Continuity mistake: After Chef rips the picture from his Playboy magazine he crouches down. In the next shot as he's taping it up he's standing. (01:12:15)
Continuity mistake: Leaving Hau Phat Clean tells the guys a story about a, "cat in the Delta" and Lance picks up his new water skies then puts them back down again while still holding on to them. In the next shot he's not touching them anymore. (01:12:30)
Continuity mistake: Clean continues to tell the guys his story and says, "One day the gook took his magazine away from him and wouldn't give them back to him," and Captain Willard's right hand goes from leaning against the machine gun to touching his head between shots. (01:12:30)
Continuity mistake: Chief Phillips gives the helm of the boat over to Chef as Clean continues with his story. Lance takes the plastic covering off of the top water-ski then in the next shot it's covered again then uncovered in the shot after that. (01:12:50)
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)
Audio problem: As Chef says to Clean, "Should've given the f***er a silver star" after hearing his story about a Vietnamese lieutenant his vocal track cuts out for a few seconds then comes back. (01:13:45)
Continuity mistake: Clean tunes in his transistor radio to AMVN and we hear it continuously playing in the background as Captain Willard goes from standing with his shirt on looking at Chief Phillips as he pilots the boat, to sitting shirtless looking at pictures of Colonel Kurtz between shots. (01:13:50)
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)
Audio problem: Clean listens to his transistor radio as the DJ says, "Â…and we have an important message for GI's who are living off base from the Mayor of Saigon". The volume that we hear stays the same but watch his thumb on the left of the radio as he turns the volume knob up. (01:14:05)
Continuity mistake: As "Satisfaction" plays in the background Clean dances behind Chief Phillips as he controls the boat. In the next long shot of Lance water skiing past the Vietnamese on the shoreline Clean is not there. (01:14:45)
Continuity mistake: Note the boat position as Chef starts to mouth the words to "Satisfaction", by tree line they are close to shore. After less than a two second pause in action, we see that the boat is no longer near the shore or any trees. (01:14:55)
Continuity mistake: As Lance water skis and Clean dances, Chief Phillips lays down to watch with his head leaning against the port side metal roof support. In the following shot he is leaning on his right elbow no longer near the roof support. (01:15:00)
Continuity mistake: As Satisfaction plays in the background, Lance is holding on to a white tow cord as he skis. Earlier as they pull out of Hau Phat we see him holding a blue tow cord before he unwraps his new water skis. Thirty seconds later in another shot he is holding the blue tow cord. (01:15:05)
Continuity mistake: As Captain Willard sits in the boat looking at pictures of Colonel Kurtz the shadow of a metal bar slowly goes from left to right across the pictures indicating that the boat is turning. In the next shot of Clean dancing to Satisfaction we can tell by looking at the sky that the boat is not turning but going straight ahead. (01:15:15)
Continuity mistake: Captain Willard sits in the sunshine near the back of the boat reading his file on Colonel Kurtz but in the wide shots of the boat as Lance is water skiing he's not there. (01:15:20)
Other mistake: Captain Willard reads to himself from Colonel Kurtz' paper titled, "Commitment and Counterinsurgency". We see a close-up of the page that he's reading from which is fine until he reads, "our conduct of the war will gain only impotence". Everything else he reads is not on the page. (01:15:35)
Continuity mistake: Captain Willard reads on uninterrupted from "Commitment and Counterinsurgency" by Colonel Kurtz who says in his report, "if they were committed this war can be won with a fourth of our present force". The weather changes from not raining to raining, back to not raining again between shots. (01:15:40)
Continuity mistake: As Captain Willard holds Colonel Kurtz' paper titled, "Commitment and Counterinsurgency" in front of the camera it changes color from yellow in the first shot to white in the second. (01:15:40)
Continuity mistake: As Captain Willard sits in the boat reading and thinking to himself about Colonel Kurtz he unwraps and eats a Hershey bar. He crumples the wrapper with both hands and throws it out of the boat, leaving nothing in his hands. In the next shot he's eating the Hershey bar again and the wrapper is back. (01:15:45)
Audio problem: When they first spot Lazaro's boat Clean says, "Chicken time" followed by some other choice hyperbole. He says more that has been edited out. (01:16:00)
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.