Question:How do we know about the structure of the Earth; in particular, that the core is solid but the outer-core is liquid?
Seismometers are positioned all around the Earth.
When an earthquake occurs the seismic waves travel out from the location of the Earthquake and are detected by these seismometers.
By looking at how long it takes for the p and s waves to reach each detector scientist can work out exactly where the earthquake took place.
If the s-wave does not arrive at some detectors then there must have been a layer of liquid in the way because only p waves can travel through liquids.
Differences in the patterns can also tell scientist about changes in the composition as refraction will cause the path of the waves to bend at interfaces between different layers of the earth.