If you thin behind the edge, obviously the first thing being thinned, is the cladding. If you are concentraing your thinning near the edge, likely the transition/wavy line will move towards the spine some. The amount it moves is related directly to how much of the cladding gets removed. If the thinning is even, it will look nice and even, if an area gets thinned more by accident/over grinding, the result will be more core exposed in that area, and look uneven. It will not likely straighten out. the wavy pattern is there because of how it was forged, from hammer blows, or from the press used.