A swan-neck arm is one continuous curve from the top of the back to the front of the seat, and it cannot be built the easy way. The obvious method is to cut it in two or three shaped sections and join them, which works structurally and always shows - the eye catches the change of direction at a joint even when the surface is smooth.
Built properly, the arm is laminated or cut from stock deep enough to take the whole sweep, so the grain runs along the curve rather than across it at any point. That costs considerably more timber and it is the only way the curve looks effortless.
Our master Indian artisans make the two arms as a pair, working them together and checking each against the other constantly, because a swan neck is the most symmetrical element on a sofa and any difference between left and right is obvious from directly in front. It is a piece of work that takes days and looks like it took none.