Continuity mistake: While discussing surfing at the beach party Colonel Kilgore says, "We'll pick your boat up and put it down like a baby where you want it." The cigarette he's smoking here gets longer between shots. (00:33:35)
Continuity mistake: During the napalm attack there's one group of men marching on the other side of the bay. The next time we look over there right after the explosions there are two groups of men marching. (00:48:55)
Continuity mistake: In the first close-up shot of the soldier wounded in the Vin Din Dop courtyard there are two medics working on him. In the next shot there is a third kneeling by his left leg in the exact spot where the cameraman just stood in the previous shot. (00:43:20)
Continuity mistake: When the soldiers lower Clean's body into the grave they're all standing. In the next shot they're all crouching. (02:01:50)
Continuity mistake: After they leave the French plantation Chief Phillips maneuvers the boat through some thick fog and says, "I said I can't see a thing captain I'm stopping this boat, I ain't riskin' no more lives". When he says this line he's looking directly at Captain Willard but in the next shot his head is turned as he looks over towards his right. (02:20:15)
Factual error: Captain Willard recalls back to lunch at General Corman's and we hear Harrison Ford's voice say as he points to a map, "Now he's crossed into Cambodia". The town he's pointing at (Ban Me Thout) and the one above it are in the Central Highlands in the middle of Vietnam and not crossover points anywhere near Cambodia. (01:18:20)
Continuity mistake: The day after the USO Playboy Bunny show, the patrol boat canopy is set afire by some pranksters from a passing boat. Yet, that night in the scene at the Do Lung Bridge, the canopy is intact. It goes through some amazing transformations during the remainder of the picture. (01:41:33)
Audio problem: Lance is singing while he's in the water with Chief Phillips' body and his lips do not match his vocalizing. (02:24:35)
Continuity mistake: Hubert de Marais continues to speak to Roxanne in French. As she looks at him intently her expression changes and she goes from having her hands at her sides to unfolding her napkin again between shots. (02:04:30)
Character mistake: After Captain Willard walks through the front door of the burnt out French plantation dock there's a close-up of Chief Phillips at the wheel the boat. He turns around and looks right at Chef and says, "Lance". (01:56:40)
Continuity mistake: As the soldiers crash the stage Bill Graham ignites the smoke canister and the flashing green running lights above and under the left door of the helicopter are off. In the next shot, as he gets in, they're on. (01:09:55)
Visible crew/equipment: As the Captain takes the binoculars from the chief and first holds them up to look at the shoreline and Colonel Kilgore you can see the reflection of a crew member in the binoculars' lenses. (00:24:35)
Audio problem: Colonel Kilgore takes the megaphone and says to the surfers, "Okay give it a try guys, one goes left and one goes right "and his lips do not match what he says. In these several shots he's saying a lot more than what we hear. (00:50:50)
Continuity mistake: When Captain Willard climbs onto the top of the boat and again when he shakes Dennis Hopper's hand we see no rips or tears on his green T-shirt. But when he says to Dennis Hopper, "Take who away?" there's a huge rip in his T-shirt under his right arm. (02:29:40)
Visible crew/equipment: After Chief Phillips is speared if you look closely at his left shoulder you can see his shirt bunched up by the harness underneath that holds both ends of the spear to his body. Later you can see a blood bag underneath his tee shirt as he lies on the floor of the boat dying. (02:22:40)
Continuity mistake: As Captain Willard tries to get Chief Phillips to stop the men from firing he says, "They're just little sticks. They're just trying to scare us". Watch the Captain's left hand closely as his rifle becomes arrows between shots. (02:22:00)
Continuity mistake: As they begin to break out of the fog bank Chef says, "watch it over here, chief, got a stump". Here he's helping Chief Phillips steer the boat with a long pole that he places back on the floor of the boat after the uses it. You can see that it's not hanging over the right side of the boat behind Chief Philips but at the beginning of the arrow attack it's back extending out over the side. Then a few shots later as Captain Willard runs to the back of the boat to stop Chef from firing it's gone completely. (02:20:55)
Continuity mistake: Chief Phillips yells, "Lance, cover the captain" and we can see the shadow of the flag waving behind him on the floor the boat. In the next shot the flag is motionless. (01:57:15)
Visible crew/equipment: As they pull away from the Do Lung Bridge, look closely and you can see water spraying up and turbulence from the propellers of the camera boat in front of them. (01:48:40)
Continuity mistake: We see a close-up of Lance's face as he talks to Chef on the front of the boat at they approach the Do Lung Bridge. The camouflage makeup around his chin and left cheek is different as he walks across the bridge with Captain Willard. (01:42:35)
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.