Theoretical minimum bow force

In his well-known paper, (Schelleng, 1973) studies what a string being bowed under static conditions (i.e. bow velocity, force, etc. held constant) must fulfill in order to self-sustain a Helmholtz oscillation. He analytically derives the conditions from a simplified model of how the bowed string works based on the work by (Raman, 1918). In that model, the bow-string interaction is completely ideal and the reflection function of the bridge, which models the body, is that of a simple dashpot that opposes to movement.

Schelleng provides two inequations that relate the model space (string impedance, friction coefficients, and dashpot impedance as a body model) and the control space (bow-bridge relative distance, bow force, and bow velocity). In those equations, bow force and bow-bridge distance have different exponents - and that is reflected on the well-known Schelleng diagram, which shows that a regime map on a logarithmic domain of these two parameters displays a diagonal band where Helmholtz motion is possible.

One of the sections in (Woodhouse, 1993) contains what is probably the most notable contribution to the study of the interplay between the body of bowed string instruments and their playability. In particular, Woodhouse builds on the work by Schelleng and studies the relationship between the reflection function of the bridge, characterized as a frequency-dependent admittance, and the minimum bow force required to sustain Helmholtz motion, all other model and control parameters being the same.

The paper goes through the inequations provided by Schelleng, by explaining its derivation, and then provides a critique to one of the several over-simplifications Schelleng makes – the assumption that the instrument body can be represented by an ideal dashpot alone. Then, following what Schelleng himself already hinted on the original paper, the author proceeds to develop the conditions for a generic, frequency-dependent driving-point admittance with a focus on the minimum required bow force. The extended model reduces to Schelleng’s version if the appropriate admittance is used, which is decreases at higher frequencies according to $Y = 1/R$, where $R$ is the resistance of the dashpot.

Woodhouse’s minimum bow force formulation

An ideal Helmholtz motion can be described as a sawtooth-shaped velocity wave, with the wave parameters given by the geometry of the string. In short, if the string is bowed faster or further away from the bridge the amplitude of the resulting motion will be higher.

A bowed string will maintain a self-sustained oscillation if the portion of it that contacts the bow does not slip while sticking before a complete cycle has elapsed. If it slips too early, assuming it will later stick, there will be higher-frequency modes.

By assuming that the bow is close enough to the bridge to consider the string in between to be straight, any movement under the bow will travel as-is to the bridge and reflect back. The contact with the bow must be strong enough to not slip when the reflected wave hits the bow, and instead “wait” until it comes back from the nut.

This wave can be described through the geometry of the ideal Helmholtz motion (which depends on the bow velocity and bow-bridge distance). The reflection on the bridge is modeled by convolving the driving-point admittance with the incoming wave in order to get the outgoing velocity. This velocity will represent a force on the bow contact, computed using the string’s wave impedance.

Finally, this incoming force must be compared with the threshold that would make the string slip, which depends on the frictional characteristics of the bow-string contact.

By putting all this together, the minimum bow force reads:

$$f_{min} = A \frac{v_b}{2 \beta^2 Y_0^2 (\mu_s -\mu_d)}$$

Where:

$$A = \frac{4}{\pi^2} \left[ \max_t \left\{ \text{Re} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} Y_1(2n\pi f_0) e^{2n \pi f_0 t} \right\} + \text{Re} \sum_{n=1}^{\infty} \frac{Y_1(2n \pi f_0)}{n^2} \right]$$

During the reflection at the bridge, each one of the harmonics gets delayed by a different amount of time, according to the phase of the admittance. Therefore, the wave that comes out may no longer have an ideal Helmholtz shape. The critical value of the incoming force into the bow will be achieved when this wave is maximum; but since the phase change is unknown, the only way to know that maximum (the amplitude of the oscillation) is to actually synthesize a period of the wave and look for its maximum. Notice how:

$$\text{Re} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} Y_1(2n\pi f_0) e^{2n \pi f_0 t}$$

Is actually the resynthesis, as per the Fourier Series definition, of our wave, only multiplied by the admittance in the frequency domain (hence convolved with it in the time domain).