The following animation is helpful in understanding the physics of a longitudinal wave. See notes below.
Well, it may not be immediately apparent, but the above animation will relate to longitudinal wave motion.
The above animation shows a particle, or object, in horizontal simple harmonic motion. It is moving to the left and right.
For our discussion in this section we will consider a wave train moving horizontally from left to right. Therefore, this above motion, being left and right, is parallel to the motion of the wave we will be considering.
Consider this yellow block above to be, say, a tiny section of a long stretched spring.
Imagine that you have horizontally compressed one section of that that spring in order to send a wave pulse down it. Imagine the wave pulse traveling from left to right.
The spring as a whole, though, is not going to move from the left to the right. The spring will horizontally vibrate some as the wave pulse goes down it.
As the wave pulse moves down the spring all the small sections of spring, symbolized by the yellow box above, move back and forth horizontally as the wave pulse moves horizontally.
Therefore, the motion of the medium, the spring, is parallel to the motion of the wave. The spring vibrates horizontally while the wave moves horizontally left to right.
This is the central concept to the definition of a longitudinal wave.
A longitudinal wave is a wave with the motion of the medium being parallel to the motion of the wave. More carefully, we would say that the direction of the velocity of the medium is parallel to the direction of the velocity of the wave.