  The Calogero model bears, in the continuum limit, collective excitations in
the form of density waves and solitary modulations of the density of particles.
This sector of the spectrum of the model was investigated, mostly within the
framework of collective field theory, by several authors, over the past fifteen
years or so. In this work we shall concentrate on periodic solutions of the
collective BPS-equation (also known as "finite amplitude density waves"), as
well as on periodic solutions of the full static variational equations which
vanish periodically (also known as "large amplitude density waves"). While
these solutions are not new, we feel that our analysis and presentation add to
the existing literature, as we explain in the text. In addition, we show that
these solutions also occur in a certain two-family generalization of the
Calogero model, at special points in parameter space. A compendium of useful
identities associated with Hilbert transforms, including our own proofs of
these identities, appears in Appendix A. In Appendix B we also elucidate in the
present paper some fine points having to do with manipulating Hilbert
transforms, which appear ubiquitously in the collective field formalism.
Finally, in order to make this paper self-contained, we briefly summarize in
Appendix C basic facts about the collective field formulation of the Calogero
model.
