Greet us with waving action (7)
I believe the answer is:
gesture
'action' is the definition.
(gesture is a kind of action)
'greet us with waving' is the wordplay.
'with waving' is an anagram indicator.
'greet'+'us'='greetus'
'greetus' anagrammed gives 'GESTURE'.
(Other definitions for gesture that I've seen before include "Expressive physical movement" , "Motion of the hands or body" , "Message through movement" , "Expressive movement of body" , "Movement, indication" .)