How to model "Doppler Distortion" of speakers?

Question

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.

The Problem

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.

Answer 1

The Doppler shift for small speeds is $\Delta f/f = \Delta v/c$, where $\Delta v$ is the (signed) speed of the source relative to the detector, and I'm using $c$ as the speed of sound.

So let's plug in some numbers. I'm going to use numbers that will produce a large effect to see how larger an effect is plausible. Let's take a woofer operating at $f = 200 \mathrm{Hz}$. Wikipedia reports that an excursion of $2.5\mathrm{in} = 6.3\mathrm{cm}$ is on the extremely high end (though there is a "citation needed" note appended to that claim). The speed of sound in air is $c = 343 \mathrm{m/s}$. Then $\Delta f/f = 0.037%, which works out to about 63 cents. Healthy adults can typically detect frequency changes of about 25 cents, so this is noticeable.

I was moderately surprised by this result; I expected it not to be noticeable. Lowering the frequency to about $80 \mathrm{Hz}$ (just barely) noticeable shift. You could probably lower the excursion by a factor of 10, which would probably make the shift undetectable by a human. On balance, I'll believe that this effect is noticeable.

That shift of $\pm 63\mathrm{cents}$ means that you will be hearing the central frequency $f_0$, and some side bands at $f_0 \pm \Delta f$. And as you allude to in the question, this won't be two discreet side bands; this will be a continuous range of frequencies. It will show up as a pulsing in the frequency, similar to beats. @Martin Drautzburg's comment reminded me that I've actually heard a form of this effect. It was a particular style of amplifier in which the cone moved by about a foot. The beats were definitively audible.

The discussion you cite by Rod Elliott looks at this from the perspective of phase shifts, rather than Doppler shifts. He seems to think it's easier to describe the phenomenon this way, and that seems true to me. Since the speed of the woofer cone is continuously changing, the Doppler shift will also change continuously. Given that, the time-domain description given by the phase shift will probably be simpler.

The speed, frequency, and wavelength are related by $c = f\lambda \Rightarrow \lambda = c/f$. Elliot discusses a tweeter frequency of $1\mathrm{kHz}$ at one point, so let me use that: $\lambda = 343/1000 = 34.3 \mathrm{cm}$. If the location of the tweeter travels by $6.3\times 2 = 12.6 \mathrm{cm}$, that will cause a phase shift of $(12.6/34.3)\times 360^\circ = 132^\circ$ between the sound the tweeter emits when the woofer cone is fully extenced and when it is fully retracted. Those phase-shifted waves will interfere in a manner that will be detected as modulation of the frequency.

This model also seems applicable to real audio speakers

I believe so. If the tweeter and the woofer are two different speakers, mounted independently, then the woofer's motion won't move the tweeter, and therefore won't cause any Doppler shift in the tweeter's output. But if you're trying to use one speaker to produce both $f_W$ and $f_T$, then modeling that speaker as a woofer with a tweeter mounted onto will probably describe the real behavior accurately enough.

Answer 2

A partial answer to where the simple linear no-Doppler model has a gap: in the coupling of the speaker cone to the air. There is a boundary condition imposed on the acoustic wave equation at the location of the speaker cone. However, the speaker cone itself is moving around, meaning that this boundary condition is imposed at different locations at different times.

A simplified problem of the same ilk would be to imagine a piston in a air-filled tube, driven to achieve a given displacement $d(t)$:

                   ~~~d(t)~~~
         ---------------------------------
                       ||
            air        ||====================
                       || 
         ------------------------------------

If you assume that the (maximum) magnitude of displacement is much less than the smallest wavelength of the resulting waves, then the maximum possible frequency (phase) change is small. If you assume the cone (piston) is amplitude $d_{max}$ and frequency limited $f_{max}$ the then constraint is $d_{max} f_{max} << c$ (here $c$ is the speed of sound).

This answer is partial in that it only identifies where the frequency (phase) modulation comes into play, but does not describe how to explicitly represent this type of boundary condition mathematically in the regime where $d_{max} f_{max}$ is not negligably small.