In a sitar wire, the type of vibration produced is a progressive longitudinal vibration.

To understand this, let`s break down the options:

Option 1: Progressive longitudinal vibration refers to a vibration that propagates in the same direction as the wave. In other words, it travels along the length of the sitar wire.

Option 2: Stationary longitudinal vibration implies that the vibration does not move along the wire and remains in a fixed position. This is not the case for a sitar wire as it produces sound through a traveling wave.

Option 3: Progressive transverse vibration involves vibrations that are perpendicular to the direction of the wave. However, in the case of a sitar wire, the vibration occurs parallel to the length of the wire, not perpendicular.

Option 4: Stationary transverse vibration suggests that the vibration does not move along the wire. However, in a sitar wire, the vibration travels from one end of the wire to the other, producing sound in the process.

Therefore, the correct answer is option 1, progressive longitudinal vibration. It accurately describes the type of vibration produced by a sitar wire.