Continuity mistake: On the beach Colonel Kilgore says to one of his soldiers, "what do you know about surfing major, you're from New Jersey" look closely at the Vietnamese prisoners walking on the opposite side of the beach. They're always walking from left to right but several times in this scene they change their starting position from further down the beach (left of screen). (00:45:50)
Continuity mistake: As Captain Willard takes a smoke from Roxanne's pipe the pillow he's lying on is light green. Later after they're finished as he lies on his back the pillow is now light red. (02:17:20)
Continuity mistake: After Chief Phillips falls to the floor of the boat Captain Willard kneels down to try to help him. Look closely at Phillips' left hand, there's blood dripping on his knuckle. In the next shot when he grabs the captain around the throat there is no blood on his left hand. (02:23:05)
Continuity mistake: As they all sit on the boat and listen to the "Arc light" B-52 strike Lance is sunbathing in direct sunlight. In the next shot he's in the shadow of the roof of the boat as he gets up and says "there they are". (00:24:15)
Continuity mistake: As Chief Phillips is speared he holds on to the roof of the boat with his left hand. The straw has been rearranged from previous shots and is no longer hanging over the side of the roof as it was in the shot before. (02:22:35)
Continuity mistake: As they approached Vin Din Dop over the beach we see the bridge on the left and no sampans with red sails are there. Fifteen seconds later when they come around the other side of the bridge there are three sampans with red sails unfurled. (00:39:40)
Continuity mistake: Before Colonel Kilgore looks at Captain Willard and says, "Someday this war's gonna end," the Vietnamese women and the soldier waving behind him to his left have moved between shots. (00:49:45)
Continuity mistake: As he looks towards the surfers, Colonel Kilgore says, "Lance, the wind, the wind" and it's quite overcast. Then in the next shot of the surfers the sun is out. This scene switches between close ups of Kilgore and the other men and the sun is out in the close-ups but not in the other shots. (00:50:10)
Continuity mistake: At General Corman's they sit down to have dinner and the ID card on his right chest pocket is attached to his shirt differently in various shots. At first it's not even on the pocket then later in the scene it's set right again. (00:11:40)
Continuity mistake: After Colonel Kilgore's helicopter flies over looking for his surfboard Lance says," Let's get this board out of my turret" but in several shots you can see the surfboard is not in his turret but wedged next to the roof of the boat. (00:54:15)
Continuity mistake: As Chef and Captain Willard get back on the boat after the tiger attack Clean is on the 60-caliber gun and the entire boat is camouflaged under a tree. Clean says, "Let's go" and you can see the river behind him without the tree there obstructing the view. (00:59:20)
Continuity mistake: When the boat stops at the Playboy bunnies' camp it's raining. It's a partly cloudy day and if you look closely it's only raining in the foreground and not in the background. (01:22:30)
Continuity mistake: As Captain Willard continues to look at his folder on Colonel Kurtz several cuts go between him and Clean with his transistor radio playing. The shots of the Captain are in bright sunlight and the first shots of Clean are overcast. (01:14:00)
Continuity mistake: After speaking to Bill Graham in the tent Captain Willard comes back and tells the guys, "I negotiated two barrels of fuel for a couple of hours with the bunnies". As he walks back to the boat the hills in the distance can't be seen because of the rain but in the next shot when Chief Phillips comes over to talk to him the hills can be seen. (01:26:25)
Continuity mistake: Captain Willard says, "the army tried one last time to bring him back into the fold" and we see a close-up of General Corman again sitting at lunch with the captain from earlier in the movie. This was not shot at the same time because the general's shirt is different. Look closely at his name, the collar covers everything except the first two letters whereas earlier in the movie, for the entire scene his name was showing and could be read. (01:18:10)
Continuity mistake: As they leave the MedEvac we see an overhead shot of the boat and the flag has changed. The back of the flag has been cut up considerably since when they arrived at the MedEvac even though they haven't been it any battles. (01:32:55)
Continuity mistake: As they approach the Do Lung Bridge Chef talks to Lance with his gun muzzle pointing out in front of him. The muzzle isn't near anything but in the next shot it's resting on the flat piece of metal on top of the machine gun. (01:40:35)
Continuity mistake: As Chef looks over the papers from the sampan crew the woman is standing under the canopy with the sun shining on her. The next time we see her several seconds later she's in the same spot but the sun is no longer shining on her. (01:35:35)
Continuity mistake: As Colonel Kurtz opens up the steel doors to the room where Captain Willard is being held we see thick black smoke engulfing the entire area behind the colonel. They cut away to the captain for few seconds then back to the colonel as he stands in the doorway, this time with no smoke at all behind him. (02:49:05)
Continuity mistake: As he talks to him in the bamboo cage Dennis Hopper gives Captain Willard a cigarette with his fingers outstretched. In the next shot his fingers are clenched in more of a fist. (02:43:20)
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.