From time to time, I see posts on various bulletin boards that make the claims that mass-loaded quarter-wave pipes are nothing but bass reflex boxes and are explained by standard box design programs. "They" state that ML TL's are simply boxes with ports, and support the claim with impedance plot that show the classic bass reflex double hump. This is nonsense.
To understand what is happening in a pipe, one must look at the frequency response. Of course the impedance response will have a double hump. The underlying impedance curve is that of the driver. It will have a single maximum at the driver resonance frequency. Superimposed on that is effect of the pipe, which like a base reflex box, will suppress the cone movement at the pipe( box) resonance frequency. Hence the double maxima. Now, let's look at the frequency response of a real mass-loaded TL, my Peerless pipe:
That sure looks like a bass reflex frequency response, doesn't it? NOT! (For reference, the red trace is the driver, the blue is the port and the black is the combined response.) Note the dip between 160Hz and 266Hz. This is where the driver and port are out-of-phase, as predicted for a quarter-wave pipe. Note the table below. Harmonic 3-17 are very close to exact multiples of 52Hz, which is the calculated fundamental.
What doe the model predict? Well -- this: (Martin King's Mathcad worksheet)
Out through 350Hz, the match is almost exact, and the wiggles are in the right place above that. In reality, once stuffed, there are no pipe features left above 350Hz.
Now, let's discuss the pipe fundamental frequency. Based on standard day calculations for the pipe length, the fundamental is 56Hz (60" pipe). Based on harmonic fit, the actual fundamental is 52Hz. But because of the pipe geometry and the subsequent mass loading, the actual pipe resonance is 39Hz. BUT THE MASS LOADING ONLY AFFECTS THE PIPE RESONANCE. All higher harmonics remain unaffected. Also note that all of this is for a bare pipe with NO stuffing. The lowering of the pipe resonance from the free air calculation in a quarter-wave pipe is solely dependant upon pipe geometry.
But what does the frequency response of a bass reflex look like? Here is the modeled plot of a golden ration box with the same volume as the Peerless TL, and with the same port area. (in both cases the port length is zero:
Combined FR in red
Driver in red, port in blue
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