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: The next day as the boat leaves the USO show at Hau Phat to continue up the river we see the battered stage from the back of the boat. The boat passes the stage by about 100 feet but in the next shot its back right next to the stage. (01:11:25)
Continuity mistake: The day after the USO show at Hau Phat, the boat leaves and there are five men working on the stage. In the next shot there are three. (01:11:35)
Continuity mistake: As the boat leaves the USO show at Hau Phat Captain Willard says, "The war was being run by a bunch of four star clowns who were gonna end up giving the whole circus away". He says this as he walks to the back of the boat moving the 60 caliber machine gun on his way so that is faces frontward. In the next shot it's been moved and is now facing sideways. (01:11:40)
Continuity mistake: As Chef shows Chief Phillips and Clean the picture in his Playboy magazine of Miss December you can see the USO stage behind them. In the previous shot of the boat leaving one of the six tall lipstick statues on the upper left of the grandstand was knocked down. Now as the boat departs all six statues are standing. (01:11:50)
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: During Clean's story about a Vietnamese lieutenant Chef puts up his picture of Miss December behind Chief Phillips and its raining and his hat is wet. In the previous and next shot it's not raining. (01:12:15)
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)
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)
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)
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.