Q:

What expression to use when a wave hits the beach and fades away?

I am looking for word or expression that refers to the moment when a wave, with all its strength, closes itself, hits the beach and fades away.

A:

After the wave breaks, it is called swash. 

Swash, in geography, is known as a turbulent layer of water that washes up on the beach after an incoming wave has broken. Swash consists of two phases: uprush (onshore flow) and backwash (offshore flow). Wikipedia

Image source: geographyas.info

A:

You could say the wave is ebbing. Typically, ebb is used to describe the tide going out, but it could also be applied to a single wave receding toward the sea.
ebb - flow back or recede

A:

I think the term is that the wave is "breaking" from Oxford Dictionary:

5.5(Of waves) curl over and dissolve into foam: the Caribbean sea was breaking gently on the shore

I think it can also be considered to be "crashing" onto the shore which is similar but seems to have a more strong or violent image.

