In this work, we study the propagation of wildfires using an advection–diffusion–reaction model which also includes convective and radiative heat loss. An existing model is discussed and a physically consistent modification of the model is proposed. Using this, the existence of travelling waves (TWs) in the one-dimensional case is investigated. Prior numerical studies reveal the existence of TWs. Under the travelling wave ansatz and certain approximation, the model is reduced to a semi-autonomous dynamical system with three unknowns which can be analyzed by a shooting algorithm. It is hypothesized that under mild wind speeds, TWs in both directions exist, and under strong tailwinds only TWs in the direction of wind are possible. The theoretical implications are investigated using both solvers for the PDE models and the shooting algorithm. The results match, and unveil the dependence of the fronts on the parameters consistent with the predictions.