If you were asking about the physical process taking place, the DC causes a galvanic reaction that erodes the top layer of the metal away. The AC current causes oxides to be deposited on the top layer.
Also I should point out that only the knife in the last 2 pics has the etch darkened with AC. The transformer I had in my etcher crapped out on me and I hadn't repaired it for the first two knives so they were only etched with DC.