  Fluctuation spectra of fluid compound membrane systems are calculated. The
systems addressed contain two (or more) almost parallel membranes that are
connected by harmonic tethers or by a continuous, harmonic confining potential.
Additionally, such a compound system can be attached to a supporting substrate.
We compare quasi-analytical results for tethers with analytical results for
corresponding continuous models and investigate under what circumstances the
discrete nature of the tethers actually influences the fluctuations. A
tethered, supported membrane pair with similar bending rigidities and stiff
tethers can possess a nonmonotonic fluctuation spectrum with a maximum. A
nonmonotonic spectrum with a maximum and a minimum can occur for an either free
or supported membrane pair of rather different bending rigidities and for stiff
tethers. Typical membrane displacements are calculated for supported membrane
pairs with discrete or continuous interacting potentials. Thereby an estimate
of how close the constituent two membranes and the substrate typically approach
each other is given. For a supported membrane pair with discrete or continuous
interactions, the typical displacements of each membrane are altered with
respect to a single supported membrane, where those of the membrane near the
substrate are diminished and those of the membrane further away are enhanced.
