  This paper is concerned with the transition of the laminar flow in a duct of
square cross-section. Like in the similar case of the pipe flow, the motion is
linearly stable for all Reynolds numbers, rendering this flow a suitable
candidate for a study of the 'bypass' path to turbulence. It has already been
shown \citep{Biau_JFM_2008} that the classical linear optimal perturbation
problem, yielding optimal disturbances in the form of longitudinal vortices,
fails to provide an 'optimal' path to turbulence, i.e. optimal perturbations do
not elicit a significant nonlinear response from the flow. Previous simulations
have also indicated that a pair of travelling waves generates immediately, by
nonlinear quadratic interactions, an unstable mean flow distortion, responsible
for rapid breakdown. By the use of functions quantifying the sensitivity of the
motion to deviations in the base flow, the 'optimal' travelling wave associated
to its specific defect is found by a variational approach. This optimal
solution is then integrated in time and shown to display a qualitative
similarity to the so-called 'minimal defect', for the same parameters. Finally,
numerical simulations of a 'edge state' are conducted, to identify an unstable
solution which mediates laminar-turbulent transition and relate it to results
of the optimisation procedure.
