Same way as you can tell a bell IS hollow, you bang hard on one side and see what happens on the other side. In the case of the earth, many things from nuclear testing, volcanoes and even seismic vibrations are strong enough to hear after they pass through the Earth. You can calculate the average density of the material by seeing how long the vibrations take to cross the distance. Taking different angles and slices to compare let’s you map out the various densities of layers.
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