Waterbag distribution ($f_0(v)=n{1}_{|v|<V}/(2V)$)

A phase-space distribution function constant inside a prescribed region and zero outside. The one-dimensional example in the title has a velocity plateau and distributional derivatives at its edges. Its simple kinetic response can support undamped collective modes, illustrating why absence of growing modes does not imply Landau damping. For electrons with fixed ions, $ϵ(\omega ,k)=1-{\omega }_p^2/({\omega }^2-k^2{V}^2)$, so collective roots lie outside the populated velocity interval.