Crossword Genius

Exude an air of sadness (4)

Ross

I believe the answer is:

sigh

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'air of sadness' is the definition.
The answer and definition can be both related to communication as well as being singular nouns.
Perhaps you can see a link between them that I don't see?

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for sigh that I've seen before include "resigned response" , "Sound of despondency" , "Yearn" , "Express relief or sorrow" , "Sound of regret" .)

I've seen this clue in The Globe and Mail and The Telegraph.
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