I became fascinated with how the overtone series, or harmonics, relates to how brass instruments function. Most trumpet players (or brass, really) should notice that as they play higher notes, they become closer together on the staff. What fewer might notice is the pattern, where each octave that one goes up, there becomes twice as many notes (played without valves), and that each newly available note comes in between the notes of the previous octave.
This led me to notice one last thing: trumpets can't play the fundamental frequency for their overtone series. The lowest note one can clearly produce (without using the valves) is a middle C, but the next note up is a G, clearly not the 2nd harmonic. This means that the lowest clear note for a trumpet is the 2nd harmonic, not the fundamental.
However, I have also played the flugelhorn, which has the same tube length as a trumpet, and the same overtone series (the 2nd and 3rd harmonics are still the same frequency C and G as on a trumpet). On a flugelhorn, I found that you can also comfortably drop an octave below middle C, which is the fundamental frequency for the instrument.
Why can a flugelhorn clearly and easily drop to the fundamental, when a trumpet can not? The main functional difference between the two instruments is that a trumpet has a cylindrical bore while a flugelhorn has a conical bore. Why does this difference in bore have such an impact on the range that each instrument can play? I know that the cylindrical bore has a "warmer sound", but I don't particularly care about that for the purposes of this question.
I did notice a section of this Wikipedia article mentions "whole tube vs. half tube" in relation to the ability of some brass instruments' ability to hit their fundamental, but I am interested in a more in-depth explanation of why that is important.
Visuals of the instruments in question
You can clearly see the difference in tubing between the two instruments.The trumpet's diameter only widens along its final approach to the bell, while a flugelhorn's diameter widens along the entire length of its tubing. Otherwise, they play nearly identically (outside of the thing that sparked this entire question).