To slowly get weaker or softer and then to disappear/end completely. E.g. "The music just petered out as the band stopped playing" "We followed the path through the forest but eventually it just petered out." "I can't talk to Jeff about anything. Our conversation always just peters out, unless we're talking about biscuits." Transcript here http://teacherluke.co.uk/phrasal-verb-a-day/119-to-peter-out/