A wave is energy moving through a medium. The medium — whether it's a spring, water, air, or rock — doesn't travel with the wave. The energy does. Your spring just showed you this twice, in two completely different ways.

The first type you made is called a transverse wave.

When you shook your end up and down, each coil of the spring moved up and then back to where it started — at right angles to the direction the wave was travelling.

The wave shape moved along the spring from your hands to your partner's, but no coil went anywhere. Transverse waves include light, the waves on the surface of water, and seismic S-waves (the side-to-side shaking waves in an earthquake).

The second type is called a longitudinal wave.

When you pushed a bunch of coils toward your partner and pulled back, you created a pulse of compression — coils bunching together — followed by a region of expansion — coils spreading apart. That pulse travelled along the spring as the compression and expansion passed from coil to coil. Again, no coil ended up at your partner's end. The energy travelled. 

The spring stayed.

Longitudinal waves are how sound travels. Right now, as you read this, the air molecules around you are being compressed and expanded by the sound waves passing through them — exactly the way the spring coils were. The compression pulse carries the energy from the source to your ears. The air molecules themselves barely move — they just push the next one along, which pushes the next, all the way to your eardrum.

The two wave types are not just different shapes — they behave differently, travel at different speeds through different materials, and carry energy in fundamentally different ways. Understanding both is the foundation of acoustics, seismology, medical imaging, and the physics of light.