Most news articles I see focus solely on the number of vaccinated people when estimating how far away countries are from herd immunity, which seems odd to me. Left out in the discussion seem to be the unvaccinated but already infected. As far as I know, prior infection confers at least some degree of immunity. It may also be the case that those inclined to be skeptical of vaccines were also the ones who exposed thems…
In practice, of course, people respond to news about Covid waves and that contributes to the wave-like dynamics. Furthermore, new variants of the virus will be able to reach people that were "cautious enough" for a previous variant. That alone would also give you wave-like dynamics, basically every new variant would affect increasingly harder to reach people.
Anyway, the point is, seeing these waves is super natural and expected and we can't deduce anything about population-level immunity from them without looking really really closely.