Trivia: When this film was made, many of the original, Russian crew members involved in this historic incident were very upset about the submarine crew being portrayed as drunkards. Oddly enough, one of the promotional items given out at premieres and to executives was a silver K-19 vodka flask.
Trivia: The 'American' destroyer seen from far away in several shots is actually the decommissioned destroyer HMCS Terra Nova.
Trivia: In real life, the K-19 was never nick-named "The Widowmaker," and its cursed history was entirely fictional, fabricated for the film. After the fact of the nuclear meltdown, the Soviet military actually nick-named the K-19 "Hiroshima," alluding to the first city ever destroyed by a nuclear weapon.
Trivia: There was never a mutiny aboard the real "K-19" as depicted in the film. In real life, the captain ordered all firearms to be thrown overboard, to avoid a mutiny.
Answer: Harrison Ford does attempt a Russian accent in this film. It's a very slight, very bad attempt but he's definitely trying. Audiences come to expect accents in films such as these for a heightened sense of immersion. Hearing American accents from supposedly Russian characters can sometimes be jarring to an audience, even if the characters are speaking English. This of course isn't always the case and plenty of films have actors speaking in their natural accents while they are playing foreign characters. The director of this film chose to have his actors speak with Russian accents, with extremely poor results pretty much all around.
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