  Interfacial instability would be aroused on a spherical liquid droplet when
it is subject to external vertical vibration. In this paper, a linear analysis
was conducted on this instability problem. The polar-angle dependent
acceleration in the spherical coordinate is strongly coupled with the temporal
and spatial component of the surface deformation displacement, which gives a
recursion equation that implicitly expresses the dispersion relation between
the growth rate and spherical mode numbers. The unstable regions (or unstable
tongues) for the inviscid fluids considering latitudinal mode (longitudinal
mode number m = 0) were derived and presented in the parameter plane. Compared
with the solution of the spherical Faraday instability under radial vibration
acceleration, the regions of harmonic unstable tongues for the mono-directional
vibration case is much narrowed and the subharmonic unstable tongues almost
become straight lines. The analysis shows that the latitudinal waves emerging
on the spherical droplet surface ought to oscillate harmonically instead of
subharmonically, which is opposite to the results for the case under radial
vibration acceleration. A corresponding experiment of a liquid droplet lying on
a vertically vibrating plate was conducted and the observations substantiate
our theoretical predictions.
