  Recent photometric observations of first-overtone classical Cepheids and RR
Lyrae stars have led to the discovery of additional frequencies showing a
characteristic period ratio of 0.60-0.65 with the main pulsation mode. In a
promising model proposed by Dziembowski (2016), these signals are suggested to
be due to the excitation of non-radial modes with degrees 7, 8 and 9 (Cepheids)
or 8 and 9 (RR Lyrae). Such modes usually have low amplitudes in photometric
data. Spectroscopic time series offer an unexplored and promising way forward.
We simulated time series of synthetic line profiles for a representative
first-overtone classical Cepheid model and added a low-amplitude non-radial
mode. We studied sets of spectra with dense sampling and without noise,
so-called 'perfect' cases, as well as more realistic samplings and
signal-to-noise levels. Besides the first-overtone mode and the non-radial
mode, also the harmonics of both modes and combination signals were often
detected, but a sufficiently high sampling and signal-to-noise ratio prove
essential. The amplitudes of the non-radial mode and its harmonic depend on the
azimuthal order $m$. The inclination is also an important factor determining
the detectability of the non-radial mode and/or its harmonic. We compared the
results obtained for the predicted high degrees with those for lower-degree
modes. Finally, we studied the sampling requirements for detecting the
non-radial mode. Our findings can be used to plan a spectroscopic observing
campaign tailored to uncover the nature of these mysterious modes.
