Simple Model w/o Doppler
I have a speaker driven by an electrical signal. The pressure at the sampling point is some linear operator acting on the input signal: $L[ s(t)]$. Where $L$ combines the linear model representing the electrical components (LRC circuit) the mechanical components (mass-spring-dashpot) and the coupling of the cone to the air (driven wave equation). I have no reason to think that there are significant non-linearities in the speaker; in fact good speakers are (usually) designed to minimize the non-linearities. Thus, I'd infer that the audio waves at my test point can be modeled using linear transfer function, and thus should only scale and/or change the phase of the components of the signal in the frequency domain.
Simple Model w/ Doppler
I have a tweeter, a speaker driven by a higher frequency ($f_T)$signal, mounted onto a larger woofer speaker cone. The woofer is driven by a much lower frequency $f_W << f_T$. If I assume an amplitude for the woofer's motion, then its relatively straightforward to compute the Doppler effect on the higher frequency signal (this can also be thought of as a phase modulation)
This model also seems applicable to real audio speakers -- the physical excursion of the cone due to lower (bass) frequencies should superimpose a time-varying Doppler shift onto the higher frequencies. This would seem to lead to the system moving energy around in the frequency domain.
I believe that the model w/ Doppler is the correct one for normal single-cone speakers: i.e. we'll observe Doppler (phase) modifications to a given high frequency signal that depend on the presence of the low frequency signal (these results are consistent although I have no way to validate their authenticity).
So what is missing from the simple linear "w/o Doppler" model that allows for, or generates, frequency modulation?
Also, although I can construct a sensible representation when presented with two frequencies (or two well separated frequency bands) how can you model the case where, in a sense, each frequency component is being Doppler shifted by all the others? I'm not sure what is the right way to look at this problem in the continuous spectrum case.