A non-premixed diffusion flame formed by co-flowing streams of air and fuel downstream of the burner is subjected to externally imposed oscillations in pressure, velocity, and density, in the transverse direction to the mean flow. The problem is investigated by means of numerical simulation. The flame is treated in a two-dimensional domain, and the response of its heat release rate fluctuations to the external excitation is examined. A relationship between a threshold amplitude and the frequency of the external excitation exists for which the flame blows off (at low frequency) or blows out (at high frequency). The classical response function curve dropping with frequency is captured when stable oscillations are observed. The response is linear with the imposed amplitude. The oscillatory combustion in the presence of the prevalent/imposed transverse velocity oscillations is coupled to the linear longitudinal acoustic modes of the duct by placing the burner somewhere along the length. Multiple longitudinal modes are simultaneously and spontaneously excited; the mode that grows predominantly depends upon the burner location in the duct. This shows that the combustion zone acts as a local nonlinearity in transferring energy between transverse modes and longitudinal modes, even when the acoustic oscillations are in the linear regime.