11.5.1 Damping models for tube resonances

The measured input impedance of a trombone without its mouthpiece, shown in the blue curve of Fig. 6 in section 11.5, gives modal fits that reveal something interesting about the usual model for energy dissipation of plane sound waves in tubes. The measured Q-factors are plotted against frequency using black stars in Fig. 1, on a log-log scale. The red circles show modal fits to the synthesised impedance (shown in the red curve of Fig. 6 of section 11.5). That synthesis model used modal Q-factors based on the usual formula for boundary dissipation in cylindrical tubes that are not extremely narrow, described in section 11.1.1 and frequently cited in the standard literature on brass and woodwind instruments, for example Fletcher and Rossing [1] and Campbell, Gilbert and Myers [2].

Figure 1. Modal frequencies and Q-factors fitted to a measured input impedance of a trombone without mouthpiece (black stars), and to a synthesised version of that impedance using the conventional model of energy dissipation (red circles). Power law trends are indicated, with exponents 0.5 (dashed line) and 0.7 (solid line).

The damping model is usually stated in terms of a spatial decay rate, so a first step is to convert that into a temporal decay rate, which in turn can be translated into a Q-factor. The argument used in this work runs as follows. The damping model gives the spatial variation of a harmonic signal at frequency $\omega$ as

$$e^{i\omega x/c-\alpha x} \tag{1}$$


$$\alpha \approx 1.2 \times 10^{-5} \sqrt{\omega}/a, \tag{2}$$

$c$ is the speed of sound and $a$ is the radius of the tube. On the other hand, in a free decay of a mode with frequency $\omega_n$ and Q-factor $Q_n$, the temporal variation is proportional to

$$e^{i \omega_n t -\omega_n t/Q_n} . \tag{3}$$

If we now assume that the frequency-wavenumber relation $k=\omega/c$ continues to hold for this complex frequency and complex wavenumber, we find

$$Q_n \approx \frac{\omega_n}{2 \alpha c} . \tag{4}$$

The Q-factors indicated by the red circles, based on this damping model, are always well within a factor of 2 of the corresponding measured results, and ordinarily this would be regarded as excellent agreement for any predictive model of damping. But in fact the plot makes it clear that this theory is missing something, because it does not predict the correct trend with frequency. The measured results show a clear and orderly pattern, which roughly follows a power law, but the power is more like 0.7 (indicated by the solid blue line) than the value 0.5 arising from the square root in equation (2) (shown by the dashed blue line).

One thing to note from Fig. 1 is that at high frequencies the measured Q-factors are higher than the predicted ones, despite the fact that the theory is for wall losses only, while the measurements will include additional damping associated with sound radiation. Indeed, a growing contribution from radiation damping might be the reason for the slight curvature of the line of black stars, detectable in the plot.

[1] Neville H Fletcher and Thomas D Rossing; “The physics of musical instruments”, Springer-Verlag (Second edition 1998)

[2] Murray Campbell, Joël Gilbert and Arnold Myers, “The science of brass instruments”, ASA Press/Springer (2021)