I've not built any but we had a pair of IMF TLS80 ( http://www.imf-electronics.com/Home/imf/speaker-range/reference-speakers/tls80 ) in a studio I was involved with which were 1/4 wave & huge. These were designed by Irving M. Fried (IMF) who is widely considered a pioneer of the Transmission line design, They are a 4 way system with a supertweeter that gave a top end that was way out of my hearing range. I loved those speakers the bass extension was great. We used to put on "bloody well right" by Supertramp, when people asked what they were like, & watch them jump out of their skin when when the stabs came in at the beginning of the track during the piano intro.
From what I understand you get 2 low resonant peaks in speakers & when you have a properly designed TL speaker it pushes one of them below audibility which makes the bass quite smoooth & gives it a lower extension than many other cabinets designs can achieve at the size of cabinet. The principal is that the rear wave from the bass speaker is killed by a lagged tapered tube. I think in practise this doesn't really happen & the tube is open but the length of the tube is tuned to mostly either 1/8th or 1/4 of the wavelength of the lowest desired frequency. I think Horn loaded speakers can sound better but are normally much bigger than a transmission line. Some people consider the TL to be a dated design that has been superceded, but in a nice sized room they sound good, at least to me. I notice that in the pic you posted the design of the folded tube is not a taper. If you look at the exploded pic of the TLS80 that I have attached you can see that the baffles are arranged so the tube tapers.
I also owned some B&W DM2 speakers from the 70's at one point (like these https://www.ebay.co.uk/itm/BW-DM2-Transmission-Line-Speakers/193289247851?hash=item2d00efdc6b:g:ldgAAOSwPCFeE51G ). For the size and price (I think I paid £50) they sounded really good (to my old ears), but the top end was a bit slow by todays standards perhaps. These were only 1/8 wave in the TL part.
I now have PMC TB2 in my mix room that I really like which are also 1/8th wave but not really a new design now but I like them & am relectant to spend any more money at the moment on replacing them.
OK people are going to say that there are better designs out there or that these are dated designs. But for the price & performance they are good.
I would say that the 1/4 wave IMF speakers need quite a big room to breathe properly though.