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Afterwards, he avoids Hawkeye: he isn't sure if he hopes Hawkeye won't notice, or fears Hawkeye will.
But he can’t avoid saying goodbye. The world still has noise: a continuous whine inside his ears, that won't go away. He's learned to sleep through it, but he can't hear through it. He's lost what Hawkeye had to say to him.
Then Hawkeye grabs him and hugs him: and Mulcahy feels him laugh. It was a joke, whatever it was.
Mulcahy tries to laugh. He watches Hawkeye walk away. A priest's life is meetings and partings. He can't hear "Goodbye", any more.