At the start of the first battle as the gun's breech mechanism is opened fumes emerge. This could only happen if the gun had just been fired.
The crew inadvertently receives an uncoded voice transmission from a German supply ship indicating its destination. In fact, German supply ships were sent to meet U-boats at prearranged meeting locations by German headquarters back in Europe using coded messages. Therefore, the crew should never have understood the transmission. Since this movie was filmed during the war, these facts were likely unknown at the time.
The radio signal about the German supply ship's position is inaccurate for a number of reasons. 1) Voice radio was used for only very short range transmissions. Morse code was used for all long range transmissions. 2) It was actually quite easy to hear most enemy radio transmissions. Decoding them was the difficult part. 3) "Garble" was used for some land based voice communications, but not with radio. Messages were coded before transmission and the encoded message was sent. Despite all this, the de-garbled radio code was a common plot device in U.S. war movies of World War II.
When Kroner is speaking with the U-Boat captain both are shown wearing caps with blue covers. Captains of German naval vessels typically wore white covers on their caps.
In the scene where Kroner speaks with the U-Boat captain about the conditions back in Germany, the roles are reversed. A supply ship would be at sea for many months, and out of touch with home. U-Boats would return to port every couple of months, so a U-Boat commander would be telling the captain of the supply ship about home.
The destroyer momentarily seen through the U-boat's periscope after the first fight was USS Preble (DD-345). The picture was obviously from well before the war, since Preble was converted to a minelayer (DM-20) in 1937. Further, she served in the Pacific, not the Atlantic.