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Figure 8. Sound radiation on the first (a) and second (b) partials. The colors correspond to the intensity of the sound
(see http://acousticstoday.org/8a.mp4 and http://acousticstoday.org/8b.mp4).
sured at the mouth of a normal (Diapason), a wide (Flute), and a narrow (Salicional or string) pipe clearly show this ef- fect (Figure 7). For the Diapason pipe, the first minimum lies at the sixth partial (Figure 7a). For the Flute pipe, the minimum occurs around the third partial (Figure 7b); for the Salizional pipe, it is shifted up to the eighth partial (Fig- ure 7c).
Different Spectral Envelopes at the Mouth
and at the Open End
It has been shown that the radiated acoustic field corre- sponds to that of two simple sources located at the open- ings of the pipe (see Figure 8). The simple sources radiate in phase for the odd partials and out of phase for the even partials. The strength is different for both sources, and the two openings radiate different spectra (Angster and Miklós, 1998). The spectra of the sound radiated at the openings are different because the standing waves in the pipe are asym- metrically located (see Figures 4 and 6). Because the end correction is inversely proportional to the area of the open- ing (Angster and Miklós, 1998), the envelope minimum oc- curs for the lower partial at the mouth than at the open end. That is, the spectral envelopes at the mouth and open end are always different.
Irregularities in the High-Frequency
Part of the Spectrum
Irregularities in the range of higher harmonics can be caused by the excitation of transverse resonances (cross-sectional eigenmodes) of the pipe. Pipe ranks may have harmonic partials in the range of transverse resonances; therefore, the transverse resonances can appear in the spectrum between the harmonic partials (Figure 7b, first transverse resonance around the eleventh partial). These resonances are excited by the high-frequency noise at the upper lip.
Irregularities in the spectrum may also be caused by wall vibrations. It has been shown that wall vibrations cannot radiate sound directly (Backus and Hundley, 1965; Angster et al., 1998). On the other hand, a linear coupling exists be- tween the air column and the pipe wall for rectangular pipes (Angster et al., 2011) and also for cylindrical pipes if the pipe cross section is not a perfect circle but is slightly elliptical or the wall is very thin (Kob, 2000). In these cases, wall vi- brations can influence the sound radiated at the openings, especially during the transient (Angster et al., 1998; Kob, 2000). If a sharp vibration mode is close to an eigenmode or harmonic partial of the pipe sound, both modes will be coupled, which leads to a slight detuning of the correspond- ing sound component. Nevertheless, such a coincidence is quite rare in practice.
Figure 9 shows the vibration diagrams recorded by a three-di- mensional (3-D) laser vibrometer of a Diapason G pair of pipes. Figure 9a (http://acousticstoday.org/9a.mp4) shows the pipes made of plain metal (tin-lead alloy) at the fifth partial (974 Hz) and Figure 9b (http://acousticstoday.org/9b.mp4) shows the pipes made of zinc at the fifth partial (same frequency). It is evident that despite the same measuring frequency, the pipes made of different materials show very different vibration mode shapes. Figure 10 shows the 3-D representation of the analyzed attack transients (attack; how the partials will be built up in time) of the same Diapason G pair of pipes (shown up to the sixth partial), and the noises between the partials are also re- corded. It is obvious that the attacks of the two pipes are very similar. Experiments showed that the differences in the record- ed attacks with pipes made of different materials are not larger than with two successive attacks of the same pipe.
Figure 9. Vibration-mode shape of the Diapason G pair of pipes. Red indicates that the pipe walls are vibrated hard and green means less vibration. a: Pipe made of plain metal (tin-lead alloy) at the fifth partial (974 Hz, see http://acousticstoday.org/9a.mp4). b: Pipe made of zinc at the fifth partial (and same frequency,
see http://acousticstoday.org/9b.mp4).
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