We present a new test of the widths of the local line profiles from th
e accretion discs of cataclysmic variable stars. Our method is based u
pon the near coincidence of the Ca II H line at 3968.5 Angstrom and th
e H epsilon line at 3970.1 Angstrom. If at any point in the disc the p
rofiles of these lines overlap, the total flux seen will in general no
t be the sum of the individual fluxes; if the profiles do not overlap,
the fluxes will add linearly. This diagnostic applies independently o
f the Doppler broadening from the disc. We estimate the 'true' H epsil
on flux from interpolation between neighbouring members of the Balmer
series. Subtracting this flux from the combined flux of the H epsilon/
Ca II H blend leaves the Ca II H flux which can be compared with the f
lux of the isolated Ca II K line. We find that the Ca II infrared trip
let lines at 8498, 8542 and 8662 Angstrom show optically thick flux ra
tios close to 1, implying that the H/K ratio should also be close to 1
, whereas the observed ratios are close to 0.5. Our results show that
although H epsilon is Separated by 120 km s(-1) from Ca II H, compared
with a sound speed of only approximate to 10 km s(-1), the lines are
overlapping so that the Ca II H emission is suppressed. Thus some form
of broadening beyond the purely thermal is required. The extra broade
ning agrees with evidence for supersonic turbulence from ultraviolet o
bservations of the dwarf nova OY Car. However, any other form of broad
ening of sufficient strength will suffice. We consider Stark broadenin
g and show that it can be significant if the lines form at baryon dens
ities above similar to 5 x 10(13) cm(-3) and temperatures above simila
r to 8000 K; it is not possible to say at present if such conditions a
re likely. The supersonic line broadening was present in all the quies
cent dwarf novae we were able to observe (other than U Gem, which we c
ould not measure because of severe blending). The broad local line pro
file implied by our data (FWHM > 240 km s(-1)) may explain the absence
of double-peaked profiles in many dwarf novae.