Re: Wave Intensity vs Energy

[James McLean <mclean@GENESEO.EDU>]

Concerning energy in, amplitude of, earthquakes, I have stumbled on what
seems to be a very helpful and informative page:
<http://www.seismo.unr.edu/ftp/pub/louie/class/100/magnitude.html>.
It describes in some detail how the magnitude of an earthquake is
measured.  Two seminal equations:
   M = log(A[mm])+3log(8*t[s])-2.92
   log(E[ergs]) = 11.8 +1.5*M
where: M = Richter magnitude
       A = amplitude of wiggle on particular kind of seismometer
       t = time delay between arrival of P and S (longitudinal and
           transverse) waves (basically, a measure of the distance
           from epicenter d)
       E = Energy *radiated* by earthquake in waves

John Denker wrote:
> We seem to be comparing applies and oranges here.  Think about the
> dimensional analysis.  The subject line mentions intensity and energy;  one
> is simply related to amplitude while the other is not.
>
> The Richter scale cannot measure both amplitude and energy;  a
> low-amplitude wave can carry arbitrarily much energy if it continues long
> enough.  There cannot be any simple relationship between magnitude and
> duration, because the waves spread out with distance.
While what John says here seems to me be unavoidably true, nevertheless
seismologists do seem to make the energy-to-magnitude connection.  I can
only guess that earthquakes have characteristic durations.

> No, I don't think there's any significant nonlinearity.
Using the equations above, I get...
  A propto d^3    (for constant E and M)
  E propto A^1.5  (at a constant distance from epicenter)
  E propto d^4.5  (for a constant observed A)
None of these make sense to me on the surface.  The first two may be
complicated by the definition of amplitude.  But I would have expected
(E propto d^2 for constant A) based simply on conservation of energy.
There must be some dispersion or absorption or other issues involved.

--
Dr. James McLean                     phone: (716) 245-5897
Dept. of Physics and Astronomy       FAX:   (716) 245-5288
SUNY Geneseo                         email: mclean@geneseo.edu
1 College Circle
Geneseo, NY  14454-1401