So as you can see only the P waves can penetrate the outer and inner core, the S waves get stopped at the core which is why you can't detect them past 105° ( about 11 000km) from the epicentre.
The P-waves can travel through everything but they also have a (smaller) shadow zone as they are refracted when they reach the outer (and inner) core. Remember that it's the surface waves (L and R) rather than the body waves (P and S) that cause all the damage as the send energy through the surface layers of the earth's crust. The behaviour of P and S waves enabled scientists to work out the inner structure of the earth.
Remember that P waves are also a lot faster than S waves, about twice as fast. This enables seismologists to use them to warn people in advance of an earthquake as the P-waves can be detected before the destructive S, L and R waves arrives. They only allow a 60-90 second warning at most but even this is better than nothing - enough to shut off gas and electricity supplies to prevent fires for example, or to leave your house or put protective padding on your head...
Thanks to Millie's Geology Blog for the seismic waves picture
Check out this site for more info on seismic waves
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