  Spiral density waves observed in protoplanetary discs have often been used to
infer the presence of embedded planets. This inference relies both on
simulations as well as the linear theory of planet-disc interaction developed
for planets on circular orbits to predict the morphology of the density wake.
In this work we develop and implement a linear framework for calculating the
structure of the density wave in a gaseous disc driven by an eccentric planet.
Our approach takes into account both the essential azimuthal and temporal
periodicities of the problem, allowing us to treat any periodic perturbing
potential (i.e. not only that of an eccentric planet). We test our framework by
calculating the morphology of the density waves excited by an eccentric,
low-mass planet embedded in a globally isothermal disc and compare our results
to the recent direct numerical simulations (and heuristic wavelet analysis) of
the same problem by Zhu and Zhang. We find excellent agreement with the
numerical simulations, capturing all the complex eccentric features including
spiral bifurcations, wave crossings and planet-wave detachments, with improved
accuracy and detail compared with the wavelet method. This illustrates the
power of our linear framework in reproducing the morphology of complicated
time-dependent density wakes, presenting it as a valuable tool for future
studies of eccentric planet-disc interactions.
