The Wikipedia page on Joseph Rochefort states:
One of the Station HYPO staff, Jasper Holmes, had the idea of faking a failure of the water supply on Midway Island. He suggested using an unencrypted emergency warning in the hope of provoking a Japanese response, thus establishing whether Midway was a target. Rochefort took the idea to Layton, who put it to Nimitz. Nimitz approved, and the garrison commander was told by submarine cable to immediately radio in [1.] "plain-language" an emergency request for water as an explosion in the water desalination system meant that they had only enough water for two weeks. [2.] An apparently "follow-up" report was to be made in one of the strip-cipher code systems that the Japanese were known to have captured on Wake. As the plan was to convince Washington, Rochefort tactfully let Fleet Radio Unit, Melbourne (FRUMEL) notify the main objects of the deception (Washington) of the Japanese message by reporting a message from the AF Air Unit saying that they had only enough water for two weeks: "This will confirm identity of AF". Rochefort then sent a reminder on Friday. 
The Japanese took the bait. Within hours they broadcast instructions to load additional water desalination equipment, confirming Rochefort's analysis.[page needed] Layton notes the instructions also "produced an unexpected bonus". They revealed the assault was to come before mid-June.
I don't understand how the Japanese believed the US so foolishly?
Why would they reckon that the US would broadcast a true emergency request on water unecrypted? It doesn't make sense for them to broadcast a weakness unencrypted. In a true emergency, wouldn't the US summon for help encryptedly, even to non-US allies like Australia and New Zealand?
Why'd they fall for the "follow-up"? The US knew that the Japanese captured this "strip-cipher code systems" on Wake. Thus why would the Japanese reckon that the US would use it unquestionably in good faith?