Continuity mistake: When they approached the French plantation both doors of the demolished building are open. Later as Captain Willard walks through them, the door on the right is closed. (01:55:35)
Continuity mistake: In one shot we see the helicopters firing on the huts along the Vin Din Dop beach and the four missiles go directly into the beach nowhere near the huts. In the next shot the middle hut blows up. (00:40:00)
Visible crew/equipment: Before we see Chef with his Bunny, the camera pans along from the rear of the helicopter to the front. First we see the reflected studio lights, then we see a great reflection of a cameraman with his equipment under a hood in the window of the copter. (01:27:50)
Continuity mistake: As Dennis Hopper leads Captain Willard up the steps we see the wall with "Apocalypse Now" written in white. In the previous shot there were a lot of sand bags there, now there is just a big boulder. (02:31:35)
Continuity mistake: When Francis Ford Coppola appears in the movie as a TV director, he asks the captain to, "Just keep on going," and a small boy is playing on a black Agent Orange drum in the background. If you watch the labeling on the drum you can see that it's been moved between shots. (00:25:50)
Revealing mistake: Dennis Hopper says to Captain Willard, "He said 'If you take my picture again, I'm gonna kill you', and he meant it". Look closely at the dead bodies behind them because they're all moving. (02:32:05)
Continuity mistake: The day after the USO show at Hau Phat, the boat leaves and there are five men working on the stage. In the next shot there are three. (01:11:35)
Visible crew/equipment: If you keep your eyes on the right side of the screen before Colonel Kilgore gives the injured VC soldier a drink you can see a crew member's hand holding a light meter pass quickly in and out of the shot. (00:29:20)
Continuity mistake: Captain Willard says, "Whatever was going to happen, it wasn't gonna be the way they called it back in Nha Trang". Note the light pole in the center of the boat, the second blue light from the top is shot out however there hasn't been any battles since the French plantation scene when all four blue lights were intact. (02:21:25)
Audio problem: On the boat while reading Kurtz' file Captain Willard narrates to himself, "dear son, I am afraid that both you and your mother well have been worried at not hearing from me during the past weeks, but my situation here has become a difficult one". At this point they show the actual letter that he's reading from and the text of the letter is not the same as what we are hearing. (01:20:40)
Visible crew/equipment: The boom mike reflection can be seen in the pilot's side window of the helicopter as Captain Willard walks up to Kilgore. (00:26:55)
Continuity mistake: In a close-up of a soldier on a helicopter gunship shooting down at the people of Vin Din Dop below we can see the bridge in the background and it's already been blown up. This doesn't actually happen till later in the scene. (00:40:05)
Continuity mistake: In the boat Captain Willard reads more about Colonel Kurtz' and fingers through his three Green Beret applications. Right after the captain says, "But when he threatened to resign, they gave it to him," he stops on a prop sheet that had the real names and dates erased and new names and dates typed in using a different typewriter that has darker numbers and letters. (00:01:25 - 01:01:25)
Continuity mistake: In the scene where they stop the junker boat that they think might be smuggling supplies, as you watch the many scene angles, you can tell that the sun is at different points at different times. Where Willard is sitting at the side of the boat the sun is nearly set, and at one point is below the horizon. At another time, you can tell, the sun is several degrees above that, and in one angle, it is practically noon. Also, the boats change angles, toward the shoreline, possibly to shoot a scene from a better light angle. (01:34:08)
Continuity mistake: The vehicle on the bridge that is attacked by the helicopter has three men in it. In the first shot of it exploding one man falls to the right and two men fall to the left. In the next shot as it falls off the bridge all three men fall to the left of the vehicle. (00:41:45)
Continuity mistake: When they leave the Do Lung Bridge the flag is visible on the roof of the boat, but several seconds later at the beginning of the next scene, it's gone. (01:48:45)
Continuity mistake: During the arrow attack as Captain Willard grabs Chief Phillips to try and make him stop shooting, the sun can be seen setting over his shoulder. Several other shots in this scene, like the next one, were filmed in the mid afternoon with the sun overhead. (02:22:20)
Continuity mistake: Captain Willard is wearing green, tan and black camouflage trousers on the PBR when they spot Kilgore's air-cav unit. They immediately proceed toward shore and as he jumps off of the boat he's wearing different camouflage trousers, which are brown and green. (00:24:15 - 00:25:15)
Audio problem: After Captain Willard says, "They're just little sticks. They're just trying to scare us", Chief Phillips has some prime words that he shouts at the captain which are edited out. (02:22:05)
Continuity mistake: As Colonel Kilgore gets into his helicopter the trumpet player is standing right there next to it blowing charge. In the next shot as the helicopter takes off he's not there. (00:34:50)
Answer: It's not any specific song; it's just the kind of generic piano music you'd hear at dinner at a high-class restaurant of the era. Think of scenes in movies, films, etc. set or made in the '50s and '60s, where the characters go to a nice dinner and there's someone at the piano playing unobtrusive music to accompany the food/conversation.