# Historical Geology/Seismic waves

In this article we shall review some important facts about the physics of seismic waves (that is, waves generated by earthquakes).

## Surface waves and body waves

When an earthquake occurs, it is the cause of seismic waves, including both waves that travel along the surface (Love waves and Rayleigh waves) and waves that travel through the body of the Earth (P-waves and S-waves, known collectively as body waves).

In this and succeeding articles we shall be interested only in body waves, since by traveling through the Earth they give us clues about the Earth's interior.

## P-waves and S-waves

P-waves are waves of compression and tension, like sound waves; indeed, they are sound waves, traveling through rock rather than through air. S-waves are waves of shear: that is, of displacement at right angles to the direction of travel of the wave, resembling the waves produced by shaking the end of a rope.

The image to the right shows the motion of P-waves (top) and S-waves (bottom) for a horizontally travelling wave. Look carefully at the picture of P-waves. If you focus on any particular vertical line, you will see that it is merely oscillating from side to side: it is the regions of compression that are moving from left to right, while the medium itself has no net motion. Similarly in S-waves, it is the displacement that moves from left to right, while the medium through which the waves move exhibits no such motion.

If we use κ to represent the incompressibility of a medium, μ to represent the rigidity, and ρ to represent the density, then the velocity of P-waves through that medium will be given by

$v_{P}={\sqrt {\frac {\kappa +{4 \over 3}\mu }{\rho }}}$

and the velocity of S-waves by

$v_{S}={\sqrt {\frac {\mu }{\rho }}}$

From these formulas we can immediately see that a P-wave will always travel faster than an S-wave through the same medium. We can also see that S-waves will not travel through liquid at all, since liquid has no rigidity and so in the case of liquids µ = 0.

## Refraction and least time

The principle of least time says that a wave traveling through a medium will take, not the shortest route as measured by distance, but rather the quickest route between two points.

The word refraction means the change of direction undergone by a wave when it passes from a material which permits travel at one speed to a material which permits travel at another speed. The existence of refraction is a direct consequence of the principle of least time: in the generalized diagram of refraction shown to the right, the path of the ray is the quickest route from point A to point B.

Using the principle of least time, we can say exactly how refraction should take place: a little simple mathematics tells us that the ratio of the velocity in medium A to the velocity in medium B should be equal to the ratio of sin θA to sin θB; or to put it another way, equal to the ratio of xA/dA to xB/dB. This is known as Snell's Law. Where the velocity changes smoothly and gradually through the medium, this will result in the ray taking a curved path.

The upshot of all this is that if we know the wave velocities associated with each point in an object then we know exactly how a wave will travel through it, since its motion is determined by the principle of least time.

## Partial reflection

When a wave encounters a sudden transition between mediums with different associated velocities, so that the wave is refracted at an angle, some of the energy will also be reflected back in a direction determined by the well-known law that the angle of incidence is equal to the angle of reflection (see the diagram to the right). This is why one can see faint reflections in windows and in water.

P-waves and S-waves behave in the same way as more familiar waves such as light, but with one difference: light is exclusively a transverse wave, and reflects back as one. When either a P-wave or and S-wave is reflected, however, the reflection will be composed of both P and S-waves.

When such reflections of P and S-waves are detected, this tells us that they are being reflected off some sort of sharp boundary between rocks having different physical properties and hence different associated velocities.

## How do we know?

The properties of body waves can be studied in the laboratory. They can also be derived from more basic principles, simply working from the fact that they are waves and must therefore obey the physics of waves. (Note that in physics a "wave" is not just something that behaves in a kind of wave-y sort of way, but something the dynamics of which can be described by the wave equation. The mere fact that body waves are waves therefore tells us quite a lot about them.)

## Seismic tomography

As discussed, the routes taken by P- and S-waves through the interior of the Earth are determined completely by the velocities at which these waves travel at each point in the interior.

If these velocities were the same for every point in the Earth, then the time it took for a wave to travel from the focus of an earthquake (the focus being the point where the earthquake occurs, not to be confused with the epicenter, which is the point on the surface directly above the focus) to an earthquake detector (a seismometer) would be proportional to the distance of a straight line drawn from the one to the other through the body of the Earth.

But this is not the case. By studying the data showing how long the waves do take to travel through the Earth, it is possible to determine the velocities of P-waves and of S-waves at each point in the Earth.

Doing so is what mathematicians know as an "inverse problem": it might be compared to trying to reconstruct the shape of an object by observing its shadow. It would obviously be much easier to work the other way round and deduce the shadow from the object; and similarly the general problem of seismic tomography — that is, of discovering the inner structure of an object by studying the passage of body waves through it — would be very difficult to solve.

But fortunately we do not have to solve this problem in the general case, but only for one particular object: the Earth. This problem has a property that makes it particularly easy to analyze.

Consider the fact that the time it takes for a P-wave or an S-wave to get from the focus to a seismometer depends to a good degree of approximation only on the angle of separation between the focus and the seismometer. This tells us that (again to a good degree of approximation) the values of vP and vS at any particular point within the Earth must depend only on the depth below the surface of the Earth and not on the longitude and latitude (in technical terms, we may say that the values of vP and vS are spherically symmetric).

Consequently, what looked like a three-dimensional problem involving finding values of vP and vS for every depth, longitude, and latitude can be reduced to the one-dimensional problem of finding values of vP and vS at every depth.

This fact makes it possible to produce a graph such as that shown to the right, relating vP and vS to depth; the figures are taken from the Preliminary Reference Earth Model (here). One fact that you should notice immediately is that the speed of vS drops to 0 in the outer core, showing that it is liquid.

I have said that the travel time of body waves depends on the angular separation between the earthquake focus and the seismometer to a good degree of approximation: but this is not exactly and perfectly the case. By studying the small variations from this rule, it is possible to detect low and high velocity anomalies within the Earth: volumes where the waves travel slower or faster than what we should expect if the Earth's interior was perfectly spherically symmetric. Work on this is ongoing.