Seismology Part V: Surface Waves: Rayleigh John William Strutt (Lord Rayleigh)

Why are surface waves important? The Earth is a finite body and is bounded by a free surface; that is, the part we live on and build buildings on. Plus, they are often the 300 pound gorilla on a seismogram! An Earthquake recorded at Binghamton, NY

The way to think about how to explain surface waves is to ask if there are any possible solutions to the wave equation that would trap energy at the surface. The trick is to see if you can make evanescent waves somehow, just like in critical refractions. Thus, we begin our discussion by recalling the free surface boundary condition: Free Surface. No displacement constraint. Traction is zero. (That’s why it’s free!) Again, if the interface is horizontal, then n = (0,0,1) and

Recall that If we concern ourselves only with motion in the (x 1, x 3 ) plane, then u 2 = 0 and d/dx 2 = 0, so (i.e.,  23 does not provide any additional constraints).

The potentials in the layer will be the sum of incident and reflected P and the potential due to a reflected SV wave: We can write the solution to the wave equations as: The signs in the arguments that correspond to the direction of propagation, and the appropriate choices of the x 3 factors. The x 1 factor is the same in every case because of Snell's law.

Recall that So, in general, we consider the displacements at the interface substituting the above expressions into the following:

In the x 1, x 3 plane this becomes: Let's see what happens when we apply the no traction boundary condition:

or Next so

Evaluating the above at x 3 = 0: We solve for A 2 /A 1 =R PP and B 2 /A 1 =R SS (for potentials!) Eliminating B 2 : Hence

Eliminating A 2 : Hence

Note that there is an angle of incidence where almost all the reflected energy is S. We can repeat the above exercise with an incident SV wave to obtain: Because a > b, there will exist an angle of incidence where the P wave “refracts” along the free surface. Note however that in this case, p > 1/  and n  is imaginary = i(p 2 – 1/  2 ) 1/2 and the total scalar potential is:

To prevent this from blowing up with large x 3, we must have A 1 = 0. If we require If A 1 = 0, then in the absence of an SV wave we have: Which means that A 2 = 0! So, these evanescent (exponentially decaying) P waves cannot propagate by themselves. To trap energy at the surface, we need to generate interference between P and SV waves, and this is what Lord Rayleigh figured out in 1887.

The way to do this is to presume that an appropriate solution might exist and then see if it really does. So, let's presume that we can trap energy at the surface by allowing both evanescent P and SV waves to exist simultaneously. In this case, the potentials are: Note that in this case p > 1/  > 1/  which means the that the horizontal apparent velocity is less than both the shear and compressional wave velocities.

Now let's apply the traction free boundary conditions:

Evaluating at x 3 = 0 Remember that and, so

Now apply the other traction condition Thus, evaluating at x 3 = 0 or

The question now is: is there a nontrivial choice for A and B that satisfies these two equations? We write them in matrix form: The above system of equations will have nontrivial solutions for A and B only if the determinate of the 2x2 matrix is 0. Thus, we require that This term is the same as the denominator of the all the reflection coefficients (R PP, R PS, R SS, R SP ), and is called the Rayleigh denominator. Note that if we were dealing with real reflected waves the amplitude would be infinite! But remember that we started off assuming that we would be dealing with evanescent waves trapped at the surface.

Let's divide this equation by  so we can talk about wavespeeds instead of elastic moduli. Also remember that factor out p 4 :

where c = 1/p is the apparent horizontal velocity. So we need to solve this for c. Factor a  2 from the above:

Multiply the above by To get

The first term on the left becomes: The second term on the left is:

Combining A solution to the above can be found for any  and .

which has roots Of these, only the last satisfied c < , and in this case c =  generally c is in the range of 0.9  to 0.95  As an example, assume a Poisson solid (  ) in which case  2 = 3  2 :

What are the particle motions? We need to solve for displacement (u): at x 3 =0. From the  33 = 0 condition above, we found that

or Recall that

The shear term for u 1 is then

Combining: The shear term for u 3 is:

Combining: Now, because of the "i" in front of u 1, the u 1 motion is 90 degrees out of phase with that of u 3 (i = e i  /2 ). Another way to see this is to consider the real parts of u 1 and u 2. u 1 will have a sine for the first term, and u 2 a cosine.

For a Poisson solid, c =  = . If we also let k=  p=  /c be the Rayleigh wave number, then and and At the surface (x 3 = 0): The motion described by the above is retrograde elliptical. At a depth of about /5, the motion goes to zero, and reverses to a prograde motion at deeper depths.

Notes: 1. There is no tangential motion (u 2 = 0) 2. The rate of decay depends on k, which means that longer wavelength waves penetrate deeper into the earth. 3. Wavespeed (c) does not depend intrinsically on frequency, so these waves are not dispersive in a half space. However, if  and/or  increase with depth, the waves will disperse with the longer wavelengths coming in first. 4. Two Dimensional spreading means these guys are large amplitude for long distances. Scorpions use them to locate prey. 5. Wave curvature is required to generate Rayleigh waves, which means that deep sources generally do not produce them.

Example of Dispersion of Rayleigh waves Rayleigh waves are not naturally dispersive, but become so if there are vertical variations in wavespeed. A simple but useful example of what happens can be had by considering what happens in a fluid layer over a half space. Let the boundary between the fluid and the solid be at x 3 = 0, and the top of the fluid at x 3 = -H. There are no S waves in the fluid, and the P wave potentials are: In the half space, we have

At the surface of the liquid, the boundary condition is

Which will be true for all p only if  I = -  R. Evaluating at x 3 = - H gives or At the interface, we require that u 3 is continuous:

So Evaluating at x 3 = 0 gives: Note that

So Continuity of the  33 gives: From above:

For the half space:

Thus Evaluating this equation at x 3 = 0 gives:

Note that So

On the right side: Hence So

Finally, we have to satisfy the shear stress condition on the interface:  is zero in the liquid, so we must have  13 = 0 at the interface. So

Evaluating the above at x 3 = 0 We now have 3 equations with 3 unknowns (A, B, and C 1 ). Writing these in matrix form:

Non trivial solutions exist when the determinate of the above matrix is zero: or solving for tan(  ):

The term in the brackets is just the (negative) Raleigh denominator again. Recalling the substitutions we made before: The term in the brackets becomes:

factor out p 4 : where c = 1/p is the apparent horizontal velocity. So the bracket term becomes

Factor a  2 from the above to get: or

(NB: For readers of the textbook by Lay and Wallace, this is their equation except that the density terms are inverted. Note that there is an extra 1/  term in their equation which is obviously wrong since the tangent MUST be dimensionless. Thus