Continuity mistake: As they pull up next to the Vietnamese sampan Clean runs to the port side of the boat in order to catch the tow rope. Chef is standing behind Captain Willard in middle of the boat and doesn't move and all. In the next shot Clean and Chef are both standing on the edge of the port side of the boat. (01:34:50)
Continuity mistake: As they pull up next to the Vietnamese sampan, Chief Phillips moves the spotlight on top of the PBR with his left hand so it's pointing at Chef as he takes the papers from the fishermen. Thirty seconds later there's a wide shot of both boats as Chef reads the papers and we see the spotlight back to where it was before facing away from Chef even though Chief Phillips doesn't touch it again. (01:35:05)
Continuity mistake: As they come along side the sampan Chief Phillips says, "Look in that forward hooch, bring the people out of there." Clean slips on something and spins the machine gun "Canned Heat" towards the sampan as he ties the boats together. In the next shot the machine gun is spun to the left pointing away from the sampan. This reversal happens several other times. (01:35:10)
Continuity mistake: The Vietnamese fishermen hand Chef the two small pieces of paper that they had in their hands. He reads the first one for a while then grabs the second one from the guy in the middle, leaving them with no papers in their hands. The next time we see Chef two shots later he has only one piece of paper in his hands and is now grabbing a new large identification paper that none of the three had before. (01:35:25)
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: 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: 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: After Chef looks over the papers handed to him by the people on the Vietnamese sampan, he hands them to Chief Phillips, who puts them under a brown flak jacket tied around a metal roof pole on his left. This brown flak jacket appears, then disappears between shots. (01:35:45)
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: Captain Willard's is not carrying a holster when he approaches the bridge and talks to Chief Phillips about him not pulling over the Vietnamese sampan or when he's tying his shoes as he sits on the front of the PBR. He never leaves the front of the boat to get his gun but as Clean starts machine-gunning the sampan he's wearing his holster. (01:36:15)
Continuity mistake: As Chef pulls the Vietnamese woman out from where she's hiding you can see the shadow of Lance's gun barrel as he stands there holding it on her. In the next shot Lance gets up from a seated position in front of the boat. (01:36:20)
Continuity mistake: As Chief Phillips orders Chef to, "Check what's in that vegetable basket," the Vietnamese woman runs past and hands him papers that he takes with his right hand. In the next shot as he pushes the woman the papers are in his left hand. (01:36:25)
Continuity mistake: The Vietnamese woman on the sampan has her hair in two ponytails when Chef shoves her. It's in disarray when she falls then in one ponytail as she is shot by Clean. (01:36:25)
Continuity mistake: When Clean fires his machine gun on the Vietnamese sampan Captain Willard ducks for cover then pulls his pistol out of his holster and holds it straight up in the air. In the next shot he takes the gun out of his holster again. (01:36:45)
Continuity mistake: After shooting up the sampan Clean closes his machine gun twice between shots. (01:37:15)
Continuity mistake: Chief Phillips Orders Clean down onto the sampan to help Chef with the injured Vietnamese woman. We hear Clean talking to Chef as they lift her body but the next time we see the Vietnamese woman Chef is the only one there and Clean is back up at his machine gun. (01:38:05)
Continuity mistake: Before Captain Willard shoots the injured Vietnamese woman on the sampan he's holding onto a metal bar with his left hand. In the next shot he's not holding onto anything. (01:38:25)
Continuity mistake: When Captain Willard shoots the injured Vietnamese woman on the sampan, the T-shirts that we saw before draped over the metal bar in front of Chief Phillips are no longer there even though no one goes near the spot. (01:38:30)
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: 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)
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.