Just as the title states. I could not find a coherent answer online.
Many thanks in advance
The acoustics of brass instruments are ridiculously complicated, and the "open end" vs. "closed end" distinction that we remember from undergraduate physics is entirely insufficient to explain it in full. I can't pretend to be an expert in the subject myself—I'm just an amateur trombone player and professional physicist with an interest in the subject. There is an extensive discussion of the acoustics of the trumpet on the UNSW website (linked in the comments above), as well as in Heller's Why You Hear What You Hear (2013). Here's a basic summary of the latter source:
As a very crude approximation, brass instruments can be thought of as a tube with one closed end (the constriction inside the mouthpiece) and one open end (the bell.) As any second-year physics major knows, the resonant frequencies of such a tube should be a fundamental frequency and the odd multiples thereof: $f_0, 3 f_0 ,5f_0, 7f_0, ...$. However, as any trumpet or trombone player knows, the "natural" frequencies of a brass instrument are the harmonic series: for a given length of tubing, there's a lowest note you can play, and then another note an octave above that, and another note a perfect fifth above that, and so forth. This implies resonant frequencies that include all multiples of a fundamental frequency: $f_0, 2 f_0, 3 f_0, ....$ There is obviously a contradiction here.
The resolution is that the mouthpiece and the bell actually shift the resonant frequencies around somewhat. The mouthpiece actually acts as a Helmholtz resonator, which when coupled to the air in the tube tends to shift the resonant frequencies of the tube to lower frequencies. And the fact that the tube of the instrument is conical, with a constantly increasing diameter, means that different frequencies "see" a the length of the tube differently, depending on whether the diameter of the tube changes quickly or slowly on the scale of the associated wavelength.
The net result of all this is that the resonant frequencies of the entire instrument (mouthpiece + horn + bell) are now tuned into the ratio $2 f'_0, 3 f'_0, 4f'_0, ...$ for a different frequency than one would assume from the length of the tube. For example, a trombone is approximately 270 cm (9 feet) long. An open tube of this length would have a fundamental frequency of about $f_0 = 31$ Hz; but for a standard trombone, all of the resonant frequencies are multiples of B♭1, with $f'_0 = 58$ Hz.
The most interesting part about the acoustics of the brass family is that they can "play" the fundamental frequency $f'_0$, even though it is not a resonance of the tube! These are called pedal tones, and they rely on a psychoacoustical trick. When most musical instruments play a note, the signal they produce doesn't just contain the fundamental frequency of the note $f$, but also contains frequencies at integer multiples $2f, 3f, 4f, ...$. This "mixture" of the base frequency with other frequencies at integer multiples of the fundamental is what gives musical notes their timbre; different amounts of these different overtones are what makes a violin (say) sound different from a clarinet even when playing the same note. A brass instrument playing a pedal tone actually only produces the overtones of the fundamental $2f'_0, 3f'_0, ...$ without playing the fundamental $f'_0$ itself. Your ear and your brain then fill in the rest, making you think that you're actually hearing a note at the frequency $f'_0$.
The issue is slightly complicated, due to the ambiguity of the terms "open pipe" vs "closed pipe" used to classify wind instruments.
Some authors use it to refer to the boundary conditions on the ends of the instrument. If the pressure of the standing waves in a pipe must be continuous with the outside pressure on both ends of the instrument, then it is considered an open pipe, whereas if the pressure is continuous with environmental pressure on one end and at a maximum on the other end, then it is considered a closed pipe. This is the definition used by this page. By this definition, brass instruments (lip reed instruments) are closed pipes.
However, some authors use it to refer to the qualities of overtone series of an instrument. If an instrument has a full overtone series, then it is considered an open pipe. If an instrument only has odd multiples of the fundamental frequency in its overtone series, then it is considered a closed pipe. By this definition, brass instruments are open pipes.
However, the second definition is not very rigorous, since the shape of the instrument plays a very important role in its overtone series. Consider the clarinet and the saxophone. They have the exact same sounding mechanism (single reed on the closed end), but the clarinet only has odd multiples of the fundamental in its overtone series, while the saxophone has a full overtone series. The reason is that the former has a cynlindrical bore, but the latter has a conical bore. This is also intuitively explained by this UNSW page. Essentially, the wavefront in a conical bore is no longer planar, but spherical, which results in different allowed or forbidden modes.
I suspect the reason why the second definition arises is the misconception that all wind instruments can be approximated by long cylindrical pipes, which would mean that their harmonic series can be classified by their open-ended-ness.
The trumpet is considered a closed-ended pipe, but this is not a straight pipe. The flared end (at the bell) modifies the overtone structure so that all the integer multiples (rather than just the odd multiples) of the fundamental are present. The fundamental is also modified by the flare. See Thomas Rossing's The Science of Sound.
The trumpet is a closed air column according to this source:
A closed-end instrument is an instrument in which one of the ends of the metal tube containing the air column is covered. An example of an instrument which operates on the basis of closed-end air columns is the clarinet. Some instruments which operate as open-end air columns can be transformed into closed-end air columns by covering the end opposite the mouthpiece with a mute. Even some organ pipes serve as closed-end air columns. As we will see the presence of the closed end on such an air column will effect the actual frequencies which the instrument can produce.
If both ends of the tube are uncovered or open, the musical instrument is said to contain an open-end air column [my emphasis].
The above line would not seem to me to apply here, except perhaps to some woodwind instruments.
I have no musical background, my answer is based on the similarly of design and operation of both the clarinet and the trumpet.
Thank you for your interest in this question.
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