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 Figure 5. Eigenresonances of a tube that is 60 cm long and 31 mm in diameter. The harmonic partials are marked by v-shaped cursors.
A Second Series of Smaller and Wider Peaks That Are Not Harmonically Related But Are Slightly Stretched in Frequency
The small, broad peaks shown in the spectrum demonstrate the presence of acoustic eigenmodes (standing waves) of the pipe. (A so-called standing wave occurs in a pipe when the sound waves reflected back and forth in the pipe are com- bined such that each location along the pipe axis has con- stant but different amplitude. The locations with minimum and maximum amplitude are called nodes and antinodes, respectively. The frequency of the standing wave is the res- onance frequency or eigenfrequency of the tube. Standing waves occur in a tube on several frequencies.) The presence of eigenmodes can be tested experimentally by using ex- ternal acoustic excitation. If a pipe is placed in the sound field generated by a loudspeaker, the pipe will amplify the frequency components that correspond to the eigenreso- nances. Placing a small microphone in the pipe and using an excitation in a wide frequency range, the eigenresonance spectrum can be determined. Such a spectrum
is shown in Figure 5 for a cylindrical tube. The eigenresonances are slightly stretched; the ei- genfrequencies are a bit higher than the har- monics of the first eigenresonance.
The stretching of the eigenfrequencies is much more pronounced in open organ pipes. In the spectrum of a Diapason pipe (Figure 4a), the ninth eigenresonance lies about halfway be- tween the ninth and tenth harmonic partials. The stretching becomes larger for larger diam- eter-to-length ratios and for smaller openings at the pipe ends. The measured spatial distribu- tion of the first, third, and fifth eigenmodes in a
fairly wide flute pipe is shown in Figure 6. It can be observed that the standing waves lay asymmetric in the pipe; they are shifted toward the mouth. Moreover, the half wavelength of the first eigenmode (and n times the half wavelength of the nth eigenmode) is longer than the length of the resonator. The difference can be regarded as an “end correction” for practical calculations.
These experimental facts can be understood by taking into account the physical properties of the organ pipe as an acoustic resonator. The air column in the pipe has several ei- genmodes (standing wave patterns) with characteristic reso- nance frequencies (eigenfrequencies). Their frequencies are not harmonically related because of the end correction (Nel- kon and Parker, 1970), which decreases with the frequency (Fletcher and Rossing, 1991). Because the end correction is proportional to the pipe diameter, the stretching of the ei- genfrequencies is larger for wide pipes than for narrow ones. Moreover, the end correction for a small opening (mouth) is larger than that of the larger open end. Therefore, the ei- genfrequency stretching of an organ pipe is larger than that of a tube with the same length and diameter. Because of the different end corrections at the openings, the standing wave is located asymmetrically inside the organ pipe (Angster and Miklós, 1998). Therefore, the sound spectra at the mouth and at the open end are different, as shown in Figure 4.
A Frequency-Dependent Baseline
The baseline of the spectrum (see Figure 4) is determined by the broadband noise at the mouth of the pipe. This noise is produced by the airflow at the flue and the upper lip (Fabre et al., 1996). Because the resonator amplifies this noise around the eigenresonances, the amplified noise may dominate the sound of the pipe in the high-frequency range,
  Figure 6. Standing waves in an organ pipe. Sound pressure distributions of the first, third, and fifth eigenmodes in a wide pipe are shown.
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