My current knowledge about how the core of the earth was discovered is the observation made from non returning/reflecting sonar signal when sent to deeper depths through the earth's crust/mantle. How was it inferred that the the core was molten?
There are basically two types of waves - shear (s) waves and primary (p) compression waves.
P waves can travel through solids and liquids but s waves do not travel in fluids.
That p waves, but not s waves, are detected from events occurring on the other side of the globe tells you that there is fluid in the way.
This is the basic picture; it is complicated/enhanced by refraction and dispersion of the waves.