over strategic locations and finding most likely. Granted, this may not be enough to win Midway battle, but at some point of time breaking of JN-25 code would eventually pay off for US.
Upvote:11
Firstly, there is nothing 'foolish' about it. Military planning is about contingencies. The Japanese listening stations had received a report, they forwarded that report using a code they believed to be secure, and the planners modified their plans for the attack on Midway to deal with the contingency that it may have been true.
The Japanese believed their main naval code, the Navy General Operational Code (dubbed JN-25 by the US codebreakers) was secure. So their listening posts simply reported what the Midway garrison was saying, using that secure code.
Of course, with the benefit of 20/20 hindsight, we know that JN-25 wasn't secure, and that Station Hypo in Hawaii was able to decrypt enough of the intercepted traffic to piece together much of the Japanese attack plan at Midway. The Japanese commanders didn't have that luxury.
There is an interesting Fact Sheet about JN-25 by Geoffrey Sinclair which provides a lot of background information about the code itself which you may find helpful if you wish to research further into this subject.
Having received the report, the Japanese commanders then sensibly modified their planning just in case. That's how contingency planning works. If there is a known contingency that you may have to deal with, you modify your plans accordingly.
In this case, if the report that the water desalination facilities at Midway were seriously damaged was true, they'd certainly need the extra desalination equipment; if it turned out that the report wasn't true, the extra equipment would do no harm and might still be useful anyway.