My PhD Thesis, PDF 3MB - Stanford University

Therefore, equation (6.5) is a dispersion relation whose roots give phase velocities and
group velocities of normal modes for a given frequency. This dispersion relation is very
simple and can be easily solved.
Once we obtain the phase velocity of a normal mode, we can calculate its eigen
function. According to equation (6.4), c p
of generality, I set c p
(1 )
(1 )
can be any non-zero value. Without any loss
( v n ) 1 . Then, all other coefficients are obtained using the
generalized R/T matrices through the following relations:
and
c
c
( j 1)
( j 1)
(1 )
c ( v ) p n ? ( 1)
R ( v ) ,
n
( v n ) ? T
( j )
( v n ) ? R
( j 1)
( v n ) ? T
( j 1)
( v n ) c
( j 1)
( v n ) ... ? T
( v n )
- 138 -
( 1)
( v n )
, j=1, 2,..., N. (6.7)
Substituting these coefficients into equation (6.2), we obtain corresponding eigen
functions. Dispersion relation (6.5) along with equations (6.6) and (6.7) are very
important for studying normal modes. In the next section, I apply this dispersion relation
to seismic attenuation estimation.
6.3 Q-VALUES OF NORMAL MODES
In the previous section, I only consider the lossless case, i.e., the Q-value for all
waves is infinite. The Q-value in real media is finite, i.e., there exists attenuation. For
an absorptive medium, the attenuation can be introduced via the complex velocity
?v ( ) v ( )[1
i
2 Q
] (6.8)