  In this note we revisit the proof of Bedrossian and Masmoudi
[arXiv:1306.5028] about the inviscid damping of planar shear flows in the 2D
Euler equations under the assumption of zero mean perturbation. We prove that a
small perturbation to the 2D Euler Couette flow in $\mathbb{T}\times
\mathbb{R}$ strongly converge to zero, under the additional assumption that the
average in $x$ is always zero. In general the mean is not a conserved quantity
for the nonlinear dynamics, for this reason this is a particular case.
Nevertheless our assumption allow the presence of echoes in the problem, which
we control by an approximation of the weight built in [arXiv:1306.5028]. The
aim of this note is to present the mathematical techniques used in
[arXiv:1306.5028] and can be useful as a first approach to the nonlinear
inviscid damping.
