To estimate the loss factor associated with bending deformation of the string, we can use yet another argument based on Rayleigh’s principle, on the assumption that material damping is small, as it invariably is for the materials that musical strings are made from. We already know the expressions for the kinetic and potential energies of the system (without damping). We also already have a good idea of the mode shapes of the undamped system: they are the sinusoidal shapes from section 3.1.1.
Now we use an argument to which we will return in rather more detail in section 10.3.3. To solve the damped problem we can replace $E$ in the expression for potential energy with a complex version $E(1+i \eta_E)$. Now Rayleigh’s principle says that given an approximation to a mode shape we can get a rather good approximation to its natural frequency by evaluating the Rayleigh quotient. The modes of the damped system will be slightly different from the modes of the undamped system, but the undamped mode shapes will still give a good approximation. So we evaluate the Rayleigh quotient using the true expression for the potential energy, with the complex modulus, but with the approximate expression for mode shape from the undamped calculation.
This gives a good approximation to the natural frequency, which will now be a complex number: $\omega_n=a+ib$, say. So the time dependence of free oscillation of this mode will be proportional to
$$e^{i \omega_n t}=e^{i(a+ib)t}=e^{iat} e^{-bt} .\tag{1}$$
The real part of the complex frequency describes the oscillation frequency as before, while the imaginary part describes an exponential decay, just as we would expect for a damped system. If we define a loss factor $\eta_{bend}$ to describe this energy loss mechanism, then from the definition of loss factor
$$b=a \eta_{bend}/2$$
so that
$$\eta_{bend} = \frac{2b}{a} =\frac{2\Im (\omega_n)}{\Re (\omega_n)} \approx \frac{\Im (\omega_n^2)}{\Re (\omega_n^2)} \tag{2}$$
in terms of the real and imaginary parts of the complex frequency, where we have assumed small damping so that $b \ll a$.
The Rayleigh quotient for the damped stiff string, from eq. (6) of section 5.4.3, is
$$\omega_n^2 \approx \dfrac{\int_0^L{P \left(\frac{\partial u_n}{\partial x}\right)^2} dx + \int_0^L{E(1+i \eta_E) I \left(\frac{\partial^2 u_n}{\partial x^2}\right)^2} dx}{\int_0^L{m u_n^2} dx} \tag{3}$$
so that
$$\omega_n^2 \approx \dfrac{P}{m} \left(\dfrac{n \pi}{L}\right)^2 + \dfrac{E(1+i \eta_E)I}{m} \left(\dfrac{n \pi}{L}\right)^4 . \tag{4}$$
Using eq. (2), we can deduce
$$\eta_{bend} \approx \dfrac{E \eta_E I \left(\dfrac{n \pi}{L}\right)^2}{P} = 2 \alpha n^2 \eta_E$$
in terms of the inharmonicity parameter $\alpha$ introduced in section 5.4.3.