As in section 7.2.1, we are interested in a monofilament string of circular cross-section of diameter $d$ and length $L$, under tension $T$ and made of material with Young’s modulus $E$ and density $\rho$. A selection criterion related to damping can be formulated using results derived earlier. The rising trend at high frequency seen in Fig. 1 (reproduced from Fig. 2 of section 7.2) is caused by the effect of material damping, acting through the bending stiffness of the string. For the two thinner strings shown in the figure, which produced acceptable musical sounds, it can be seen that the data points run out at roughly the same value of loss factor, around $10^{-3}$.
This suggests that the useful bandwidth of a given plucked string might be associated with a threshold value of loss factor, and specifically of the component of loss factor associated with bending stiffness. A model for that loss factor was derived in section 5.4.4. Allowing for damping, Young’s modulus becomes a complex value $E(1+i \eta_E)$, and the loss factor of the $n$th string overtone is then given by
$$\eta_{bend}=\dfrac{EI}{T}~\left(\dfrac{n \pi}{L} \right)^2 \eta_E = \left[\dfrac{nd}{L} \right]^2 \dfrac{\pi^2 E}{64 \rho \gamma^2} \eta_E \tag{1}$$
after substituting the expressions for $T$ and $I$ from section 7.2.1, where $\gamma=Lf_1$ as before. The factor $n^2$ in this expression describes the rising trend. The criterion we want takes the form of a threshold value of $\eta_{bend}$, and for any particular string this will result in a threshold value of $n$ because the other factors in eq. (1) will be known.
Of course there is not a crisply-defined threshold for damping, but for the purposes of a selection criterion with the right order of magnitude, a threshold value $\eta_{bend} \approx 2 \times 10^{-3}$ will be used. Furthermore, we will require that a “musically acceptable” string should have at least 10 overtones with damping lying below this threshold. The appropriateness of these choices will be confirmed shortly for the three strings shown in Fig. 1, and they have been further supported by a wide range of case studies described in reference [1]. The material loss factor $\eta_E$ is fixed for a given string and tuning: measurements on a range of nylon strings [1,2] suggest a value $\eta_E \approx 0.04$.
The criterion can be expressed in graphical form. From eq. (1), it can be seen that the value of $\eta_{bend}$ for a given material depends on two parameters relevant to string choice: $nd/L$ and $\gamma$. For material of a given density, the value of $\gamma$ determines the stress:
$$\sigma = 4 \rho \gamma^2. \tag{2}$$
This is important, because it was shown in reference [2] that the value of $E$ for nylon strings varies significantly with stress through the effect of strain stiffening, described in section 7.2. In other words, $E$ is a function of $\gamma$. It is straightforward to draw a contour map of $\eta_{bend}/\eta_E$ in the $(\gamma, nd/L)$ plane. The result is shown in Fig. 2, for nylon strings. Contours of $\eta_{bend}/\eta_E$ have been plotted at intervals of 0.01 up to the value 0.1. Beyond that value the string overtones will surely be too highly damped to be of interest: the suggested threshold value is 0.05, in the middle of the plotted range.
Points corresponding to particular strings can be calculated and added to the plot, but because of the presence of $n$ in the quantity plotted on the $y$-axis, a given string gives a point for every relevant overtone. These overtones all have the same value of $\gamma$, so they make a regular vertical column in the plot. The open symbols show the three strings from Fig. 1. For each string, the lowest plotted symbol shows the fundamental $n=1$, and then to indicate the pattern without cluttering the plot with too many points, symbols are plotted above it for $n=10, 20, 30…$
To interpret the plot, consider first the thinnest string of the set in Fig. 1, indicated by square symbols towards the right-hand side of Fig. 2. Locating the contour corresponding to $\eta_{bend}/\eta_E=0.05$, it can be seen that the closest square symbol to that contour marks the value $n=50,$ so the prediction is that this string should have about $50$ overtones with damping lower than the chosen threshold. The middle string from Fig. 1 is indicated by circular symbols, and because this string had the same length and the same tuning as the thinnest string, they appear at the same value of $\alpha$. However, the circular symbols are wider apart, and the $\lambda=0.05$ contour passes between the two symbols marking $n= 20$ and $30$. So for this string, roughly $25$ overtones should have damping below the threshold. Comparing the two, the prediction is that the bandwidth of lightly-damped string modes should be roughly twice as big for the thinner string. Looking at where the plotted points run out in Fig. 1, this prediction matches the observations quite well. Both these strings exceed the requirement of at least 10 overtones with light damping.
The thickest string from Fig. 1 is indicated in Fig. 2 by diamond symbols, towards the left-hand side. For this string, even the symbol corresponding to $n=10$ lies above the $\lambda=0.05$ contour, so the prediction is that this string should have too few lightly-damped overtones to comply with the suggested criterion for acceptability. Furthermore, remember that the criterion underlying this plot captures only the damping due to viscoelasticity: for a very thick string like this, the damping due to air viscosity takes over at low frequency while the viscoelastic loss is still quite high, so that in fact the model predicts that this string should have no modes at all with low damping. That is exactly what the measurements in Fig. 1 revealed.
Based on this information, a damping criterion can be plotted on the string selection chart developed in section 7.2. Because the vertical axis depends on $d$ rather than on $nd/L$ as in Fig. 2, the length $L$ will make a difference. For a given value of $L$, it is easy to take each value of $\gamma$ and use the expression for $\eta_{bend}/\eta_E$ to calculate the threshold value of $d$ for which a string of that length would have at least 10 overtones with damping lower than the chosen value. This leads to the rising curving lines in the selection chart (Fig. 6 of section 7.2).
[1] J. Woodhouse and N. Lynch-Aird “Choosing strings for plucked musical instruments”. Acta Acustica united with Acustica 105, 516-529, (2019).
[2] N. Lynch-Aird and J. Woodhouse “Mechanical properties of nylon harp strings”. Materials 10, 497, (2017).