Earl's paper on directivity is an excellent one. The issue with most "horns" are that they are poorly designed, as Earl's paper points out. I.e. uneven directivity over frequency and narrowing at high frequencies to a pencil beam width. Most horns vertical directivity is much narrower than horizontal directivity with typical patterns like 90 degrees wide and 40 or 50 degrees vertical.
However, modern waveguides like what is used in the
JBL M2 with it's "constant directivity" waveguide with a pattern of 110 x 100 does not have narrowing at high frequencies or any "horn honk" whatsoever. Have a look at the "
spinorama" to see what I mean. If you can find a pair (or other JBL's with the image control waveguide) to listen to, you will hear it first hand. Earl's waveguides are also constant directivity using an oblate spheroid and if you look at Earl's whitepaper, you can also see a Polar Map of his waveguide, which is pretty dang good. Note that constant directivity waveguides also require a
special type of eq compensation.
There has been a lot of trickle down tech from the pro field into consumer loudspeakers and you will notice that D&D 8c and even the Revel Salon 2 have their tweeters mounted in a "waveguide" for more of a constant directivity control as their polar plots and spinoramas reveal. Both these products have excellent, constant off-axis frequency response at the mid and high frequencies. Almost all pro monitors also have the tweeter loaded in a waveguide...
Unfortunately, the Avant Garde system mention above, along with the pic of the "Smith" horn and bullet tweeter are not constant directivity devices (aside from mismatching directivity between devices). Having heard both in the past, they are not a good representation of what a properly designed constant directivity system should sound like. In fact, the whole idea is that the off-axis response is as good as on the on axis response. My JBL 4722's use a constant directvity waveguide and I get an even mid to high frequency response across my 3 seat, 6 foot couch. No beaming here, but it is an older design compared to the M2. The math is really tricky to design a proper constant directivity horn/waveguide with excellent on and off axis response. But there is some good software at this thread that does it right:
https://www.diyaudio.com/forums/multi-way/338806-acoustic-horn-design-easy-ath4.html