When it comes to seismic waves, there are two categories, body waves, and surface waves. They each do many different thinks, and interact in different ways.
The first kind of body wave is the primary wave, or p-wave for short, it acts as a longitudinal wave and arrives first during earthquakes because they are the fastest waves. They pass through the solid and liquid parts of the earth.
The next body wave is calles the secondary wave, or s-wave, these waves act like transverse waves and are the second waves to arrive because they are slower than p-waves. They can only travel through the liquid part of the earth.
Surface waves are iconic for destroying buildings and only occur at the surface, hence the name. For our first surface wave, we have the love wave, these waves move in a serpentine motion and we're mathematically discovered by A.H.E Love, a British mathematician.
Our last wave is the Rayleigh wave, this wave acts like a water wave and causes a lot of destruction, this wave can get pretty big, this wave was discovered by John William Strutt, or Sir Rayleigh.