Two of the kinds of waves, P waves and S waves form because of the way rock springs back when it is stretched. Rock can get pulled like a rubber band and spring back. It can also get bent like a diving board with a diver standing on the end and spring back like the way the board will spring back when the diver dives off. Both of these kinds of springing occur during earthquakes, so that explains two of the kinds of waves. The third kind, called surface waves, occurs because when the other two types of waves reache the surface they cannot just keep going on up into the air. Instead, their energy causes the surface waves to form so that the energy can keep moving away.
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