Mr. Maloney soon requested permission to cruise along the Rockaways shore and sounded quite relaxed, even uncharacteristically so by pilot standards. Twice, upon receiving authorization to fly below 500 feet along the shoreline, the pilot replied, “Roger,” but stretched it out and pronounced it, “rod-jaaah,” according to the recording. He also told the controller, who was nearby at Kennedy Airport monitoring outgoing flights, “Just let us know if we’re up in your grill, you know?”

At some point, Ms. Protter became sick, and Mr. Maloney made an unusual request to the controller. He said he wanted to bring a “pastor” to Kennedy “who’s doing some medical mission work. Where would I drop them off at your airport?”

“That would be up to the Port Authority,” the controller replied, referring to the Port Authority of New York and New Jersey, which runs the airport. Whether Mr. Maloney contacted the Port Authority on another frequency, or what he meant by “medical mission,” was unclear. Mr. Maloney’s family declined to comment on the matter on Tuesday, and his passengers did not respond to voice and e-mail messages.

A short while later, Mr. Maloney presented his next idea to the controller.

“Hey tower, I’ve got a question for you,” he said. “This might be crazy, but are we allowed to land on the beach?”

“I don’t think so, unless it was an emergency,” the controller replied. The recording was posted to the Web by LiveATC.net, which tracks air communications.