component, the higher-order modes are negligibly small, and almost the whole of
the CPSD is made up of the fundamental mode. The same condition holds for the
large subcritical system. The higher-order mode effect in the correlated component
is minor even in the large subcritical system. In the uncorrelated component, the
higher-order mode effect is significant even in the nearly critical system. In the
large subcritical system, the higher-order mode effect is much more significant.
Thus, fitting the uncorrelated component to Eq. (12.9) yields an inaccurate α value
unless the system is nearly critical. For example, in the large subcritical system we
obtain α ¼ 789 (s
À1 ) for the true fundamental mode α value of 940 (s
À1 ) [15] from
the uncorrelated component. On the other hand, we obtain α ¼ 900 (s
À1 ) from the
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1
10
100
1000
|CPSD (w)| (/s)
Frequency (Hz)
Monte Carlo
Theory
Fig. 12.6 Amplitude of cross-power spectral density (CPSD) in the nearly critical system by
Monte Carlo simulation and theoretical value
–5.0E+05
0.0E+00
5.0E+05
1.0E+06
1.5E+06
2.0E+06
2.5E+06
3.0E+06
3.5E+06
10
100
1000
Frequency (Hz)
Fundamental
1st higher
2nd higher
3rd higher
4th higher
5th higher
Total
Re[CPSD
UN (w)] (/s)
Fig. 12.7 Mode components of the uncorrelated component in the CPSD in the nearly critical
system (real part)
126
T. Yamamoto