Yearned to possess sink (3,4)
I believe the answer is:
die down
'sink' is the definition.
'yearned to possess' is the wordplay.
'yearned' becomes 'died' (dying is a kind of yearning).
'to possess' becomes 'own' ('own' can be a synonym of 'possess').
'died'+'own'='DIE DOWN'
(Other definitions for die down that I've seen before include "Subside" , "For ease" , "Become less loud" , "Quieten" , "Become less loud/strong" .)