  [Abridged] I report the discovery of modulations in radial velocity (RV)
curves of four Galactic classical Cepheids and investigate their impact as a
systematic uncertainty for Baade-Wesselink distances. Highly precise Doppler
measurements were obtained using the Coralie high-resolution spectrograph since
2011. Particular care was taken to sample all phase points in order to very
accurately trace the RV curve during multiple epochs and to search for
differences in linear radius variations derived from observations obtained at
different epochs. Different timescales are sampled, ranging from cycle-to-cycle
to months and years. The unprecedented combination of excellent phase coverage
obtained during multiple epochs and high precision enabled the discovery of
significant modulation in the RV curves of the short-period s-Cepheids QZ
Normae and V335 Puppis, as well as the long-period fundamental mode Cepheids l
Carinae and RS Puppis. The modulations manifest as shape and amplitude
variations that vary smoothly on timescales of years for short-period Cepheids
and from one pulsation cycle to the next in the long-period Cepheids. The order
of magnitude of the effect ranges from several hundred m/s to a few km/s. The
resulting difference among linear radius variations derived using data from
different epochs can lead to systematic errors of up to 15% for
Baade-Wesselink-type distances, if the employed angular and linear radius
variations are not determined contemporaneously. The different natures of the
Cepheids exhibiting modulation in their RV curves suggests that this phenomenon
is common. The observational baseline is not yet sufficient to conclude whether
these modulations are periodic. To ensure the accuracy of Baade-Wesselink
distances, angular and linear radius variations should always be determined
contemporaneously.
