If we have measured the bridge admittance $Y(\omega)$ of a particular cello, we can use that in an enhanced version of the Raman-Schelleng calculation of minimum bow force, set out in section 9.3.1. As before, we start by assuming that the string is executing Helmholtz motion. If the bow speed is $v_b$, the string has tension $T$, length $L$ and impedance $Z_0$,the bowing position is $\beta$ and the fundamental frequency is $f_0$ Hz, then the sawtooth waveform of bridge force can be written as a Fourier series in the form:
$$f_{br}(t)=\sum_{n=1}^\infty{a_n \sin 2 \pi n f_0 t} \tag{1}$$
with
$$a_n = (-1)^{n+1} \dfrac{Tv_b}{n \pi f_0 \beta L} . \tag{2}$$
The displacement response at the bridge can then be written in terms of $Y$ and the real part of a complex form of the Fourier sine series as
$$\Re \sum_{n=1}^\infty{\dfrac{(-1)^n Z_0v_b}{n^2 \pi^2 f_0 \beta} Y(2 n \pi f_0) e^{2n \pi i f_0 t}} . \tag{3}$$
We can now use the same approximation as in section 9.3.1 to deduce the perturbation force at the bow, by assuming that $\beta$ is small so that we can approximately treat the section of string between bow and bridge as being straight:
$$f_{pert}(t)=\dfrac{T}{\beta L}\Re \sum_{n=1}^\infty{\dfrac{(-1)^n Z_0v_b}{n^2 \pi^2 f_0 \beta} Y(2 n \pi f_0) e^{2n \pi i f_0 t}} + K$$
$$=\dfrac{2 v_b Z^2_0}{\pi^2 \beta^2}\Re \sum_{n=1}^\infty{\dfrac{(-1)^n }{n^2} Y(2 n \pi f_0) e^{2n \pi i f_0 t}} + K \tag{4}$$
where $K$ is a constant. Its value is determined in the same way as before, by requiring that $f_{pert}(\pm 1/2f_0)=0$, so that
$$K=-\dfrac{2 v_b Z^2_0}{\pi^2 \beta^2}\Re \sum_{n=1}^\infty{\dfrac{Y(2 n \pi f_0)}{n^2} } . \tag{5}$$
We can immediately deduce the new expression for the minimum bow force: the criterion is that this perturbation force is just sufficient that, at some stage in the cycle, it takes the friction force up to the limiting value. The result is
$$f_{min}=\dfrac{2 v_b Z^2_0}{\pi^2 \beta^2 (\mu_s – \mu_d)} \times \left[ \Re \sum_{n=1}^\infty{\dfrac{Y(2 n \pi f_0)}{n^2} } \right.$$
$$\left. – \max \left\lbrace \Re\sum_{n=1}^\infty{\dfrac{(-1)^n }{n^2} Y(2 n \pi f_0) e^{2n \pi i f_0 t}} \right\rbrace \right] \tag{6}$$
where “max” means the maximum value of this function of time, taken over a complete cycle of the periodic waveform.
This expression looks very complicated, but we can relate it back to Schelleng’s original formula by looking at an important special case. In our study of the wolf note, we were mainly interested in a single body resonance, and in the case where the cellist bows the note coinciding exactly with that resonance. Suppose this resonance is described in terms of an oscillator with mass $m$, stiffness $k$ and dashpot strength $c$. The admittance then takes the simple form
$$Y(\omega)=\frac{i \omega}{k + i \omega c – \omega^2 m} . \tag{7}$$
At the resonance frequency, $k=\omega^2 m$ so that $Y=1/c$. Furthermore, looking at the two Fourier series summations in equation (6) we see that, with the factor $1/n^2$, they both converge very rapidly. So we probably get a reasonable approximation to this resonant case if we simply keep the fundamental term $n=1$. Equation (6) then reduces to
$$f_{min} \approx \dfrac{2 v_b Z^2_0}{\pi^2 \beta^2 (\mu_s – \mu_d)} \max \left\lbrace Y(2 \pi f_0) (1+e^{2 \pi i f_0 t} \right\rbrace$$
$$ = \dfrac{4 v_b Z^2_0}{c \pi^2 \beta^2 (\mu_s – \mu_d)} . \tag{8}$$
This looks exactly like Schelleng’s formula (equation (9) of section 9.3.1), with $c$ in place of his assumed resistance $R$, and a numerical factor $8/\pi^2 \approx 0.8$.