A vibrating object will sweep particles together in front of it and leave fewer particles behind. The sweep is clear in the animation below. As the particles are compressed together they spring apart, compressing the next group of particles which themselves then spring apart.
This animation of sound being made by a vibrating red piston in a tube is courtesy of Dr. Dan Russell, Kettering University. his site is linked from the animations.
It shows a longitudinal waveform
The picture below links to an applet which also describes longitudinal waves
This can also be seen at this site.
Watch the red markers for an effective understanding of longitudinal waves but beware that the browser may freeze: just cntl - alt - del to break out. Worth the inconvenience. Other useful sites include fear of physics.
For a two dimensional effect, an animation also from Dr Russell:
Another animation is on this site.
It is possible to model this with a slinky spring. Hold both ends firmly and move one end to and fro. The video is 130 k.
A wave of compression travels along the spring. If this is done repeatedly, standing waves can be observed at definite frequencies. Standing waves appear to stand still arise form a dynamic process of interferance.