This thesis investigates the thermoacoustic behaviour of the PB1–HX3 subsystem—comprising a preburner and heat exchanger—within SABRE (Synergetic Air-Breathing Rocket Engine), a single-stage-to-orbit propulsion concept by Reaction Engines Ltd. SABRE integrates advanced heat exchangers with conventional turbomachinery to operate in air-breathing and rocket modes. PB1–HX3 is essential for low-Mach operation, where a hydrogen-fuelled micro-mix combustor provides thermal energy to a helium-based closed cycle via a micro-tube heat exchanger. To analyse this subsystem, new models were developed to study its thermoacoustic response. The work advances the field on three fronts: (i) analytical and semi-analytical approaches for mean-flow estimation; (ii) a model for the flame transfer function (FTF) of hydrogen diffusion flames; and (iii) a hybrid model to assess heat-exchanger acoustics. A quasi-one-dimensional open-source solver was created to compute mean flows with heat transfer, friction, or both, validated against CFD and used to assess the mean flow under four operating conditions. Flame dynamics were captured via a variable-density, variable-diffusivity mixture-fraction model coupled to simplified momentum equations, validated against benchmark cases and reproducing steady flame shapes and dynamic responses over a wide parameter range. The acoustic response of HX3 was investigated using a hybrid model incorporating temperature gradients and viscous damping, validated against solutions to the Linearised Navier–Stokes Equations and used to obtain the acoustic transfer matrix and absorption coefficient, revealing opposing effects of negative thermal gradients and viscous damping. Finally, a low-order network model integrating all components was constructed to assess the linear stability of the full PB1–HX3 assembly. The system was found to exhibit potentially unstable modes, with growth rates and frequencies dependent on flame characteristics, heat-exchanger power, and other key design parameters. A secondary contribution is the release of open-source tools to support design and research into thermoacoustic instability in hybrid rocket systems.