That sound is so close to the tie fighter sound that I had to refresh myself on how it was made. According to this article on Popular Mechanics, the source isn’t this film:

With their flat-panel wings, the Empire’s TIE Fighters seem to be ferociously cutting through space, the noise they emanate resembling a high-tech buzzsaw. It’s a sound that triggers memories of the wailing dragons or pterodactyls we’ve seen in other movies—and like those ancient beasts, the TIE Fighter is a ceaseless killing machine.

Joe Johnston, who won an Academy Award for the effects work on Raiders of the Lost Ark and later directed such films as Captain America: The First Avenger, drew storyboards for A New Hope, later recalling the inspiration behind the TIE Fighters. “In World War II the super dive-bombers had an artificially created siren wail created by air ducts,” he once said. “They didn’t serve any purpose except to create this noise, which would terrify people. It was intended that the TIE should achieve the same effect with just a menacing appearance.”

But Burtt figured out how to make them sound just as frightening. He turned to The Roots of Heaven, a 1958 adventure film directed by John Huston, which starred Errol Flynn as part of a team trying to save African elephants from extinction. Burtt sampled the movie’s elephant noises and slowed them down, but then he hit upon the idea of mixing them with the sound of cars on wet pavement.

“Swoosh, the car would come by, and you heard this car plowing through the water,” Burt later recalled. “I took that sound still thinking that I was making a laser of some kind.” But when he tried the combination of elephants and cars with the TIE Fighter footage, the rest of the Star Wars brain trust flipped for it. “I’d really put it in because I had no other alternative,” Burtt admitted, “but it got great reviews, so naturally it became the sound of the TIE Fighters.”