It is useful to find a way to focus on a formant “hill” without the distracting details of the individual body modes. This can often be done by using an approach known as “Skudrzyk’s mean value method” [1]. The starting point is eq. (12) from section 2.2.7, expressing the admittance in terms of modal parameters. The relevant version of that equation is:

$$Y(\omega) =\sum_n \dfrac{i \omega u_n^2}{\omega_n^2+2i\omega \omega_n \zeta_n-\omega^2} \tag{1}$$

where the $n$th mode has amplitude $u_n$ at the bridge, frequency $\omega_n$ and modal damping ratio $\zeta_n$.

Skudrzyk was able to deduce from this equation that the height of an isolated modal peak is proportional to $1/\zeta_n$, while the level at the bottom of an *antiresonance* dip is proportional to $\zeta_n$. It follows that the mean level of a logarithmic plot follows the geometric mean of these two, and is thus independent of damping. If the damping were increased, the peaks and dips would blur out and all admittance curves would tend towards smooth “skeleton” curves representing the logarithmic mean of the original curves, in other words the mean trend in a decibel plot.

There is a physically appealing way to visualise the effect of increasing the damping. When a force is applied at a point on the structure, it generates a “direct field” consisting of outward-travelling waves. In time these will reflect from the various boundaries and return. Modal peaks will occur at frequencies where the reflections combine in phase-coherent ways. Antiresonances occur when the sum of reflected waves systematically cancels the original direct field. But at an “average” frequency, where neither of these coherent phase effects occurs, the reflected waves from the various boundaries tend to arrive in random phases and to cancel each other out, leaving the direct field to dominate the response. If damping is increased, the influence of reflections decreases. In the limit of high damping, the desired skeleton of the admittance is given by the direct field alone.

The effect is thus the same as if the boundaries had been pushed further away until the system becomes *infinitely large*. This gives a simple recipe to find skeleton curves for some idealised systems. For a plate-based system, we need to consider an infinite plate with the same material properties and thickness. The vibration of a point-driven infinite plate has a simple closed-form solution. If the plate has thickness $h$, and is made of isotropic material with Young’s modulus $E$, Poisson’s ratio $\nu$ and density $\rho$, the result is

$$Y_\infty^{p} (\omega)=\frac{1}{4h^2} \sqrt{\frac{3(1-\nu^2)}{E \rho}} . \tag{2}$$

We will also look at the behaviour of the banjo, so it is useful to note the corresponding result for a membrane. But this case has a snag: if a point force is applied to an ideal membrane, the linear governing equation predicts that the response will be *infinite*. To avoid this issue, the force must be applied through a finite footprint. If an infinite membrane with tension $T$ and mass per unit area $\sigma$ is driven through a circular region of radius $a$, the admittance is

$$Y_\infty^{m} (a,\omega)=\frac{ic}{2 \pi a T} \frac{H^{(2)}_0 (\omega a/c)}{H^{(2)}_1 (\omega a/c)} \tag{3}$$

where $c=\sqrt{T/\sigma}$ is the wave speed and $H^{(2)}_N$ is the *Hankel function* of order $N$, describing outgoing waves.

[1] E. Skudrzyk: The mean-value method of predicting the dynamic response of complex vibrators. *Journal of the Acoustical Society of America* **67**, 1105–1135 (1980).