The calculation of the QTN in Bernstein waves [Sentman, 1982] can be generalized to the case where the dispersion equation has multiple solutions. In that case, using (22) of [Meyer-Vernet, Hoang and Moncuquet, 1993], we deduce an approximate expression for the noise measured by an antenna in the plasma frame
where 
denotes the multiple solutions of Bernstein's dispersion
equation for the angular frequency 
, L is the antenna length (35 m),
is the angle between the antenna and 
, 
is the antenna
response to Bernstein waves, 
is the real part of the dielectric
function, and 
is Boltzmann's constant.
Since the antenna moves with respect to the plasma,
we must add to (2) an integration over the direction of with
respect to 
, involving the solutions of the Doppler-shifted dispersion
equations [Moncuquet, Meyer-Vernet and Hoang, 1995]. The term 
is the (small) range in parallel wave vector for which the hot
population makes a dominant contribution to the QTN. This term
vanishes at gyroharmonics where thermal electrons damp Bernstein waves,
resulting in the well-defined
noise minima observed at gyroharmonics during the Io torus traversal
[Meyer-Vernet, Hoang and Moncuquet, 1993].
Equation (2) does not hold near resonant solutions where
.
In this case, the spectral density increases until the first-order approximation
of 
used to derive (2) breaks down [Sentman, 1982], then the maximum voltage is set by the second
derivative 
. Here we shall not try
to compute 
at resonances (below we show it is not necessary),
but we note that since the energy flux is
expected to remain constant, the noise level should continue to increase as the
group velocity vanishes, reaching a maximum at resonances. Equation (2) and
the above remark allow us to summarize two important properties of the
QTN, which the observed spectra should exhibit:
(1) In the upper hybrid gyroharmonic band, the spectral density
should reach high levels at each resonance and in some frequency
band around it,
since the 
and 
peaks are
smoothed out by the Doppler shift corresponding to different
directions. Note that since vanishes for small argument
( 
), the peak at 
(where 
) could be
attenuated,
and if the range of variation of is large enough,
the noise should be spin modulated.
(2) The signal should plummet in the forbidden bands
between the largest Doppler-shifted 
of the considered harmonic
band and 
(or
slightly below that gyroharmonic, because of the Doppler shift as
explained in section 2).
