  We propose a continuum model to predict long-wavelength vibrational modes of
empty and liquid-filled tubules that are very hard to reproduce using the
conventional force-constant matrix approach based on atomistic ab initio
calculation. We derive simple quantitative expressions for long-wavelength
longitudinal and torsional acoustic modes, flexural acoustic modes, as well as
the radial breathing mode of empty or liquid-filled tubular structures that are
based on continuum elasticity theory expressions for a thin elastic plate. We
furthermore show that longitudinal and flexural acoustic modes of tubules are
well described by those of an elastic beam resembling a nanowire. Our numerical
results for biological microtubules and carbon nanotubes agree with available
experimental data.
