<p>tldr; <strong>Increased density corresponds with decreased P-wave velocity</strong></p>

<p>The P-wave velocity for an isotropic medium is:</p>

<p><span class="math-container">$$
V_p = \sqrt\frac{K+4/3\mu}{\rho}
$$</span></p>

<p>where <span class="math-container">$K$</span> is the Bulk Modulus, <span class="math-container">$\mu$</span> is the shear modulus, and <span class="math-container">$\rho$</span> is the density (<a href="https://en.wikipedia.org/wiki/Elastic_modulus" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Elastic_modulus</a>, see <span class="math-container">$M$</span> 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 <strong>increased density corresponds with decreased velocity</strong>. </p>

<p>In the mantle, <span class="math-container">$\rho$</span> increases with depth, but the numerator <span class="math-container">$K+4/3\mu$</span> increases faster than the <span class="math-container">$\rho$</span>, so the net effect is an increase in velocity. </p>

<p>In the <strong>fluid</strong> outer core, <span class="math-container">$\mu = 0$</span>. This is because fluids do not support shear stresses. Correspondingly, <span class="math-container">$V_p$</span> drops precipitously.</p>

<p>In the <strong>solid</strong> inner core, <span class="math-container">$\mu$</span> is nonzero hence there is a jump in velocity at the inner/outer core boundary. The combination of <span class="math-container">$K$</span>, <span class="math-container">$\mu$</span>, and <span class="math-container">$\rho$</span> in the inner core is such that <span class="math-container">$V_p$</span> doesn't exceed the lower mantle values.</p>

CC BY-SA 4.0
2019-11-26T14:35:27.517Z
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
CC BY-SA 4.0
2019-11-23T01:41:03.977Z
2019-11-23T01:41:03.977Z
dvoytan
<p>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: <a href="https://www.researchgate.net/figure/Earths-P-wave-velocity-S-wave-velocity-and-density-profiles-as-a-function-of-depth_fig8_280835886" rel="nofollow noreferrer">https://www.researchgate.net/figure/Earths-P-wave-velocity-S-wave-velocity-and-density-profiles-as-a-function-of-depth_fig8_280835886</a></p>

<p>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. </p>

<seismic><waves>
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?
Greg Walsh
# 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?

(Asked by: username_0 on 2019-11-21)

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.

**Tags:** seismic, waves


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## 1 answers

### Answer by username_1 on 2019-11-23. Score: 3

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.

> **Comments:**
> **username_2**: To supplement the informative answer @username_1 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).  

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