  Aims. Estimates of coronal wave energy remain uncertain as a large fraction
of the energy is likely hidden in the non-thermal line widths of emission
lines. In order to estimate these wave energies, many previous studies have
considered the root mean squared wave amplitudes to be a factor of $\sqrt{2}$
greater than the non-thermal line widths. However, other studies have used
different factors. To investigate this problem, we consider the relation
between wave amplitudes and the non-thermal line widths within a variety of 3D
magnetohydrodynamic (MHD) simulations.
  Methods. We consider the following 3D numerical models: Alfv\'en waves in a
uniform magnetic field, transverse waves in a complex braided magnetic field,
and two simulations of coronal heating in an arcade. We applied the forward
modelling code FoMo to generate the synthetic emission data required to analyse
the non-thermal line widths.
  Results. Determining a single value for the ratio between the non-thermal
line widths and the root mean squared wave amplitudes is not possible across
multiple simulations. It was found to depend on a variety of factors, including
line-of-sight angles, velocity magnitudes, wave interference, and exposure
time. Indeed, some of our models achieved the values claimed in recent articles
while other more complex models deviated from these ratios.
  Conclusions. To estimate wave energies, an appropriate relation between the
non-thermal line widths and root mean squared wave amplitudes is required.
However, evaluating this ratio to be a singular value, or even providing a
lower or upper bound on it, is not realistically possible given its sensitivity
to various MHD models and factors. As the ratio between wave amplitudes and
non-thermal line widths is not constant across our models, we suggest that this
widely used method for estimating wave energy is not robust.
