Just wave!

Assume we have a driver with an fs of 50 Hz and it is installed at the closed end of a straight TL tuned to 50 Hz, the simplest form of TL design. The TL will produce standing waves at the 1/4, 3/4, 5/4, … frequencies of 50 Hz, 150 Hz, 250 Hz, … as expected. Neglect the acoustic impedance at the open end or terminus and assume the TL is completely empty so there is no damping of any kind. At a standing wave resonance, the back pressure on the driver cone will attenuate the motion, almost stopping the driver like in a BR design, and almost all of the SPL output will be from the terminus, like the port in a BR enclosure. The SPL and phase of the outputs will behave as follows.

Well below 50 Hz – as the driver moves into the TL it displaces a volume of air and an equivalent volume of air is pushed out of the terminus. The driver and the terminus are 180 deg out of phase and the SPL almost cancels producing a 24 dB/octave roll-off below 50 Hz.

At 50 Hz – the 1/4 fundamental standing wave is excited, the driver motion is significantly attenuated, and most of the SPL output comes from the terminus. The driver and the terminus are 90 degrees out of phase.

At 100 Hz – the driver and terminus are now in phase. There is no standing wave and the SPL from the driver and terminus are equal. This is because sound radiated from the back of the cone is the same as from the front of the cone but 180 degrees out of phase, the sound traveling down the TL is constrained so it does not decrease with distance, and the distance it travels is equal to a half of a wavelength (another 180 degrees). Theoretically, the system SPL will be 6 dB greater than the driver’s SPL in an infinite baffle.

At 150 Hz – the 3/4 standing wave is excited, the driver motion is again significantly attenuated, and most of the SPL output comes from the terminus. The driver and the terminus are 270 degrees (-90 degrees) out of phase.

At 200 Hz – the driver and terminus are now out of phase. There is no standing wave and the SPL from the driver and terminus are equal. This is because sound radiated from the back of the cone is the same as from the front of the cone but 180 degrees out of phase, the sound traveling down the TL is constrained so it does not decrease with distance, and the distance it travels is equal to a full wavelength (another 360 degrees). The driver and the terminus are 180 deg out of phase and the SPL almost cancels producing a deep null, maybe this is what the measurements are showing.

The phase and SPL pattern repeat as frequency increases and moves through the higher quarter wave frequencies. Below is a list of key frequencies and the phase differences between the driver and terminus.

10 Hz – 180 deg SPL --> 0 dB

50 Hz – 90 deg fundamental 1/4 wave and SPL mostly from terminus

100 Hz – 0 deg SPL + 6 dB

150 Hz – 270 deg (-90 deg) 3/4 wave and SPL mostly from terminus

200 Hz – 180 deg SPL --> 0 dB

250 Hz – 90 deg 5/4 wave and SPL mostly from terminus

300 Hz – 0 deg SPL + 6 dB

350 Hz – 270 deg (-90 deg) 7/4 wave and SPL mostly from terminus

400 Hz – 180 deg SPL --> 0 dB

The pattern repeats in steps of 50 Hz. I hope that is clear.

Thank you very much Martin for taking the time. Your last post makes sense to me.

I was having a hard time with the second half of this sentence:

"You cannot produce a half wave resonance and the output will never be in phase with the driver output. "

But this from your last post makes sense to me:

"At 100 Hz – the driver and terminus are now in phase."

So at 100 Hz (where the line length is equal to 1/2 wavelength) there is NO standing wave resonance, BUT the outputs from the driver and terminus are IN PHASE.

Have I finally got that right?

And "At 200 Hz – the driver and terminus are now out of phase."  (But there is no standing wave resonance.)  So IN THEORY at least, assuming a lightly-damped line, couldn't we get a cancellation notch in the summed response in that region?

Duke

Thank you once again Martin!  I REALLY appreciate your taking the time to go through and explain this to me and correct my misunderstandings.  Your example above of the 50 Hz line is extremely educational for me.  I had failed to appreciate that a standing wave occurs every time the line length puts its output at phase quadrature relative to the cone.  Imo this is valuable information. 

So now you have my wheels turning about making the enclosure deeper and putting the terminus on the back and using the path-length-difference (relative to the listening position) to help mitigate that one-wavelength cancellation notch.  What imo makes this approach promising is that, on either side of the one-wavelength frequency, we have standing wave resonances which shift the output primarily to the terminus, so there would be insufficient output from the cone itself for a cancellation notch.  I'd have to do some math to optimize it, but for now it looks like a possible window of opportunity.  Of course this isn't the only thing that would need to be juggled.

Duke

https://pmc-speakers.com/technology/atl

The commentary says it's tapered, but the cut-away doesn't look tapered to me.  Mixed in with the advertising copy, there are little technical nuggets dotted throughout the 3 mins.