  This paper presents the linear theory of adjoint equations as applied to
thermo-acoustics. The purpose is to describe the mathematical foundations of
adjoint equations for linear sensitivity analysis of thermo-acoustic systems,
recently developed by Magri and Juniper (J. Fluid Mech. (2013), vol. 719, pp.
183--202). This method is applied pedagogically to a damped oscillator, for
which analytical solutions are available, and then for an electrically heated
Rijke tube with a mean-flow temperature discontinuity induced by the compact
heat source. Passive devices that most affect the growth rate / frequency of
the electrical Rijke-tube system are presented, including a discussion about
the effect of modelling the mean-flow temperature discontinuity.
