Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle
$\unicode[STIX]{x1D703}=0$
) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation
$|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$
, where
$z=\unicode[STIX]{x1D714}/k_{\Vert }c$
is the phase speed and
$\unicode[STIX]{x1D6FD}_{\text{A}}c$
is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function,
$z^{2}W(z)$
, which is negative for
$|z|<z_{0}$
and has a sharp maximum at
$|z|=z_{\text{m}}$
, with
$1-z_{\text{m}}<1-z_{0}\ll 1$
. We give numerical estimates for the maximum of
$z^{2}W(z)$
and for
$z_{\text{m}}$
and
$z_{0}$
for a one-dimensional Jüttner distribution. The L and A modes reconnect, for
$z_{\text{A}}>z_{0}$
, to form the O and Alfvén modes for oblique propagation (
$\unicode[STIX]{x1D703}\neq 0$
). For
$z_{\text{A}}<z_{0}$
the Alfvén and O mode curves reconnect forming a new mode that exists only for
$\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$
. The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around
$\unicode[STIX]{x1D703}\approx 0$
. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.