Can we surf a seismic surface wave?


‘For every complex problem there is a solution that is clear, simple and wrong’, is a Mencken quote* that applies to many geophysical problems.

In geoscience we have no alternative to simplifying the problems we deal with: the complexity of the Earth, with its constitutive laws and heterogeneities is too large for our small minds and computers. We have to build models, describe the complex nature with simpler, manageable systems and objects.

But the devil is in the details. Sometimes we oversimplify and lose the essence, and we model waves that cannot be surfed.

In the post’s title there’s a steep, breaking wave, which cannot be described with a simple linear model. The image is of course drawn starting from The Great Wave Off Kanagawa, the masterpiece of Hokusai ( Simple models are not sufficient to describe that rogue wave that is dramatically threatening three fishing boats; simple models cannot even describe the familiar ocean waves that we can see on the coast.

The simple linear Airy wave is valid when the surface and the associate boundary condition doesn’t move too much. It is easy to describe and clear when drawn: a circular particle motion, a sinusoidal surface… rather similar to a Rayleigh wave in a homogeneous halfspace. If you start from Rayleigh, you just need to add a turtle, change the particle motion from elliptical to circular (in deep water), and reverse it from counterclockwise to clockwise.


The Rayleigh waves are actually more complicated: in the previous figure you can see the red orbit changing direction, at a certain depth. And if the subsurface is not homogeneous (it never is!) the velocity variations produce geometric dispersion, and multiple modes: the trajectory becomes a tilted ellipse.

All this can be predicted by a model, just a bit more complicated than the previous one. But there is another detail (hosting a devil): if the seismic source has a finite size, if it’s not a point source, and if it has strong harmonics, everything complicates further. When you record the particle motion, you don’t observe a tilted ellipse, as in the 5 true data examples on the right part of the figure below. In the data, the particle trajectories are like the recorded GPS tracks of particles that have raced 50 laps on different, bizarre formula1 circuits. And the five receivers of this experiment are only a few meters apart.


Modeling the seismic surface waves can be complicated: basically because the medium in which they propagate is very heterogeneous, unlike water. But water waves are not easy either. Feynman wrote that water waves are the usual but worst example of waves in elementary physics courses: they have all complications that waves can have.

The linear waves are not steep enough. To be able to surf, you need to model shallow-water waves, with a higher degree of non-linearity and the Stokes-drift. When moving into shallower water, then the wave velocity decreases, the wavelength shortens and increases the wave height: and the faster top overtakes the slower bottom, and spills forward. The break!


Coming back to the initial question: can you surf a seismic wave? No.





The true Menken quote is more general: Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.



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