The nonlinear interaction of two disturbances excited successively in
a two-dimensional Couette flow is shown to lead to a transient energy
growth. This phenomenon, which is called the echo effect and exists in
several other physical systems, is interesting because the energy gro
wth appears long after the energy associated with the original disturb
ances has decayed. Here, the echo effect is studied analytically and n
umerically in a situation where the nonlinear response has the same or
der of magnitude as the two excitations. A system of amplitude equatio
ns describing the nonlinear interactions between three sheared modes i
s derived and employed to examine the physical mechanism of the echo.
The qualitative validity of this system is confirmed by numerical simu
lations. The influence of viscous dissipation on the echo effect is al
so considered. (C) 1998 American Institute of Physics.