Purdue University

EAS 557

Introduction to Seismology

Robert L. Nowack

Lecture 15A

Seismic Surface Waves

 

 

      We now investigate seismic surface waves which have the following properties.

 

1)   Surface waves propagate parallel to the Earth’s surface.

 

2)   Amplitude as a function of depth is stationary with horizontal distance (apart from spreading and anelastic viscoelastic attenuation).  The amplitude decay from geometric spreading for different waves is

 

surface waves         (cylindrical spreading)

 

body waves          (spherical spreading)

 

refracted head waves     

 

      Thus, at long ranges, surface waves will be larger in amplitude than the body waves.

 

3)   Long period surface waves give information about earth structure as well as the source mechanism.

 

4)   In seismology, the most important interface is the Earth’s free surface. 

 

5)   Except for first arriving P and S waves, surface waves will be the most prominent ground disturbance seen on seismograms.  Also, their character is very different than body waves.

 

 

 

 

As a first example, we’ll look at the antiplane (SH) case for a layer over a half space.  The antiplane guided waves are called Love waves.  Consider, the elastodynamic equation with fi = 0.  For an isotropic solid

 

 

and

 

 

 

Consider an SH wave traveling in the x1 direction, but with particle motion in the x2 direction.

 

 

Substituting this into the elastodyamic equation then,

 

 

For SH particle motion in a layer over a half space with , then a horizontally propagating wave, called a Love wave, can exist,

 

 

 

 

where  and .  Within the layer,

 

 

In the halfspace,

 

 

Now, we will use the trial solution

 

 

The boundary conditions of the problem are

 

     at the interface on plane x3 = H

 

     at the interface on plane x3 = H

 

     on the free surface

 

 

            What kind of solution are we looking for?

 

1)   We want surface waves, i.e., waves whose displacements are confined, “close” to the free surface x3 = 0 and don’t increase in amplitude with increasing depth.

 

2)   We want a wave whose horizontal velocity,  is the same for all depths x3.  (In particular, the horizontal speed should be the same for the layer and the half space.)

 

3)   The phase velocity  could possibly be frequency dependent.

 

This is not the same problem as plane waves incident on a boundary at some angle .  We are trying to find a solution to the equation of motion which is a wave of a different kind with the properties listed above.

 

            We seek a solution propagating along the x1 axis with horizontal phase velocity  and time dependence , where  and .  Use the solution form  for the layer and a similar form for the half space and put these into the equation of motion.  Then

 

 

or,

 

 

for the layer and a similar equation for the half space.  A solution for h(x3) can be written

 

     in the layer for

 

     in the half space for x3 > H

 

where

 

    for 0 < x3 < H

 

    for x3 > H

 

We now have unknowns .

 

            We now apply the boundary conditions to solve for the unknowns.

 

1)   First, we will assume that there are no sources at  and thus no up-going waves in the lower half space.  Thus,  in the lower half space.

 

2)   If  or , then  and .  We thus want an evanescent wave in the half space, i.e., an exponentially decreasing function with depth in the halfspace.

 

3)   At the free surface,  at x3 = 0.  Then,

 

 

      This gives

 

 

      We are then down to three unknowns

 

4)   We must now satisfy continuity of displacement and traction at the interface x3 = H.

 

            For displacement     

 

            For traction             

 

 

            Now, recall

 

 

and

 

 

Now, the above two equations can be written

 

 

 

then,

 

 

This gives two equations for  if we know k1.  The equality between the right two terms gives an equation .  This is called a dispersion relation for .  Thus,

 

 

where the vertical wavenumbers for the layer and half space can be written.

 

 

 

 

 is an implicit equation to find  or .  The solution is called an eigenvalue for a given frequency .  Writing this in terms of  gives

 

 

where  results in an evanescent wave in the lower half space.

 

            We will use a graphical method to solve this for .  For , then there are real roots to

 

 

where

 

 

 

 

 

 

 

At X = 0, .  For , , and

 

 

 

The following comments can be made.

 

1)   Real roots are limited to lie between .

 

2)   For any given frequency , there will be multiple roots (but only a countable number of them).

 

3)   For , the tangent curves will expand giving only the n = 0 curve between the  and  lines.

 

4)   For slightly larger , the n = 1 tangent curve will enter the real region from the right.  It enters when  is such that

 

 

5)   For larger , more of the tan curves enter from the right.  The nth curve enters when

 

 

      This is called the cutoff for the nth higher mode.  The nth higher mode can only propagate for .

 

            Ex)  The continental crust beneath Indiana has a crustal thickness of about 35 km.  Assume

 

 = 3.5 km/sec     = 4.5 km/sec

 

      then,

   where    = 0.08 Hz

 

      or the period of the cutoff for the 1st higher mode is  = 13 sec = .  Thus, for this velocity structure, only the n = 0 fundamental mode Love wave can propagate for T > 13 sec (or  cycles/sec).

 

6)   At each mode’s cutoff frequency, then .

 

7)   As ,  for all modes and a large number of them can propagate.

 

 

      A graph of the horizontal velocities as a function of frequency is shown below.  (This is often also plotted as a function of period.)

 

 

 

 

The change of  with  is called the dispersion curve for a given mode number n.

 

 

            Now with the eigenvalues derived  or ) which satisfy the dispersion relation, the corresponding vertical eigenfunction h(x3) can be found as

 

 

 

 

where

 

 

 

The vertical eigenfunction is a sinusoidal standing wave in x3 in the layer and exponentially decreasing in the lower half space.  The first three eigenfunctions for a frequency near the cutoff frequency and at a larger frequency are shown below.

 

 

 

 

Different mode numbers correspond to a different number of zero crossings in the layer.  Also, each mode for each frequency travels at a different horizontal speed.

 

            If a source is at one of these nodal depths, then the source will not be very efficient in generating that mode or vertical eigenfunction for that frequency.

 

For an actual source, we must superpose all allowable modes to satisfy the given source condition.

 

            For more layers than one, we must use a computer program to solve the dispersion relation for k1 or , as well as to find the vertical eigenfunctions.  The steps include:

 

1)   Solve the dispersion relation for  for a given . 

 

2)   Solve for the vertical eigenfunction  for each  for a given .

 

3)   Superpose the different eigenfunctions to satisfy a source condition for each frequency.

 

4)   Superpose all frequencies for the source excitation.

 

5)   If this is done correctly, this will simulate the surface wave contributions to the seismic signal.

 

 

 

Next, we will look at the P-SV case for in plane particle motion.