The use of fire is one of the oldest engineering disciplines known to man, and the field of application is vast, so it is impossible to mention all aspects of the effects of sound on combustion. Those effects fall roughly into the following categories:
First there is the periodic air displacement associated with sound. This effect is comparable to forced convection, except that the speed and perhaps direction of flow is variable. In general, convection improves transfer processes like heat transfer, evaporation on surfaces, mass transfer and mixing. These phenomena are crucial to many combustion processes and sound can have a considerable effect in this way. A major advantage of sound over classic convection is that the average air flow is independent of the degree of convection; sound can increase convection without increasing flow rates through the burner.
A second category of effects is linked to unstable forms of combustion. We refer here to instability in the flame structure itself, not so-called instable or pulsating burner operation. Many flames take the form of a jet, where a boundary region forms between the flame and the surrounding medium. This boundary layer has an unstable speed profile, and waves can develop similar to tearing off of boundary layers from a wall. This is the periodic shedding of vortices as observed on a flickering candle flame. As these instabilities have natural frequencies, obviously sound can have the effect of increasing or decreasing the stability, depending on frequency, and of changing the vortex shape and size.
Related to the previous two is the effect of sound on turbulence. Unfortunately there is as yet no general theory of the effect of turbulence on combustion, and the mechanism through which sound affects turbulence remains unknown. But the structure and scale of turbulence in boundary layers is affected by sound, boosting energy of the smaller vortices, usually improving mass transfer and mixing and so improving combustion.
Finally there are effects of energy transfer between sound and flame. Sound causes a periodic variation of pressure, density and temperature, leading to periodic change in the speed of the chemical processes of combustion. If the chemical time constant is such that more heat is released during the half-period of higher pressure, energy will be transferred from the combustion to the sound. This phenomenon can cause havoc in rocket engines, but can be applied usefully to maintain a sound level in a burner without having to inject the sound.
In conclusion, sound influences combustion, and vice versa, and it is possible to improve the quality of combustion by use of sound.