  Context : Star formation takes place in cold dense cores in molecular clouds.
Earlier observations have found that dense cores exhibit subsonic non-thermal
velocity dispersions. In contrast, CO observations show that the ambient
large-scale cloud is warmer and has supersonic velocity dispersions. Aims : We
aim to study the ammonia ($\rm NH_3$) molecular line profiles with exquisite
sensitivity towards the coherent cores in L1688 in order to study their
kinematical properties in unprecedented detail. Methods : We used $\rm NH_3$
(1,1) and (2,2) data from the first data release (DR1) in the Green Bank
Ammonia Survey (GAS). We first smoothed the data to a larger beam of 1' to
obtain substantially more extended maps of velocity dispersion and kinetic
temperature, compared to the DR1 maps. We then identified the coherent cores in
the cloud and analysed the averaged line profiles towards the cores. Results :
For the first time, we detected a faint (mean $\rm NH_3$(1,1) peak brightness
$<$0.25 K in $T_{MB}$), supersonic component towards all the coherent cores in
L1688. We fitted two components, one broad and one narrow, and derived the
kinetic temperature and velocity dispersion of each component. The broad
components towards all cores have supersonic linewidths ($\mathcal{M}_S \ge
1$). This component biases the estimate of the narrow dense core component's
velocity dispersion by $\approx$28% and the kinetic temperature by
$\approx$10%, on average, as compared to the results from single-component
fits. Conclusions : Neglecting this ubiquitous presence of a broad component
towards all coherent cores causes the typical single-component fit to
overestimate the temperature and velocity dispersion. This affects the derived
detailed physical structure and stability of the cores estimated from $\rm
NH_3$ observations.
