## PSN-L Email List Message

Subject: Re: Wavelength of Surface Waves
From: John Hernlund hernlund@.......
Date: Mon, 10 Apr 2000 10:08:35 -0700 (MST)

```On Fri, 7 Apr 2000, Paul Jebb wrote:
> Dear Members,
> I am trying to getting a feeling for the  S- waves that our Lehman
> seismometer will detect.

Do you mean surface waves or do you mean S-waves?  These are two different
concepts in that an S wave travels through the interior of a body while a
surface wave travels along the surface of a body because of the way it snaps
back when waves arrive at the edge.

When an earthquake source occurs, waves are given off at many different
frequencies and different amplitudes corresponding to each frequency.  If you
made a plot of the amplitude vs. the frequency of the source (the fourier
transform) then you would see some type of specific shape.  As the distance
these waves travel increases, the amplitude vs. frequency plotted at each
distance will begin to keel over to the right so that higher frequencies get
lower and lower faster than lower frequencies do.  If you use a logarithmic
scale this tilt to the right should be like tilting a straight line.  As the
distance from the source increases then the straight line has a more negative
slope.

The shape of the initial amplitude vs. frequency plot is not know in general
from the earthquake source.  To simplify matters a little, we often use
"impulse" sources in time.  This is like a sharp hammer strike with a very
short duration.  So the activity on the fault itself would only have a spike
if you looked at displacement vs. time.  It turns out that the amplitude vs.
frequency plot of an impulse source is a flat line where all the frequencies
have the same amplitude.  This means that the straight line will begin to tilt
to the right as the waves travel away as described above.

Now this scenario is simplified too much to account for everything that is
seen in seismograms.  So instead of using a single impulse, we can approximate
the true source by adding a bunch of impulses that occur in different
locations and at different times along the fault.  In this way, a very
accurate model can be made.  So since we can model this source with a bunch of
spikes, and we know how the magnitude vs. frequency changes over the distance
travelled, you might have guessed that we can get a pretty accurate idea of
what the actual seismogram should look like at any sight along the wave path.
Because the amplitude-frequency is a fourier spectrum, you can perform an
inverse fourier transform on it to see how the amplitude changes with time,
which is what a seismogram is.

These records are called synthetic seismograms, and can be used to compare
with actual records to see how accurate your estimates for the source really
are.  There will be some variations in the real seismogram that cannot be
accounted for by the source.  These variations give you specific information
about the material the waves travelled through.

Also, the amount that the spectrum tilts to the right over a given distance is
a measure of the "attenuation" of the wave.  As you can imagine, people have
come up with all kinds of line fitting methods to measure how much tilt has
occurred.  This is another way of mapping out the material the waves are
passing through.  Generally, hot materials attenuate more (tilt the frequency
spectrum more per a given distance) than cool materials.  This is because hot
materials deform inelastically more easily than cool ones and take energy away
from the seismic waves passing through them by rearranging some of their
atoms.  If you look in the literature, you can find a lot of interesting
studies on this topic as well as maps of attenuation (also referred to a Q or
quality factor, though high Q implies low attenuation).

Indeed, you can take real seismograms and back up time to the initial source
to get an estimate of what the real source actually looks like.  This is a
wonderful way to study active faults.  You can tell where the most slip has
occurred and even when it occurred.  This has been done beautifully for the
Taiwan earthquake.

> If the period of the boom must be adjusted to around 20 seconds or greater,
> what is the wavelength of these waves.  My back-of-an-envelope estimate
> (v=fl) is in the order of 100 km.  Is that reasonable?

Yes, this estimate always works, provided you know the velocity well enough.
Mostly though, the velocity changes quite a bit in the intervening material,
but again, this can also be measured by compiling many records from different
locations.  An interesting thing to think about is the way in which waves with
different wavelength interact with objects they are passing through when the
objects diameter is around the wavelength of the wave.  If the diameter is
much smaller, than the wave does not "see" the object other than its
contribution to the mechanics of the whole interval.  Higher frequencies can
then see smaller objects, because the wavelength is shorter.  But as waves
travel, we have seen that the high frequencies die out.  So what if this wave
has travelled through the core?  Then the shortest detectable wavelength is
hundreds of kilometers, and no object smaller than that will be very
detectable.  This is a challenge for studying the structure and properties of
the region just outside of the liquid outer core (also called D").

> What is a typical amplitude for such a wave?
> Paul Jebb
> Newcomb Central School

Depends on the frequency...  You get a blast of various frequencies, but the
lower ones become most dominant as the distance travelled increases.

This is probably more info than you needed, but perhaps it can explain how
complicated the real world actually is and how we can use seismograms to
explain it.

John Hernlund
E-mail: hernlund@.......
WWW: http://www.public.asu.edu/~hernlund/

******************************************************************************

__________________________________________________________

Public Seismic Network Mailing List (PSN-L)

```