Continuity mistake: As Chef and Clean sit on the back of the PBR reading their mail, we see Lieutenant Kilgore's surfboard wedged between the machine guns. It's there until the shot when we see Clean after he's been killed, where it disappears. (01:51:40)
Continuity mistake: Just after the rocket attack that kills Clean starts there's a close-up of Lance manning the machine gun in the front of the boat. The Vietnamese spray this area with machine gun fire and chip away a lot of the green paint to his left under the radar dome. The next time we see this spot the paint is untouched. (01:52:00)
Continuity mistake: The Vietnamese fishermen hand Chef the two small pieces of paper that they had in their hands. He reads the first one for a while then grabs the second one from the guy in the middle, leaving them with no papers in their hands. The next time we see Chef two shots later he has only one piece of paper in his hands and is now grabbing a new large identification paper that none of the three had before. (01:35:25)
Continuity mistake: After the boat docks at the Medevac Captain Willard gets out and walks around and it's clear from the wide shot that there's only one helicopter there. When he comes back to tell the guys about the arrangement with the bunnies we see a close-up of this helicopter and it has yellow tipped rotors. The helicopter that Chef and Miss May are in later in the scene is different as it has white tipped rotors. (01:31:50)
Continuity mistake: As the boat docks at Hau Phat it's not raining and we see several motorcycles parked at the right of the dock. In the next shot it's raining and the handlebars of the motorcycles have been covered. (01:04:25)
Continuity mistake: Colonel Kurtz sits at the door to the dark room where Captain Willard is being held reading articles to him from Time magazine, and it's obvious that the light is coming from the door in front of him. The door doesn't close nor does the captain stop looking forward, but the lighting changes throughout the scene. In some of the close-up shots his entire face and neck are lit up and in other shots shadows fall across his face and neck. (02:51:15)
Continuity mistake: One minute into the attack on Vin Din Dop one of the helicopters flies over the schoolyard. This aerial shot is taken around mid-day as there are no shadows visible in schoolyard, unlike the first shot of the schoolyard at the beginning of this scene, which was done when the shadows were long. (00:40:20)
Continuity mistake: As Chef and Captain Willard walk through the jungle we get a couple of good close-ups of the band-aid on his cheek and it's different between shots. In the close-up when he first hears the tiger, the right side of the band-aid is pointing towards the top of his ear lobe. Then just seconds before the tiger attacks we see it pointing directly into the middle of his ear. (00:57:50 - 00:58:45)
Continuity mistake: During Clean's story about a Vietnamese lieutenant Chef puts up his picture of Miss December behind Chief Phillips and its raining and his hat is wet. In the previous and next shot it's not raining. (01:12:15)
Continuity mistake: Roxanne continues to prepare the pipe for Captain Willard and says, "he would rage and he would cry, my lost soldier, and I said to him there are two of you". In the shot of them from the foot of the bed Roxanne's head is almost touching the captain's but in the next shot over the side of the bed she's about a foot away. (02:16:55)
Continuity mistake: Clean points right at Chef, at the wheel of the boat, as he continues to tell his story and says, "don't do her like that". In the next shot he's not pointing anymore and his right hand is resting in front of his face. (01:12:55)
Continuity mistake: When the bunnies are dancing there's a close up of playmate Sandra Beatty and as a soldier yells, "take it off" it's raining. In the next shot it's not. (01:09:10)
Continuity mistake: Just before the Medevac helicopter lands in the Vin Drin Dop plaza, a soldier machine guns 13 urns sitting next to the building on the left of the courtyard and the building also explodes. In the next shot of the medics working on the injured soldier the 13 urns are back and the building hasn't exploded yet. (00:43:05)
Continuity mistake: Dennis Hopper says, "Zap 'em with your siren, man. Zap 'em with your siren," as they get close to shore. At first there's no smoke and unarmed natives behind him and the smoke changes to red in the close-up. Then in the next close up there's orange smoke and armed troops behind him. (02:29:15)
Continuity mistake: Chief Phillips Orders Clean down onto the sampan to help Chef with the injured Vietnamese woman. We hear Clean talking to Chef as they lift her body but the next time we see the Vietnamese woman Chef is the only one there and Clean is back up at his machine gun. (01:38:05)
Continuity mistake: When the helicopters first attack Vin Din Dop we see eight in formation attacking from the beach with the bridge on the left. One second later we're in a helicopter attacking from the other side of the bridge and we can see the entire beach. From this angle we should be able to see all eight helicopters, but we don't and we also don't see any smoke on the beach from the explosions in the previous shot. (00:39:55)
Continuity mistake: At dinner with the general, Harrison Ford walks over to the captain with the picture of Colonel Kurtz and he has a pencil in his right hand. In the next shot as he drops the folder on the floor the pencil is not there. (00:12:25)
Continuity mistake: During the beach party as Captain Willard is saying, "He was one of those guys that had that weird light around him." Note he is holding a cigarette and drinking a Schlitz, with four cans of Budweiser in front of him. In his next shot, just seconds later, there is another can of beer and the cigarette is gone despite the fact he has not moved at all. (00:32:00)
Continuity mistake: In the beginning of the movie when Captain Willard hears the helicopter flying over and walks to his window the blinds are dark. In the next shot when you see him looking through them they're lit up. (00:04:15)
Continuity mistake: Before Captain Willard shoots the injured Vietnamese woman on the sampan he's holding onto a metal bar with his left hand. In the next shot he's not holding onto anything. (01:38:25)
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.