Types of longitudinal waves

Answer 1

An excellent discussion of wave types can be found here: http://www.glenbrook.k12.il.us/gbssci/Phys/Class/waves/u10l1c.html
In one sentence, a longitudinal wave is any wave where the quantity that is oscillating is along the direction of propagation of the wave. There is no oscillation which is associated with any direction perpendicular to the direction of propagation.

We speak of two types of waves, longitudinal and transverse, because almost all waves that we encounter can be categorized as one or the other. (Not all, but almost all waves are one or the other.)

For completeness, a transverse wave is one which exhibits the oscillations to be occurring in a direction perpendicular to the direction of motion.

Example 1. Sound

The sound wave generated by a vibrating string is a longitudinal wave, as are all sound waves. The quantity that is oscillating in a sound wave is the pressure. For a pure tone, the sound wave is a series of peaks and troughs of pressure extending along the line that is the direction the wave is traveling. The pressure does not have a direction and the increases and decreases in pressure are associated with no direction perpendicular to propagation. That is enough to make it a longitudinal wave, i.e. it is enough that it is not a transverse wave.

The usual argument that a sound wave is a longitudinal wave utilizes the direction of motion of the air caused by the wave. At any particular point in space, as the wave move past, there is motion of each region of air as it moves sinusoidal forward and backward along the direction of motion. Oscillating movement along the direction of motion certainly meets the criteria necessary to be longitudinal.

Example 2. Spring Compression

In another example, a wave is traveling along a spring where the wave pattern is produced by the coils of the spring moving closer together and further apart and those motions traveling along the spring in a regular sinusoidal pattern. This is directly analogous to the properties of a sound wave and it is, therefore, an example of a longitudinal wave.

Example 3: Earthquake Elastic Wave

In Earthquakes, they talk about S and P waves. The P waves are longitudinal and are the propagation of the compression of the Earth material followed by rarefaction followed by compression, etc. It is an elastic wave and the physics of the wave is just like a sound wave or just like the compression waves in the spring in example 2. The S waves are transverse, like water waves on a surface, but they occur inside the Earth, not on a surface.

Example 4: Imaginary Color Wave

One could imagine a rope of lights where the colors at each point varied continuously and sinusoidal as the oscillating pattern moved along a wire. (Thing of each point changing color from red to green to blue to green to red etc.) This color wave also has no direction associated with the oscillating character, so it too would be longitudinal.

Example 5. Quantum

It is not discussed much, but in the theory of quantum mechanics, there are many examples of waves that have the characteristic that there is no direction associated with the thing that is waving. In fact, so called "matter waves" are such. It is a little tricky to discuss and there may be other opinions, but since they are waves and have no direction of oscillation, they do fit the definition of longitudinal wave. In that theory though they tend to call them scalar waves which has a more accurate technical meaning.

Answer 2

Some include sound waves, which are the most well known, and surface waves (like ocean waves), which are both longitudinal and transverse waves.