Webster's dictionary defines a wave as:

"A disturbance or variation that transfers energy progressively from point to point in a medium and that may take the form of an elastic deformation or of a variation of pressure, electric or magnetic intensity, electric potential, or temperature."

http://www.gmi.edu/~drussell/Demos.html

A Longitudinal sound wave behaves similar to the traveling wave.

http://www.gmi.edu/~drussell/Demos.html

The animation above shows one dimensional longitudinal plane propagating down a tube. Watch a single particle's motion.

Another type of wave is the Transverse wave.

A Transverse wave is, "a traveling wave that causes the particles of the disturbed medium to move perpendicular to the wave motion."

Serway*Beichner Physics for Scientists and Engineers.

http://www.gmi.edu/~drussell/Demos.html

The animation above shows a one dimensional transverse plane wave that is propagating from left to right. The particles are simply moving up and down. They are not moving along the wave. Note that none of the particles ever