5.4.3 Natural frequencies of a stiff string

To estimate the natural frequencies of a string with non-zero bending stiffness, we can use Rayleigh’s principle. In order to do that we need expressions for the potential and kinetic energies of the stiff string. Because kinetic energy is conventionally denoted $T$ in textbooks, for this particular section we will denote the tension of the string by $P$ to avoid confusion.

The kinetic energy is, as usual, the sum of “$\frac{1}{2}m v^2$” for all small elements of the string, so that

$$T=\frac{1}{2}\int_0^L{m \left(\frac{\partial w}{\partial t}\right)^2} dx \tag{1}$$

for a string with mass per unit length $m$.

Figure 1. A small element of deflected string (exaggerated vertical scale).

The potential energy is the sum of two terms, one representing the potential energy of an ideal flexible string and the other being the term we have already seen (in section 3.3.1) for a bending beam. To find the expression for the string component, consider the small element of string sketched in Fig. 1. This element started with length $\delta x$, but it stretches slightly to a length $\delta l$ after the string has deflected. By Pythagoras’s theorem

$$\delta l = \sqrt{\delta x^2 + \delta w^2}=\delta x [1+(\delta w / \delta x)^2]^{1/2}$$

$$\approx \delta x \left[1+ \frac{1}{2}\left(\dfrac{\partial w}{\partial x}\right)^2 \right]\tag{2}$$

by the binomial theorem. Work is done during this stretch, against the existing tension $P$ of the string. So the stored energy in this element is approximately

$$\frac{P}{2}\left(\dfrac{\partial w}{\partial x}\right)^2 \delta x \tag{3}$$

and the total potential energy of the string is obtained by integration along the length. Adding in the term for the bending beam, the total potential energy of the stiff string is

$$V=\frac{1}{2}\int_0^L{P \left(\frac{\partial w}{\partial x}\right)^2} dx + \frac{1}{2}\int_0^L{E I \left(\frac{\partial^2 w}{\partial x^2}\right)^2} dx . \tag{4}$$

For a monofilament string with a circular cross-section of radius $r$ made of material with Young’s modulus $E$, the bending rigidity coefficient has the value

$$EI=\dfrac{E \pi r^4}{4} \tag{5}$$

but for a string with multi-layer wrapped construction the factors $E$ and $I$ have no useful independent meanings, and the combined coefficient $EI$ is best regarded as a single constant, to be fitted to measurements of the behaviour of the string. We will see an example in section ?.

Now to use Rayleigh’s principle to estimate frequencies, we need an approximate expression for the mode shape; but the mode shapes are not much affected by stiffness, so we can simply use the ideal shapes $u_n(x)=\sin(n \pi x/L)$ for this purpose. We can then evaluate the Rayleigh quotient to obtain

$$\omega_n^2 \approx \dfrac{\int_0^L{P \left(\frac{\partial u_n}{\partial x}\right)^2} dx + \int_0^L{E I \left(\frac{\partial^2 u_n}{\partial x^2}\right)^2} dx}{\int_0^L{m u_n^2} dx} \tag{6}$$

so that

$$\omega_n^2 \approx \dfrac{P}{m} \left(\dfrac{n \pi}{L}\right)^2 + \dfrac{EI}{m} \left(\dfrac{n \pi}{L}\right)^4 . \tag{7}$$

For any realistic musical string, the effect of stiffness is relatively small, and in particular the fundamental frequency is always well approximated by neglecting the effect of stiffness:

$$\omega_1^2 \approx \dfrac{P \pi^2}{mL^2} . \tag{8}$$

We can then approximate eq. (7) by

$$\omega_n^2 \approx n^2 \omega_1^2 \left[1 + 2 \alpha n^2 \right] \tag{9}$$

so that

$$\omega_n \approx n \omega_1 \left[1 + \alpha n^2 \right] \tag{10}$$

where

$$\alpha= \dfrac{EI \pi^2}{2P L^2} . \tag{11}$$