a background flow opposing its direction of travel. Quantitatively, when the waves are well separated, we find that the taller and smaller waves travel at speeds $c_B=0.231077=0.994049c_{B,0}$ and $c_C=0.226153=0.991531c_{C,0}$, respectively. The relative speed is then $(c_B-c_C) = 1.12526(c_{B,0}-c_{C,0})$. Thus, $$\label{eq:ineq}
  \frac{\pi-d}{c_B-c_C}  < \frac{T}{2} <
  \frac{\pi-d}{c_{B,0}-c_{C,0}} < \frac{T_0}{2} = \frac{\pi}{c_{B,0}-c_{C,0}},
$$ with numerical values $531.5<554.1<598.0<717.9$. This means that both effects together have overestimated the correction needed to obtain $T$ from $T_0$. This is because the relative speed slows down as the waves approach each other, which is expected since the amplitude of the trailing wave decreases and the amplitude of the leading wave increases in this interaction regime. Indeed, the average speed of the waves is $$\label{eq:average:speed}
  \overline{c_B} = \frac{\theta/2 - d/2}{T/2} =   0.993388c_{B,0}, \qquad
  \overline{c_C} = \frac{\theta/2 + d/2 - \pi}{T/2} =   0.991737c_{C,0},$$ which are slightly smaller and larger, respectively, than their speeds when well separated. Note that $T/2$ in (\[eq:ineq\]) may be written $T/2=(\pi-d)/(\overline{c_B} - \overline{c_C})$. We used $\theta/2=2.54258+40\pi$ in (\[eq:average:speed\]) to account for the 20 times the waves cross the domain $(0,2\pi)$ in time $T/2$ in addition to the offset shown in Figure \[fig:evol:kdv3\].

Comparison with KdV {#sec:kdv}
===================

In the previous section, we observed two types of 