Continuity mistake: As the Captain goes looking for Colonel Kilgore, there's a sampan docked on the shore with it's sail unfurled. In the previous shot, as the PBR approached the shore, the sail had been folded up. (00:25:30)
Continuity mistake: In the rear close-up shot of the Vietnamese machine gun nest the soldier in the dark green shirt is to the right of the soldier in the tan shirt who is firing it. In the next shot the soldier in the dark green shirt is behind and to the left of the soldier in the tan shirt. (00:40:45)
Continuity mistake: In the first close-up shot of the soldier wounded in the courtyard the medic behind him tries to give him some water as the camera pans away to two Vietnamese civilians for a second. In the next shot the medic is still holding the canteen but the injured soldier now has something hanging out of his mouth, that looks like a piece of string, which wasn't there in the previous shot. (00:43:25)
Continuity mistake: As they approach Kurtz' compound the dead body hanging from the tree changes between shots. At first it's facing outward, then as Dennis Hopper climbs on board it's facing the opposite direction and it's an entirely different body wearing a different jacket. (02:29:30)
Continuity mistake: As they carry Clean's body to the grave there's a lot of smoke blowing through the area. The smoke is in each shot except the close ups of Clean's flag draped body when there is no smoke at all. (02:00:50)
Continuity mistake: At the Playboy bunnies camp Captain Willard jumps off the boat with his rifle in his left hand. In the next shot as he turns around it's in his right. (01:23:30)
Continuity mistake: As they stand on the French plantation dock we see the front of the boat a lot and the chipped paint from the rocket attack in the previous scene has been repaired. There's also a camouflage shirt under the radar dome that sometimes has a T-shirt next to it and sometimes doesn't. (01:57:25)
Continuity mistake: The cigarette that Dennis Hopper hands to Captain Willard is lit and in perfect shape, however in close-up the cigarette is bent near the filter. In the next shot it is once again in perfect condition. (02:43:20)
Continuity mistake: Hubert de Marais says, "Oh I'm sorry, Captain," as Roxanne makes him apologize for carrying on their conversation in French. If you look at her hair it goes from being in the sunlight, to not, than back again several times in this scene. (02:04:40)
Continuity mistake: The lights on the Do Lung Bridge keep changing between shots. There are many more burned out light bulbs in the shot of Captain Willard and Lance as they leave the boat then there are when we first see the bridge or when the bridge is in the background as they walk past it. (01:42:15)
Continuity mistake: Docked next to the sampan Chief Phillips yells at Chef to, "Get on that boat," there's a shirt on the pole with the blue lights on it in the front of PBR. This shirt is in some of the shots and not in others throughout this scene. (01:36:00)
Continuity mistake: During Willard's bout of insanity in the opening scene, we see him doing karate or something in front of his bed and there's blood visible on the sheets. Yet that is not supposed to be there until he smashes the mirror (half a minute later) with his hand and then runs the blood over the bed and his face. (00:06:03)
Continuity mistake: As they leave the USO show the Green Beret security guard gets into the helicopter before Bill Graham. In the next shot he gets in again. (01:10:10)
Continuity mistake: When he first opens the package from his mom Clean is holding a brown envelope with a cassette tape in it. Three shots later as he reads the letter from his mom and listens to the tape, the package has changed to a white carton. (01:51:05)
Continuity mistake: Kilgore walks towards the beach and notices that the wind is starting to blow on shore and ruin the surf, and intense red green and yellow smoke canisters appear on his left between shots. (00:50:00)
Continuity mistake: On the dock as he talks to the sergeant about getting some Panama Red, Chef's mustache is combed downward and shaved more around his upper lips than it was in the shot of him before they docked at Hau Phat where it's bushier and combed off to the sides. (01:05:00)
Continuity mistake: As the people on the Vietnamese sampan hand their papers over to Chef, Clean is holding the 60 caliber machine gun with both hands. In the next shot he's holding the bullets in his left hand. (01:35:35)
Continuity mistake: When they first meet, Colonel Kurtz says to Captain Willard, "Have you ever considered any real freedoms?" and the Captain's shadow appears on the stone column to the right of the Colonel. The Captain is kneeling down and hasn't moved from his spot, but when we saw this column before his shadow was not anywhere near it. (02:39:25)
Continuity mistake: Before they board the sampan Lance says, "It's clear, Chief," and we see a wide shot of the sampan and PBR. No one goes near it, but for just one shot the dress on the clothesline is rearranged. The two straps it hangs from are pushed together. In all the other shots the straps are wide apart. (01:35:40)
Continuity mistake: In the close-up of Dennis Hopper giving the Captain some water he holds the ladle between his first and second finger. In the next shot he's holding it between his thumb and first three fingers. (02:42:45)
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.