# 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). Nov 26 '19 at 14:35