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)
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)
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.
Continuity mistake: When Willard has a severed head dropped on his lap, the stump is clean and flat like it was sliced by a samurai sword. When the head falls to the ground, the stump is bumpy and ragged like it was hacked with an ax.
Continuity mistake: As Colonel Kurtz reads his poetry out loud we see Dennis Hopper gesturing with both hands as he speaks with the captain. When the colonel gets mad and throws the bowl of bananas at them Dennis Hopper is holding his camera in his left hand. His bandana is also rolled up differently with the corner sticking up in one shot and down in the other. (02:56:20)
Continuity mistake: As Captain Willard tries to get Lance away from Colonel Kilgore the band-aid on his right cheek angles upward as it nears his eye. Later we see it up close as they pull him into the boat with the surfboard and it does not angle upward but goes straight across his face. (00:50:40 - 00:51:45)
Continuity mistake: As they all read their mail from home Lance ignites a canister of purple smoke and it's sunset. In the next shot of Clean grabbing his tape recorder it's the afternoon and the sun isn't even shining. (01:50:55)
Continuity mistake: As they approached the French plantation the partially crushed fuel drum on the back the PBR that we saw after Clean was shot, has been replaced with one that is only slightly dented. (01:56:15)
Continuity mistake: As they depart from the USO show at Hau Phat Chef shows everyone his picture of Miss December and there's a helmet hanging on a poll in the rear of the boat with the top facing Chief Phillips. Each time we see this helmet in the next few shots it moves sometimes facing the back of the boat and sometimes facing the front. (01:11:55)
Continuity mistake: As they approached the French plantation there are two distinct shots of this area, one is from further back then ten seconds later there's a close-up. The clouds in the background are entirely different in these two shots and the clouds change again when the Frenchmen show up. (01:55:50 - 01:59:20)
Continuity mistake: As Roxanne sits down at the table Captain Willard looks at Hubert de Marais. In the next shot he's smiling and looking at Roxanne. (02:04:10)
Continuity mistake: As Captain Willard takes a smoke, the shadow of the pipe is on Roxanne's face but in the next shot as she looks at him the pipe's shadow has disappeared. (02:17:30)
Continuity mistake: As Roxanne unties the curtains around the bed the scarves draped over the light on the left of Captain Willard are not the same ones we just saw in a previous shot as the captain took a smoke from the pipe. The ones before were red and yellow but now the middle one is primarily green. (02:19:05)
Continuity mistake: As he says, "so when you ask me why we want to stay here, Captain" Hubert's face and jacket go from being in the light to being in a shadow between shots. (02:21:30)
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.