The brass simulations in section 11.5 used the simplest possible model to represent the action of the player’s lips; essentially the same as the model used earlier for reed instruments except with a sign change to give the “lip reed” the character of an opening reed rather than a closing reed. This model gave qualitatively plausible predictions, and even achieved reasonable quantitative agreement with a set of waveforms published by Boutin, Smith and Wolfe [1]. However, in that paper Boutin et al. highlighted an aspect of the observed lip vibration which is not included in the modelling so far. They suggested that this extra effect might have important consequences, and the purpose of this section is to present a preliminary investigation of that suggestion.

The first ingredient is an additional component of the volume flow rate into the instrument. We already have the flow rate associated with air flow through the gap, but there is also a component generated directly by the mechanical motion of the “reed”, or in this case of the player’s lips. By analysing video recordings of a trombone player’s lips (making use of a transparent mouthpiece), Boutin et al. were able to estimate this “sweeping flow” contribution. They found that the magnitude was significant, and also found that the complicated motion of squashy lips resulted in a phase difference between the “opening” and “sweeping” motions: they described the result as “swimming motion” of the lips. Furthermore, when a professional trombonist adjusted their embouchure in order to raise or lower the pitch by “lipping up” or “lipping down”, they seemed to be changing this phase relationship.

Finally, the authors calculated that this phase difference makes a significant contribution to the energy budget of the oscillation, which might affect the blowing pressure threshold as well as the pitch of the played note. These are the effects we now want to investigate, by extending the simulation model to allow a simple representation of “swimming motion”, including the possibility of an adjustable phase shift.

Back in section 11.3.1, when we first described a model for flow through a reed mouthpiece, we already mentioned the “sweeping flow” effect. Equation (14) from that section gave the total volume flow rate into the instrument as

$$v=-w(H_0-y) \sqrt{2 |\Delta p|/\rho_0} \mathrm{~sign} (\Delta p)+A_r \dot{y} \tag{1}$$

where the first term describes the air flow through the gap, and the second term describes the sweeping flow. This second term is proportional to the tip velocity of the reed, $\dot{y}$, with a coefficient $A_r$ which is an effective area of the vibrating reed. The other variables in this equation are the pressure difference $\Delta p = p-p_m$, the density of air $\rho_0$, the width $w$ of the lip opening (idealised as rectangular), and the initial opening $H_0$. The time-varying pressure inside the mouthpiece is $p(t)$, while $p_m$ is the player’s mouth pressure, assumed constant for this simple model.

This formulation of sweeping flow is quite natural for a woodwind reed mouthpiece, but it feels rather artificial when describing lip vibration: if you visualise the trombone being held horizontally as usual, the opening displacement $y$ is predominantly in the vertical direction, but the sweeping flow depends on predominantly horizontal motion of the lips. It is this distinction between vertical lip opening and horizontal motion which allows the possibility of a phase difference: the squashy nature of lips allows the vertical and horizontal components of motion to be out of step. Boutin et al. observed the two things separately by analysing high-speed video recordings of fully-developed periodic motion of a professional trombonist’s lips: we will see their results shortly.

So the first step towards an enhanced model is to represent the horizontal and vertical motions separately, with tip displacements $x$ and $y$ as sketched in Fig. 1, measured from an equilibrium position determined by the player’s embouchure and mean flow rate. The sketch only shows the upper lip: I will assume that the motion of the two lips is symmetrical, so that the combined opening height is $2y+H_0$ where $H_0$ is the opening in the equilibrium state, as in the reed case. An approximate revised version of equation (1) is thus

$$v=-w(H_0+2y) \sqrt{2 |\Delta p|/\rho_0} \mathrm{~sign} (\Delta p)+A_r \dot{x} \tag{2}$$

where $A_r$ now represents a combined effective area of the two vibrating lips.

Now we introduce another factor suggested by Boutin et al. [1]. They point out that this effective area is likely to vary slightly through the vibration cycle of the lips. If that variation in area is somewhat out of phase with the horizontal motion described by $x$, net work can be done by the steady component of the pressure difference during a cycle of vibration, an effect that might change the pressure threshold for producing a note on the trombone. A simple idealisation of this effect is to use the vertical displacement $y$ to modulate the area: we can express

$$A_r \approx w (L_{lip} – \lambda y) \tag{3}$$

where $L_{lip}$ is an effective combined length of the two vibrating lip “reeds”, and $\lambda$ is a dimensionless constant to be investigated by numerical experimentation once we have a complete model. The maximum plausible value of $\lambda$ would be 2, which would correspond to the effective length $L_{lip}$ being reduced by the entire opening displacement $2y$.

The dynamics of the “reed” is determined, in this simple idealisation, by the horizontal displacement $x$: a suitable new version of the governing equation from section 11.3.1 (equation (3) there) can be written

$$M_r \ddot{x} + C_r \dot{x} + K_r x=-\Delta p \dfrac{A_r}{w L_{lip}}=-\Delta p \left(1-\dfrac{\lambda y}{L_{lip}}\right) . \tag{4}$$

The term involving $\lambda$ arises because the variables $M_r$, $C_r$ and $K_r$ were defined per unit area, so we now need to use the equilibrium values of those variables on the left-hand side of the equation, and then adjust the right-hand side to take account of the (small) variation in effective area.

In order to perform simulations using equations (2–4), we need a relation between $x$ and $y$. The simplest approximation would be to assume proportional motion, so that $x=\alpha y$ with some (positive) value of the constant $\alpha$. This assumption does not give us any direct control over the phase, but we can cheat to obtain a simple model for the purposes of this preliminary investigation. The experimental data only tells us about the final periodic part of the note, so we can concentrate on that in our modelling. We can achieve a phase difference by calculating $y$ from a *delayed* value of $x$ (the measurements of Boutin et al. [1] always show the $x$ motion leading the $y$ motion). This would correspond to the tip of each “lip reed” moving along an elliptical path, with an orientation determined by the constant $\alpha$ and the ellipticity determined by the delay. This gives a simple representation of the observed “swimming motion”.

Varying the delay will be (roughly) the same as varying the phase shift, so this simple model can specify the relative phase of the opening motion and the sweeping motion arbitrarily, and by adjusting the values of $L_{lip}$, $\alpha$ and $\lambda$ we can also control the magnitudes of the various contributing factors. The delay strategy does not really qualify as a physical model of lip vibration, but it will allow us to make controlled investigations over a range of amplitudes and phases.

We have three targets for this investigation. The first is to see how close we can get to reproducing the three sets of measured waveforms from Boutin et al. [1]. The second is to investigate the specific proposal that inclusion of “swimming motion” with an appropriate phase shift might allow successful simulations with a lower Q factor for the lip resonance. In section 11.5 we showed results using Q factors in the range suggested by Doc et al. [2], with values 7, 15 and 30. But even Q=7 seems implausibly high for a resonance of soft flesh, suggesting that there may be something important missing from the model — perhaps swimming motion of the lips will be the answer? The third effect concerns the pitch of the played note. The initial model from section 11.5 predicted that the player, by varying lip tension, could raise the pitch of the note, but never lower it below the relevant resonance of the tube — see Fig. 18 of section 11.5. But in reality, a trombonist can “lip down” just as easily as they can “lip up”, so this again points to a deficiency in the initial model.

With these aims in mind, we will concentrate our efforts on a small region of the pressure — lip resonance diagram, illustrated in Fig. 2. This version of the diagram shows results for Q=7, and is colour-shaded to indicate the length of transient following a cold start. The chosen region, outlined with a green box, is concentrated around the “nose” of the feasible zone for the lowest normal note of the trombone, based on the second tube resonance. The three measured cases of Boutin et al. fall near the centre of this region, and if swimming motion can have an influence that allows lower Q factors to be used, we would expect to see the threshold pressure reduce — in other words, the “nose” should move to the left.

The chosen range has blowing pressure up to 2 kPa, and lip resonance in the range 80–130 Hz. This region has been covered by a $10 \times 10$ grid of points, and sets of simulations have been run. The grid has been repeated with 5 different values of the delay: one case with no delay, and four cases with progressively longer delays. The results will be shown in a plotting format illustrated in Fig. 3. The top left plot is the case with no delay. It corresponds to the original data as shown in Fig. 2, and forms our reference comparison for all the other plots. After that, the delay increases along the top row, and continues to increase on the second row. The constant $\alpha$ is set (rather arbitrarily) to the value 2 so that the elliptical path of the lips is angled at $45^\circ$. All simulations use a width $w$ of the (assumed rectangular) lip opening of 12 mm (a value recommended by Doc et al. [2]). The parameter $\lambda$ governing the area modulation effect is set to zero for this initial case so that the effective area is then given by

$$A_r=w L_{lip} \tag{5}$$

in terms of the “effective reed length” $L_{lip}$. The simulations in Fig. 3 use the value $L_{lip}=1 \mathrm{~mm}$ so that the sweeping flow term is very small, and the results for zero delay are almost identical to those seen in Fig. 2.

Figures 4 and 5 show closely-related plots, with different parameter values. Figure 3 had the area modulation effect switched off, and the sweeping flow term rendered negligibly small, so that only the direct effects of the phase delay between $y$ and $x$ were highlighted. In Fig. 4 a larger value is used for $L_{lip}$, 8 mm. We will see shortly that this value gives an approximate match to the magnitude of the sweeping flow as measured by Boutin et al. In Fig. 5 the area modulation effect is turned on in addition, with the largest credible value $\lambda =2$.

Comparing Figs. 3, 4 and 5 proves very revealing. Figure 4 reveals a very strong effect of the delay: as the delay is increased, the region of coloured pixels grows and the “nose” of the region moves to the left. Figure 5 shows that adding in the sweeping flow term has a small but clear effect in reversing these changes: the coloured region is slightly smaller in every case, and the nose retreats to the right. Finally, Fig. 5 shows that switching on the area modulation effect has virtually no effect. The differences between Figs. 4 and 5 are so small that you have to look very carefully to see them. So we see a clear hierarchy among the three effects we have added in to the original model: the phase difference between $x$ and $y$ (and therefore between $x$ and pressure) has a very strong effect, the sweeping flow term has a modest effect, and the area modulation has a negligible effect. This conclusion has been confirmed by computed results for other cases.

In order to see whether this model addresses our three targets, we need to dig into the results in more detail. Figure 4 strongly suggests that we could surely get away with a lower Q-factor for the lip resonance once we have included a significant phase delay, and simulation experiments confirm this. For the remainder of this section we will turn to results with a modestly reduced value $Q=5$, but it seems likely that even lower values could be tolerated within a model of this general type — an investigation that is a task for the future.

Figure 6 shows some results with this reduced value of Q, this time colour-shaded to show the actual phase lag, in degrees, between $y$ and $x$. The plot range has been restricted to $0-90^\circ$, because the measured phase lags from Boutin et al. fall in this range: they found a lag by $36^\circ$ for normal playing, rising to $61^\circ$ for lipping down, and falling to $31^\circ$ for lipping up.

To see whether that reported pattern is consistent with the model predictions, Fig. 7 shows the same set of results colour-shaded to indicate the playing frequency over a range of two semitones above and below the nominal frequency. Careful inspection of this figure, in conjunction with Fig. 6, is very encouraging. The shortest non-zero delay, shown in the middle of the top row, gives a frequency pattern rather similar to the original model: lots of yellow and white indicating the possibility of lipping up to play sharp, but no darker reds indicating the possibility of lipping down.

But as the delay is increased, all the colours shift progressively towards the red end. With the larger values of the phase lag, the model does indeed predict a continuous pattern from lipping up, through the nominal pitch, to lipping down. Comparing with the colours in Fig. 6, we can see that the second step of increasing delay (top right plot) gives phases around the values measured by Boutin et al. for normal playing or lipping up, while the next step (bottom left plot) is about right for their lipped-down measurement.

The simulation model is predicting a pattern of behaviour that matches, qualitatively at least, two of our three targets. It remains to see if we can find among these results credible matches to the measured waveforms for the three cases studied by Boutin et al. To guide this search, we can use versions of the pressure — lip resonance diagram with a different choice of colour shading.

Boutin et al. gave phasor diagrams, showing the relative phases of the fundamental Fourier component of the various measured waveforms. Inspired by this, we can determine the corresponding phases from each simulation that yields a final periodic waveform (i.e. a “note” rather than “silence”). These can be compared with the measured values and combined to give a metric of “overall phase deviation” from a particular choice among the three measured cases. Specifically, I have normalised all the phases by the phase of the opening area, then subtracted the measured value from the simulated value for each of the pressure, volume flow rate and sweeping flow rate waveforms, then added the absolute values of these three deviations together.

The results are plotted in Figs. 8, 9 and 10 corresponding to the three cases from Boutin et al.: Fig. 8 shows “lipping down”, Fig. 9 shows normal playing, and Fig. 10 shows “lipping up”. The best match of this phase deviation metric is found where the colours are closest to white. Selected points for the three cases are marked by circles, and the same three points were also marked in Figs. 6 and 7. These selections were governed by a combination of criteria. First, the blowing pressure in the experiments was close to 1 kPa in each case, so I only considered cases with blowing pressure near that value. Second, each case falls in a region of pale colour in the corresponding plot of the phase deviation metric. But I also had an eye on the correct phase difference in Fig. 6, and the expected frequency behaviour as revealed by Fig. 7.

The comparison of waveforms of the three selected simulations with the corresponding measurements is shown in Figs. 11, 12 and 13. The measured waveforms are shown in dashed lines, the simulations in solid lines of the corresponding colour. The top panel of each plot shows the pressure, the middle panel shows the total volume flow rate (in blue) and the sweeping flow contribution to that rate (in green). The bottom panel shows the open area of the lips.

All these comparisons are very encouraging. Every waveform is recognisably close to the corresponding measurement, both in absolute level and in shape. Furthermore, some trends in waveform shape between the three cases are reproduced. In particular, look at the general shape of the blue waveforms in the middle panels. There is a progressive change from trending upwards, through more or less level, to trending strongly downwards as we move through the three cases. These qualitative changes are well captured by the simulations.

The results from this preliminary investigation of the effect of “swimming motion” of a brass-player’s lips are very encouraging: we have seen a reasonable qualitative match for all three targets. The rather *ad hoc* delay model has allowed us to probe the influence of separate ingredients of the model, which a more complete and physical model might not have allowed. The conclusion was that by far the most important effect to include in an improved model of lip vibration was the potential for a phase difference between the $x$ and $y$ motions to produce the “swimming” effect. By comparison, the effects of the sweeping flow term in equation (2) and of modulation of the effective lip area over which the pressure acts are very minor.

However, there is no doubt that further development of the lip model would be desirable to give a better physical basis. There are some suggestions already in the literature, involving two or more degrees of freedom to represent additional aspects of the lip motion: see for example the discussion in section 6.4 of Campbell, Gilbert and Myers [3]. But we would need to be cautious in exploring further down that line. The problem is that progressively more complicated models introduce more and more parameters. Without a genuinely physical model in the background to inform the choice of values for these parameters, there are too many dimensions to explore comfortably by a purely computational approach. Also, there are not very many detailed measurements against which to benchmark the models: for example, there is nothing as detailed as the experimental Guettler diagrams we were able to use in chapter 9 when comparing different models for bowed-string motion. We have already seen that a credible match to the only published measurements can be obtained with the simple delay model.

[1] Henri Boutin, John Smith and Joe Wolfe, “Trombone lip mechanics with inertive and compliant loads (‘lipping up and down’)”, *Journal of the Acoustical Society of America* *147*, 4133—4144 (2020)

[2] J.-B. Doc, C. Vergez and J. Hannebicq, “Inverse problem to estimate lips parameters values of outward-striking trumpet model for successive playing registers”, *Journal of the Acoustical Society of America* 1**53**, 168—178 (2023).

[3] Murray Campbell, Joël Gilbert and Arnold Myers, “The science of brass instruments”, ASA Press/Springer (2021)