11.3.2 Impedance, impulse response and the reflection function

In order to complete our simulation model of a reasonably realistic “one-note clarinet” we need a suitable reflection function. In order to obtain this, we need to clarify the relationships between four different acoustical quantities that can be used to characterise the linear acoustics of a tube: the impedance, the impulse response, the reflection coefficient and the reflection function.

Suppose we inject a sinusoidal pressure wave into the tube at the mouthpiece end. This wave travels down the tube, and generates a reflected wave travelling back in the reverse direction. As measured at the mouthpiece, we can denote these $P_R e^{i \omega t}$ and $P_L e^{i \omega t}$ respectively (we are envisaging the outgoing wave as travelling to the right, and the reflected wave travelling to the left). The two wave amplitudes are linked by

$$P_L=R(\omega) P_R \tag{1}$$

where $R(\omega)$ is the reflection coefficient. Note that $P_R$, $P_L$ and $R$ may all be complex quantities, because they contain phase information as well as amplitude information.

We can use a result from section 4.1.1 to obtain the corresponding complex amplitudes of the two travelling waves of volume flow rate: these will be

$$V_R=P_R/Z_t,\mathrm{~~~~}V_L=-P_L/Z_t \tag{2}$$

in terms of the characteristic impedance of the tube, $Z_t$. Now we can write down an expression for the input impedance $Z$:

$$Z(\omega) = \dfrac{P_R+P_L}{V_R+V_L}=\dfrac{Z_t(P_R+P_L)}{P_R-P_L}=Z_t \dfrac{1+R}{1-R}. \tag{3}$$

Inverting this relation gives

$$R=\dfrac{Z-Z_t}{Z+Z_t}. \tag{4}$$

Both $Z$ and $R$ are defined in the frequency domain. The inverse Fourier transforms of these two quantities are both of some interest to us. We know from section 2.2.8 that the inverse transform of any input-output frequency response of a linear system is the corresponding impulse response: the response of the output variable when the input is a unit delta function $\delta(t)$. So the inverse transform of the impedance is the impulse response $g(t)$, the pressure response at the mouthpiece to a pulse of volume flow. The inverse transform of the reflection coefficient is the reflection function $r(t)$ that we need: the reflected pressure wave when a pressure spike is injected into the tube.

It is time to see examples. The starting point is the impedance of a clarinet-like tube, which we can calculate as described in section 11.1.1. We choose a tube of length 0.66 m and internal diameter 15 mm, matching the dimensions of a clarinet. Strictly the length should be augmented by suitable end corrections, without which our “instrument” will play a bit sharp, but this is not important for the purpose of representing the essential physics of a clarinet.

The resulting impedance is shown in the red curve of Fig. 1. We want to use an inverse FFT to calculate the impulse response, and in order to avoid numerical artefacts we need to reduce the amplitude by the highest frequency. This has been done in the blue curve. The damping of the tube has been artificially raised, starting at 6 kHz and increasing thereafter. The result is that the curve has settled down to a flat line at the highest frequency plotted. This flat line corresponds to the characteristic impedance $Z_t$, appropriate to a semi-infinite tube of the same diameter. (With the decibel scale of the plot, this illustrates Skudrzyk’s theorem: see section 5.3.2.)

Figure 1. The scaled input impedance $Z/Z_t$ of a clarinet-like cylindrical tube, normalised by the characteristic impedance. The red curve shows the result according to the method described in section 11.1.1, the blue curve shows a modified version with increased damping at higher frequencies.

Although this increase of damping is entirely artificial, something of the kind happens in a real clarinet because of the influence of the bell. The efficiency of sound radiation at the bell rises with frequency, so that the impedance does indeed tend to flatten out. Figure 2 shows an illustration. This is a measured input impedance of a clarinet fingered for its lowest note, with all the tone-holes closed, taken from Joe Wolfe’s web site. (We should mention a small complication: the measurement is on a $B\flat$ clarinet, which is a transposing instrument. The note in question is written as $E_3$, but it sounds at $D_3$, 147 Hz.) This measured impedance shows peaks reducing in amplitude by the highest frequency plotted here, 4 kHz. This real effect is far more drastic than the artificial one imposed in Fig. 1.

Figure 2. The measured input impedance of a clarinet with all tone-holes closed, taken from https://newt.phys.unsw.edu.au/music/clarinet/E3.html and reproduced by permission of Joe Wolfe. Note that the frequency range here only runs up to 4 kHz, much shorter than in Fig. 1.

Taking the inverse FFT of the impedance plotted in blue in Fig. 1 gives the impulse response shown in Fig. 3. The initial delta function can be seen, appearing as a single digital sample of height 1 in this discrete-time representation. The initial pulse travels down the tube and returns, inverted by the reflection at the end. It then reflects at the mouthpiece end and travels off down the tube again, not inverting at this reflection because the mouthpiece is a closed end. After another delay, it returns after another inversion at the far end so that it now appears as a positive pulse. The pattern then repeats, with alternating negative and positive pulses getting gradually lower and more rounded.

Figure 3. Impulse response of an idealised clarinet tube, obtained by inverse FFT of the blue impedance from Fig. 1.

If we process the impedance with equation (4) to obtain the reflection coefficient, then take the inverse FFT of that, we obtain the result shown in Fig. 4. This is the reflection function we need for the simulation model. It is clear from the plot that the processing has had the desired effect: the multiple reflections have all been removed, just leaving the single negative pulse representing a single reflection. This is the function used in the simulations described in section 11.3. One thing to note: the pulse seen in Fig. 3 is symmetrical, but in reality this would not quite be true because of a factor neglected in this analysis. Interactions with the thermal and viscous boundary layers in the tube affect the wave speed as well as the damping, causing wave dispersion and leading to an asymmetric pulse. For details, see for example Fletcher and Rossing [1], section 8.2.

Figure 4. Reflection function corresponding to the impulse response in Fig. 3, obtained by processing the impedance with equation (4) then taking an inverse FFT.

For this reflection function, we find numerically that

$$\int_0^\infty{r(t) dt}=-0.966$$

so that it does not satisfy the unit-area condition given in equation (5) in section 8.5.3. In section 11.3 we also showed simulations done with a corresponding Raman model, and it is now easy to describe how that model was created. Raman’s model has a delta-function reflection function

$$r_{Raman}(t)=-\lambda \delta(t-\tau)$$

with amplitude $\lambda \le 1$ and delay $\tau$. So to create a Raman model similar to the reflection function in Fig. 4, we choose $\tau$ to match the timing of the (negative) peak response, and $\lambda=0.966$ to match the area. In section 11.3 we also showed some results with the lossless version of the Raman model: that model uses the same delay, but with $\lambda=1$.

[1] Neville H Fletcher and Thomas D Rossing; “The physics of musical instruments”, Springer-Verlag (Second edition 1998)