Cylindrical pipe with a conical end

Most real wind instruments have a complex geometry. Many of them, however, such as the trumpet or the clarinet, have a bore that can be well approximated by joining two sections, one cylindrical and one conical. Staying within the scope of model instruments, it is therefore interesting to study the response of pipes with this geometry by varying separately their two possible parameters, namely

1. the relative length of the cylindrical and conical sections, with the end radius of the cone fixed
2. the width of the mouth of the cone, with the position of the junction on the cylinder fixed.

The graphs below show the result of an approximate calculation of the response, and allow us to draw some conclusions.

First of all, it is clear that joining the two sections produces varied and interesting responses compared with the simpler geometries, which explains, at least in part, the variety of timbre that can be obtained from wind instruments. In both cases, a continuous change of a single parameter affects the whole spectrum, not just one component. This suggests that designing the sound of a pipe "on paper", however simple its geometry, is no simple matter. In particular, even small changes can affect both the position of the fundamental and the ratios between the frequencies of the higher harmonics and the fundamental.

The first effect is not particularly important, since it shifts the basic tuning of the instrument, a shift that can always be remedied by correcting the overall length of the pipe.

The second effect is more interesting, since it can affect the degree of inharmonicity of the instrument. This, in turn, affects the timbre of the instrument both physically and through the mechanism of perception, which is able to detect subtly whether a complex sound is harmonic or not. For a discussion of this topic, however, see the relevant pages in the section devoted to sound perception.

Geometry	Acoustic response	Resonances
By varying the relative length of the cylindrical section with respect to the conical one, we pass gradually from a purely conical instrument to a purely cylindrical one through a series of intermediate steps. We already know the spectra of the two extremes, and in particular we know that they have the same resonance frequencies (see cylindrical pipe and conical pipe). Now, however, we can appreciate that the transition of the frequencies is not uniform. The fundamental rises and then returns to its initial value, non-linearly as a function of the position where the conical section begins, while the higher harmonics oscillate, passing through a number of maxima equal to the order of the harmonic plus one. The first harmonic has two maxima, the second three, and so on. There are points where the series is more harmonic than at others, but only at the extremes is it exactly harmonic.

Geometry	Acoustic response	Resonances
The effect of the bell is particularly important in brass instruments, and cannot be fully described in terms of our simple model. Nevertheless, even for the simplified geometry of the drawing, we can make some useful observations. First, the bell matches the impedance of the pipe to the impedance of free air, thus making it easier for the sound to be radiated. This radiation is especially favoured at high frequencies, which helps give brass instruments a particularly brilliant sound, even when the emitted fundamental is rather low, as in large instruments (trombone, bass tuba, etc.). As the opening of the bell increases, the impedance at the end of the pipe decreases, favouring the attenuation of the even harmonics, and the response of the instrument becomes less and less harmonic. Note, however, that modern brass instruments, unlike the corresponding early instruments, do not have conical bells, but bells shaped according to more complex curves, which require a more accurate analysis.