# Why do P waves have a higher velocity in the lower mantle than in the core even though the core has a much higher density?

I've just started teaching 9th grade Earth science and my students and I have noticed something about P and S wave depth vs velocity graphs. In the typical graph, eg: https://www.researchgate.net/figure/Earths-P-wave-velocity-S-wave-velocity-and-density-profiles-as-a-function-of-depth_fig8_280835886

With the p wave's velocity plotted from a depth of 0 km to a depth of 6371 km it shows the p wave velocity reaching about 14 km/sec by the time it reaches the bottom of the mantle. Then of course it drops precipitously through the mantle/outer core boundary. However, even when the wave reaches the inner core, it's velocity never gets close to that 14 km/sec speed, it's speeds in the inner/outer core range only from 8 to 12 km/sec even though the core has a much higher density than the mantle. You would expect a much higher wave velocity.

tldr; Increased density corresponds with decreased P-wave velocity

The P-wave velocity for an isotropic medium is:

$$V_p = \sqrt\frac{K+4/3\mu}{\rho}$$

where $$K$$ is the Bulk Modulus, $$\mu$$ is the shear modulus, and $$\rho$$ is the density (https://en.wikipedia.org/wiki/Elastic_modulus, see $$M$$ the P-wave modulus). Isotropy just means that the strain on the rock caused by a stress is irrespective of the stress orientation. Examining the equation you can see that increased density corresponds with decreased velocity.

In the mantle, $$\rho$$ increases with depth, but the numerator $$K+4/3\mu$$ increases faster than the $$\rho$$, so the net effect is an increase in velocity.

In the fluid outer core, $$\mu = 0$$. This is because fluids do not support shear stresses. Correspondingly, $$V_p$$ drops precipitously.

In the solid inner core, $$\mu$$ is nonzero hence there is a jump in velocity at the inner/outer core boundary. The combination of $$K$$, $$\mu$$, and $$\rho$$ in the inner core is such that $$V_p$$ doesn't exceed the lower mantle values.

• To supplement the informative answer @dvoytan provided, I'll add some more insight in regards to the density idea. Namely, greater density translates to greater inertia for a given unit volume - be it fluid or solid. This relates to Newton's part of the wave equation and not as much Hooke's (which deals with the continuum mechanics terms relayed in the answer).
– nate
Nov 26 '19 at 14:35