A Preliminary Study of the Burgers Equation with Symbolic ...

234 DERICKSON AND PIELKE, SR.
FIG. 6. The time evolution of the 1st harmonic. Displays backscatter, Bn1-n2, and outscatter, Om1-m2, resulting
from interactions between specific harmonic pairs, (n1,n2) and (m1,m2), and damping due to viscosity. Example
B1-3 represents backscatter due to interaction between fundamental and 2nd harmonic.
damping serve to diminish the amplitude of the 1st harmonic, and damping increases with
time because the harmonic is growing.
When Re =∞, there is no damping, but production and backscatter increase by only a
small amount, as shown in Fig. 8. Like the fundamental, the 1st harmonic is affected more
by inviscid mechanisms than by viscosity throughout the time evolution of the symbolic
computation.
Bypassing the 2nd harmonic, an analysis is made of the 3rd harmonic due to its greater
complexity. While the 2nd harmonic is created solely by interactions between the fundamental
and the 1st harmonic, the 3rd harmonic is created from outscatter produced by two
distinct pairs of interactions: (a) the fundamental and the 2nd harmonic, and (b) the selfinteractions
of the 1st harmonic. The two production mechanisms are represented by O1−3
and O2−2 in Fig. 7, in which O1−3 is nearly three times as large as O2−2 for all seven time
steps. It may seem counterintuitive that O1−3 should be that much greater than O2−2,given
that the 1st harmonic is straddled by an interaction that produces a greater effect than its
own self-interaction. The explanation lies in the large amplitude, F1, of the fundamental,
such that |F1F3| > 1
2 F 2 2 , (see discussion in Appendix B).