  We study the spectral properties of the thermal force giving rise to the
Brownian motion of a continuous mechanical system -- namely, a nanomechanical
beam resonator -- in a viscous liquid. To this end, we perform two separate
sets of experiments. First, we measure the power spectral density (PSD) of the
position fluctuations of the resonator around its fundamental mode at its
center. Then, we measure the frequency-dependent linear response of the
resonator, again at its center, by driving it with a harmonic force that
couples well to the fundamental mode. These two measurements allow us to
determine the PSD of the Brownian force noise acting on the structure in its
fundamental mode. The PSD of the force noise extracted from multiple resonators
spanning a broad frequency range displays a "colored spectrum". Using a
single-mode theory, we show that, around the fundamental resonances of the
resonators, the PSD of the force noise follows the dissipation of a blade
oscillating in a viscous liquid -- by virtue of the fluctuation-dissipation
theorem.
