Continuity mistake: At dinner, the Frenchmen sitting to the right of Captain Willard says, "Why don't you Americans learn from us, from our mistakes?" When he says this there's a shadow falling on his face but in the next shot his face is lit up again by the sun. (02:10:10)
Continuity mistake: After the little French boy kisses Hubert de Marais he walks away and we watch from behind as Captain Willard's head turns to the right as he looks at him. In the next close-up of the Captain's face he does the exact same thing again. (02:03:05)
Continuity mistake: After Colonel Kilgore looks at Captain Willard and says, "someday this war's gonna end" he gets up to walk away and the soldier to his right straps his rifle around his right shoulder twice between shots. (00:49:55)
Audio problem: When the natives start shooting arrows at the boat, Chief is imploring his crew to "fire...fire...FIRE.". Then if you listen to the scenes at Do Lung Bridge, they have reused the sound of this battle as a background sound. Its just after Willard says he needs some information, and to pick him up on the other side of the bridge. It's very clear on the Apocalypse Now (2 CD version) soundtrack, track 1 at 3:06. (01:42:14)
Continuity mistake: Note the boat position as Chef starts to mouth the words to "Satisfaction", by tree line they are close to shore. After less than a two second pause in action, we see that the boat is no longer near the shore or any trees. (01:14:55)
Continuity mistake: When Captain Willard looks out his window at the streets of Saigon you can tell by the shadows that the sun is overhead. But in the next shot the sun is coming directly through the window and shining on his face. (00:04:20)
Continuity mistake: As the helicopters approached Vin Din Dop Colonel Kilgore said to Lance, "We'll come in low, out of the rising sun," but when the helicopters fly over the beach their shadows are directly below them. (00:39:40)
Continuity mistake: After the woman tries to feed Captain Willard some rice he looks to his left and sees Colonel Kurtz standing there with one thin ray of sunlight coming down on his head. We saw this spot in the wide shot when the three soldiers placed the captain near the window and it's in the shade, plus the sun is not coming into the room from above but sideways from the right. (02:54:10)
Continuity mistake: Three soldiers carry Captain Willard into a room by a window where a woman gives him some water from a ladle. As first the sunlight falls across his waist then in the next shot as she gives him the water the sunlight is now on his face and chest. (02:53:35)
Continuity mistake: In a shot from behind his head Captain Willard sits and looks at Roxanne as the man in the tan jacket to the left of her says, "Look, Captain". In the next shot the captain's right hand is raised shielding his eyes from the sunlight. (02:07:15)
Continuity mistake: When the soldiers carry Clean's flag draped body to its final resting place there's no sunlight from behind falling on the flag. The next time we see it it's bathed in sunlight coming from the opposite direction. (02:00:30)
Character mistake: When Martin Sheen is on the boat, looking over Col. Kurtz's school papers, there is a dissertation by Kurtz on the history of the Philippines until 1905. The word Philippines is spelled "Phillipines"
Visible crew/equipment: The boom mike, camera and lights can be seen in Lieutenant Kilgore's sunglasses as he gets out of his helicopter. (00:27:05)
Continuity mistake: When Chief throws the binoculars to Captain Willard, Clean puts on his flak vest. 7 seconds later, Clean is not wearing the flak vest, then he puts it on again. (00:24:37)
Continuity mistake: As Lieutenant General Corman walks to the table, all of the hutch cabinet doors are closed. 17 seconds later, the left end and right end hutch cabinet doors are open. (00:11:40)
Continuity mistake: After Captain Willard smashes the mirror, there is no blood on his right hand and chest. 5 seconds later, there is.
Continuity mistake: After Clean is killed, the blood splatter on the PBR's metal plate changes. (01:52:34 - 01:52:58)
Other mistake: As Captain Willard gets up, he holds the Schlitz beer can upside down. (00:32:17)
Continuity mistake: When Colonel Kilgore is telling Lance about surfing in the front of the water, the sun changes between shots.
Continuity mistake: Kilgore is talking about how he wants to surf the beach. The lighting changes significantly in the space of seconds from the shots of the surf to the shots of the fighting. The shots of the surf are later in the day while the angle of the sun in the fighting shots is higher indicating afternoon.
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.