  In this theoretical and numerical paper, we derive the adjoint equations for
a thermo-acoustic system consisting of an infinite-rate chemistry diffusion
flame coupled with duct acoustics. We then calculate the thermo-acoustic
system's linear global modes (i.e. the frequency/growth rate of oscillations,
together with their mode shapes), and the global modes' receptivity to species
injection, sensitivity to base-state perturbations, and structural sensitivity
to advective-velocity perturbations. We then compare these with the Rayleigh
index. The receptivity analysis shows the regions of the flame where open-loop
injection of fuel or oxidizer will have most influence on the thermo-acoustic
oscillation. We find that the flame is most receptive at its tip. The
base-state sensitivity analysis shows the influence of each parameter on the
frequency/growth rate. We find that perturbations to the stoichiometric mixture
fraction, the fuel slot width, and the heat-release parameter have most
influence, while perturbations to the P\'eclet number have least influence.
These sensitivities oscillate: e.g. positive perturbations to the fuel slot
width either stabilizes or destabilizes the system, depending on the operating
point. This analysis reveals that, as expected from a simple model, the phase
delay between velocity and heat-release fluctuations is the key parameter in
determining the sensitivities. It also reveals that this thermo-acoustic system
is exceedingly sensitive to changes in the base state. The
structural-sensitivity analysis shows the influence of perturbations to the
advective flame velocity. The regions of highest sensitivity are around the
stoichiometric line close to the inlet, showing where velocity models need to
be most accurate.
