# How do we know the speed of S-waves (shear waves) in the Earth's inner core? Has this been measured?

Wikipedia's Seismic wave includes the plot below of the speed of S and P waves as a function of depth in the Earth. In the region of the outer core the speed of S-waves or shear waves is shown to be zero but really they are evanescent and so the velocity is probably complex.

Then at the inner core boundary the speed of S-waves is shown again down to the center of the Earth, fairly flat near 4 km/sec.

Is the speed of shear waves in the Earth's inner core a measured quantity or is this just deduced? If it's measured, how do these shear waves get to the core if they can not propagate through the outer core?

• – uhoh Apr 4 '20 at 2:34
• Perhaps your question is based on one of the following two assumptions: that once the S-wave is killed in the outer core, we will not either be able to create an S-wave in the inner core, or that we won't be able to see what kind of S-wave was present. It turns out that a P-wave moving from outer to inner core generates new S-waves in the inner core, and that S-waves in the inner core can create new P-waves that travel to the surface again. See how the green P-wave front generates red S-wavefronts in the core youtube.com/watch?v=j7eoxizmC1I . Those things can be measured! – Erik Apr 4 '20 at 6:16
• @Erik that's a beautiful video, thank you! I noticed that as long as the P-wave is away from precisely normal incidence on a liquid to solid interface there is indeed a new S-wave produced. The arrow shows a gap in the S-wave front near the line of symmetry i.stack.imgur.com/WRe9x.jpg Consider writing your comment up as an answer? Thanks! – uhoh Apr 4 '20 at 6:28
• @uhoh: Have a look at this paper: science.sciencemag.org/content/362/6412/329.full. I haven't read it because time, but it seems to contain some background. – user20217 Apr 4 '20 at 8:29
• @uhoh that's right and well-observed, indeed at normal incidence there is no mode conversion from P to S waves. The angle-dependent relation for the amplitudes is given by the Zoeppritz equations, en.wikipedia.org/wiki/Zoeppritz_equations , which are rather complicated! – Erik Apr 4 '20 at 8:44

If waves would travel only through a medium while keeping their form (i.e., P-waves remain P-waves and S-waves remain S-waves), you would obtain something a bit like this video: https://www.youtube.com/watch?v=YctV5crEXyM .

However, standard models for wave propagation are more complicated than that. Upon hitting an interface, P-waves both reflect off and refract into the other material. And if this happens at a bit of an angle, it even generates a reflected and a refracted S-wave (if the material allows it!). Something like this image:

What you see is a P-wave that comes towards an interface between two different media (Vp=P-wave velocity, Vs=S-wave velocity, \rho=density). After hitting the interface, it creates 4 new waves! Two P-waves, two S-waves. A similar figure can be made for an S-wave hitting an interface.

So we can follow a hypothetical wave-path as follows:

1. There is a P-wave generated at the Earth's surface that travels down from the crust to the mantle (creating 4 new wave modes as in the figure above; we follow the transmitted P-wave that goes down),
2. That wavefront travels from the mantle to the outer core (which creates 3 wave modes because there is no transmitted S-wave! We follow the transmitted P-wave the goes down).
3. That wavefront travels from the outer core into the inner core (which creates 3 wave modes, because there is no reflected S-wave; We follow the transmitted S-wave that goes down).
4. The S-wave travels through the center of the Earth back up to the interface between inner core and outer core (creating 3 wave modes, because there is no transmitted S-wave; We follow the P-wave that goes up).
5. The P-wave travels through the outer core and hits the interface with the mantle (which creates 3 wave modes because there is no reflected S-wave; We follow the P-wave that goes up).
6. This P-wave further travels to the Earth's crust and surface (creating more wave-modes in the process), and is measured there.

You can see some of these effects in this nice video: https://www.youtube.com/watch?v=j7eoxizmC1I that I put in the comments also. It shows P-waves in green and S-waves in red. You can see how complicated the total wavefield becomes because every interface creates new waves...so it'll be a bit hard to see the particular wavefront I described above, but you can see the general picture.

On this snapshot you see the P-wave (in green) generating an S-wave (in red) in the inner core.

So -- S-waves can exist in the inner core even when the outer core doesn't support them (because waves modes convert, as it's called, at interfaces), and their presence can be measured at the surface. I think that answers your question mostly.

Then, of course, there is still the question of how you'd measure the speed of the S-waves in the inner core, which is rather complicated. These days, it is done by assuming some seismic model of the earth (choosing V_p, V_s, and \rho everywhere within the Earth), and modeling seismic events such as on the linked YouTube video. Then you compare real recordings against your modeled data. If there are particular misfits, you create a new model and you iterate this procedure until your model fits the recorded data 'well'.

So is the shear-wave in the inner core a measured quantity or a deduced quantity? That requires a very careful definition of those terms, so I won't answer that question! Regardless, there exists a shear-wave velocity for the inner core that describes and explains recorded data very well.

• Wow, thank you for the very thorough answer! – uhoh Apr 4 '20 at 10:31