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Woodwinds

We now consider models of woodwind instruments, focusing on the single-reed case. In §9.5.4, tonehole modeling is described in some detail.

Single-Reed Instruments

A simplified model for a single-reed woodwind instrument is shown in Fig. 9.38 [431].

Figure 9.38: A schematic model for woodwind instruments.
\includegraphics[width=\twidth]{eps/fSingleReedAI}

If the bore is cylindrical, as in the clarinet, it can be modeled quite simply using a bidirectional delay line. If the bore is conical, such as in a saxophone, it can still be modeled as a bidirectional delay line, but interfacing to it is slightly more complex, especially at the mouthpiece [37,7,160,436,506,507,502,526,406,528], Because the main control variable for the instrument is air pressure in the mouth at the reed, it is convenient to choose pressure wave variables.

To first order, the bell passes high frequencies and reflects low frequencies, where ``high'' and ``low'' frequencies are divided by the wavelength which equals the bell's diameter. Thus, the bell can be regarded as a simple ``cross-over'' network, as is used to split signal energy between a woofer and tweeter in a loudspeaker cabinet. For a clarinet bore, the nominal ``cross-over frequency'' is around $ 1500$ Hz [38]. The flare of the bell lowers the cross-over frequency by decreasing the bore characteristic impedance toward the end in an approximately non-reflecting manner [51]. Bell flare can therefore be considered analogous to a transmission-line transformer.

Since the length of the clarinet bore is only a quarter wavelength at the fundamental frequency, (in the lowest, or ``chalumeau'' register), and since the bell diameter is much smaller than the bore length, most of the sound energy traveling into the bell reflects back into the bore. The low-frequency energy that makes it out of the bore radiates in a fairly omnidirectional pattern. Very high-frequency traveling waves do not ``see'' the enclosing bell and pass right through it, radiating in a more directional beam. The directionality of the beam is proportional to how many wavelengths fit along the bell diameter; in fact, many wavelengths away from the bell, the radiation pattern is proportional to the two-dimensional spatial Fourier transform of the exit aperture (a disk at the end of the bell) [308].

The theory of the single reed is described, e.g., in [102,249,308]. In the digital waveguide clarinet model described below [431], the reed is modeled as a signal- and embouchure-dependent nonlinear reflection coefficient terminating the bore. Such a model is possible because the reed mass is neglected. The player's embouchure controls damping of the reed, reed aperture width, and other parameters, and these can be implemented as parameters on the contents of the lookup table or nonlinear function.

Digital Waveguide Single-Reed Implementation

A diagram of the basic clarinet model is shown in Fig.9.39. The delay-lines carry left-going and right-going pressure samples $ p_b^{+}$ and $ p_b^{-}$ (respectively) which sample the traveling pressure-wave components within the bore.

Figure 9.39: Waveguide model of a single-reed, cylindrical-bore woodwind, such as a clarinet.
\includegraphics[width=\twidth]{eps/fSingleReedWGM}

The reflection filter at the right implements the bell or tone-hole losses as well as the round-trip attenuation losses from traveling back and forth in the bore. The bell output filter is highpass, and power complementary with respect to the bell reflection filter [500]. Power complementarity follows from the assumption that the bell itself does not vibrate or otherwise absorb sound. The bell is also amplitude complementary. As a result, given a reflection filter $ H_r(z)$ designed to match measured mode decay-rates in the bore, the transmission filter can be written down simply as $ H_t(z) = 1 - H_r(z)$ for velocity waves, or $ H_t(z) = 1 +
H_r(z)$ for pressure waves. It is easy to show that such amplitude-complementary filters are also power complementary by summing the transmitted and reflected power waves:

\begin{eqnarray*}
P_t U_t + P_r U_r &=& (1+H_r)P \cdot (1-H_r)U + H_r P \cdot (-H_r)(-U)\\
&=& [1-H_r^2 + H_r^2]PU \;=\; PU,
\end{eqnarray*}

where $ P$ denotes the z transform transform of the incident pressure wave, and $ U$ denotes the z transform of the incident volume-velocity. (All z transform have omitted arguments $ (\exp(j\omega T)$, where $ T$ denotes the sampling interval in seconds.)

At the far left is the reed mouthpiece controlled by mouth pressure $ p_m$. Another control is embouchure, changed in general by modifying the reflection-coefficient function $ \rho(h_{\Delta}^{+})$, where $ h_{\Delta}^{+}
\isdeftext p_m/2 - p_b^{+}$. A simple choice of embouchure control is an offset in the reed-table address. Since the main feature of the reed table is the pressure-drop where the reed begins to open, a simple embouchure offset can implement the effect of biting harder or softer on the reed, or changing the reed stiffness.


A View of Single-Reed Oscillation

To start the oscillation, the player applies a pressure at the mouthpiece which ``biases'' the reed in a ``negative-resistance'' region. (The pressure drop across the reed tends to close the air gap at the tip of the reed so that an increase in pressure will result in a net decrease in volume velocity--this is negative resistance.) The high-pressure front travels down the bore at the speed of sound until it encounters an open air hole or the bell. To a first approximation, the high-pressure wave reflects with a sign inversion and travels back up the bore. (In reality a lowpass filtering accompanies the reflection, and the complementary highpass filter shapes the spectrum that emanates away from the bore.)

As the negated pressure wave travels back up the bore, it cancels the elevated pressure that was established by the passage of the first wave. When the negated pressure front gets back to the mouthpiece, it is reflected again, this time with no sign inversion (because the mouthpiece looks like a closed end to a first approximation). Therefore, as the wave travels back down to the bore, a negative pressure zone is left behind. Reflecting from the open end again with a sign inversion brings a return-to-zero wave traveling back to the mouthpiece. Finally the positive traveling wave reaches the mouthpiece and starts the second ``period'' of oscillation, after four trips along the bore.

So far, we have produced oscillation without making any use of the negative-resistance of the reed aperture. This is merely the start-up transient. Since in reality there are places of pressure loss in the bore, some mechanism is needed to feed energy back into the bore and prevent the oscillation just described from decaying exponentially to zero. This is the function of the reed: When a traveling pressure-drop reflects from the mouthpiece, making pressure at the mouthpiece switch from high to low, the reed changes from open to closed (to first order). The closing of the reed increases the reflection coefficient ``seen'' by the impinging traveling wave, and so as the pressure falls, it is amplified by an increasing gain (whose maximum is unity when the reed shuts completely). This process sharpens the falling edge of the pressure drop. But this is not all. The closing of the reed also cuts back on the steady incoming flow from the mouth. This causes the pressure to drop even more, potentially providing effective amplification by more than unity.

An analogous story can be followed through for a rising pressure appearing at the mouthpiece. However, in the rising pressure case, the reflection coefficient falls as the pressure rises, resulting in a progressive attenuation of the reflected wave; however, the increased pressure let in from the mouth amplifies the reflecting wave. It turns out that the reflection of a positive wave is boosted when the incoming wave is below a certain level and it is attenuated above that level. When the oscillation reaches a very high amplitude, it is limited on the negative side by the shutting of the reed, which sets a maximum reflective amplification for the negative excursions, and it is limited on the positive side by the attenuation described above. Unlike classical negative-resistance oscillators, in which the negative-resistance device is terminated by a simple resistance instead of a lossy transmission line, a dynamic equilibrium is established between the amplification of the negative excursion and the dissipation of the positive excursion.

In the first-order case, where the reflection-coefficient varies linearly with pressure drop, it is easy to obtain an exact quantitative description of the entire process. In this case it can be shown, for example, that amplification occurs only on the positive half of the cycle, and the amplitude of oscillation is typically close to half the incoming mouth pressure (when losses in the bore are small). The threshold blowing pressure (which is relatively high in this simplified case) can also be computed in closed form.


Single-Reed Theory

Figure 9.40: Schematic diagram of mouth cavity, reed aperture, and bore.
\includegraphics[scale=0.9]{eps/fReedSchematic}

A simplified diagram of the clarinet mouthpiece is shown in Fig. 9.40. The pressure in the mouth is assumed to be a constant value $ p_m$, and the bore pressure $ p_b$ is defined located at the mouthpiece. Any pressure drop $ p_{\Delta}= p_m-p_b$ across the mouthpiece causes a flow $ u_m$ into the mouthpiece through the reed-aperture impedance $ R_m(p_{\Delta})$ which changes as a function of $ p_{\Delta}$ since the reed position is affected by $ p_{\Delta}$. To a first approximation, the clarinet reed can be regarded as a spring flap regulated Bernoulli flow (§B.7.5), [249]). This model has been verified well experimentally until the reed is about to close, at which point viscosity effects begin to appear [102]. It has also been verified that the mass of the reed can be neglected to first order,10.18 so that $ R_m(p_{\Delta})$ is a positive real number for all values of $ p_{\Delta}$. Possibly the most important neglected phenomenon in this model is sound generation due to turbulence of the flow, especially near reed closure. Practical synthesis models have always included a noise component of some sort which is modulated by the reed [431], despite a lack of firm basis in acoustic measurements to date.

The fundamental equation governing the action of the reed is continuity of volume velocity, i.e.,

$\displaystyle u_b+u_m= 0$ (10.35)

where

$\displaystyle u_m(p_{\Delta}) \isdef \frac{p_{\Delta}}{R_m(p_{\Delta})}$ (10.36)

and

$\displaystyle u_b\isdef u_b^{+}+ u_b^{-}= \frac{p_b^{+}-p_b^{-}}{R_b}$ (10.37)

is the volume velocity corresponding to the incoming pressure wave $ p_b^{+}$ and outgoing pressure wave $ p_b^{-}$. (The physical pressure in the bore at the mouthpiece is of course $ p_b=p_b^{+}+p_b^{-}$.) The wave impedance of the bore air-column is denoted $ R_b$ (computable as the air density times sound speed $ c$ divided by cross-sectional area).

In operation, the mouth pressure $ p_m$ and incoming traveling bore pressure $ p_b^{+}$ are given, and the reed computation must produce an outgoing bore pressure $ p_b^{-}$ which satisfies (9.35), i.e., such that

0 $\displaystyle =$ $\displaystyle u_m+u_b= \frac{p_{\Delta}}{R_m(p_{\Delta})} + \frac{p_b^{+}-p_b^{-}}{R_b},$ (10.38)
$\displaystyle p_{\Delta}$ $\displaystyle \isdef$ $\displaystyle p_m-p_b= p_m- (p_b^{+}+p_b^{-})$  

Solving for $ p_b^{-}$ is not immediate because of the dependence of $ R_m$ on $ p_{\Delta}$ which, in turn, depends on $ p_b^{-}$. A graphical solution technique was proposed [151,244,308] which, in effect, consists of finding the intersection of the two terms of the equation as they are plotted individually on the same graph, varying $ p_b^{-}$. This is analogous to finding the operating point of a transistor by intersecting its operating curve with the ``load line'' determined by the load resistance.

It is helpful to normalize (9.38) as follows: Define $ G(p_{\Delta}) = R_b
u_m(p_{\Delta}) = R_bp_{\Delta}/R_m(p_{\Delta})$, and note that $ p_b^{+}-p_b^{-}=2p_b^{+}-p_m-(p_b^{+}+p_b^{-}-p_m)\isdeftext p_{\Delta}-p_{\Delta}^{+}$, where $ p_{\Delta}^{+}\isdeftext
p_m-2p_b^{+}$. Then (9.38) can be multiplied through by $ R_b$ and written as $ 0=G(p_{\Delta})+p_{\Delta}-p_{\Delta}^{+}$, or

$\displaystyle G(p_{\Delta}) = p_{\Delta}^{+}-p_{\Delta},\qquad p_{\Delta}^{+}\isdef p_m- 2p_b^{+}$ (10.39)

The solution is obtained by plotting $ G(x)$ and $ p_{\Delta}^{+}-x$ on the same graph, finding the point of intersection at $ (x,y)$ coordinates $ (p_{\Delta},G(p_{\Delta}))$, and computing finally the outgoing pressure wave sample as

$\displaystyle p_b^{-}= p_m- p_b^{+}- p_{\Delta}(p_{\Delta}^{+})$ (10.40)

An example of the qualitative appearance of $ G(x)$ overlaying $ p_{\Delta}^{+}-x$ is shown in Fig. 9.41.

Figure 9.41: Normalized reed impedance $ G(p_{\Delta }) = R_bu_m(p_{\Delta })$ overlaid with the ``bore load line'' $ p_{\Delta }^{+}-p_{\Delta }= R_bu_b$.
\includegraphics[width=\twidth]{eps/fReedRelations}

Scattering-Theoretic Formulation

Equation (9.38) can be solved for $ p_b^{-}$ to obtain

$\displaystyle p_b^{-}$ $\displaystyle =$ $\displaystyle \frac{1-r}{1+r}p_b^{+}+ \frac{r}{1+r}p_m$ (10.41)
  $\displaystyle =$ $\displaystyle \rho p_b^{+}+ \frac{1-\rho}{2} p_m$ (10.42)
  $\displaystyle =$ $\displaystyle \frac{p_m}{2} - \rho \frac{p_{\Delta}^{+}}{2}$ (10.43)

where
$\displaystyle \rho(p_{\Delta}) \isdef \frac{1-r(p_{\Delta})}{1+r(p_{\Delta})}, \qquad r(p_{\Delta})\isdef \frac{R_b}{R_m(p_{\Delta})}$     (10.44)

We interpret $ \rho(p_{\Delta})$ as a signal-dependent reflection coefficient.

Since the mouthpiece of a clarinet is nearly closed, $ R_m\gg R_b$ which implies $ r\approx 0$ and $ \rho\approx1$. In the limit as $ R_m$ goes to infinity relative to $ R_b$, (9.42) reduces to the simple form of a rigidly capped acoustic tube, i.e., $ p_b^{-}= p_b^{+}$. If it were possible to open the reed wide enough to achieve matched impedance, $ R_m=R_b$, then we would have $ r=1$ and $ \rho=0$, in which case $ p_b^{-}= p_m/2$, with no reflection of $ p_b^{+}$, as expected. If the mouthpiece is removed altogether to give $ R_m=0$ (regarding it now as a tube section of infinite radius), then $ r=\infty$, $ \rho=-1$, and $ p_b^{-}=
-p_b^{+}+ p_m$.


Computational Methods

Since finding the intersection of $ G(x)$ and $ p_{\Delta}^{+}-x$ requires an expensive iterative algorithm with variable convergence times, it is not well suited for real-time operation. In this section, fast algorithms based on precomputed nonlinearities are described.

Let $ h$ denote half-pressure $ p/2$, i.e., $ h_m\isdeftext p_m/2$ and $ h_{\Delta}^{+}\isdeftext p_{\Delta}^{+}/2$. Then (9.43) becomes

$\displaystyle p_b^{-}= h_m- \rho(p_{\Delta})\cdot h_{\Delta}^{+}$ (10.45)

Subtracting this equation from $ p_b^{+}$ gives
$\displaystyle p_b^{+}+ \rho h_{\Delta}^{+}- h_m$ $\displaystyle =$ $\displaystyle p_b^{+}- p_b^{-}= p_{\Delta}- p_{\Delta}^{+}$  
$\displaystyle \,\,\Rightarrow\,\,
\rho h_{\Delta}^{+}$ $\displaystyle =$ $\displaystyle \underbrace{h_m- p_b^{+}}_{h_{\Delta}^{+}} + p_{\Delta}- 2h_{\Delta}^{+}= p_{\Delta}-h_{\Delta}^{+}$  
$\displaystyle \,\,\Rightarrow\,\,
\rho$ $\displaystyle =$ $\displaystyle \frac{p_{\Delta}}{h_{\Delta}^{+}}-1$ (10.46)

The last expression above can be used to precompute $ \rho$ as a function of $ h_{\Delta}^{+}\isdef h_m- p_b^{+}= p_m/2 -p_b^{+}$. Denoting this newly defined function as
$\displaystyle \hat\rho (h_{\Delta}^{+}) = \rho(p_{\Delta}(h_{\Delta}^{+}))$     (10.47)

(9.45) becomes

$\displaystyle p_b^{-}= h_m- \hat\rho (h_{\Delta}^{+})\cdot h_{\Delta}^{+}$ (10.48)

This is the form chosen for implementation in Fig. 9.39 [431]. The control variable is mouth half-pressure $ h_m$, and $ h_{\Delta}^{+}=h_m-p_b^{+}$ is computed from the incoming bore pressure using only a single subtraction. The table is indexed by $ h_{\Delta}^{+}$, and the result of the lookup is then multiplied by $ h_{\Delta}^{+}$. Finally, the result of the multiplication is subtracted from $ h_m$. The cost of the reed simulation is only two subtractions, one multiplication, and one table lookup per sample.

Because the table contains a coefficient rather than a signal value, it can be more heavily quantized both in address space and word length than a direct lookup of a signal value such as $ p_{\Delta}(p_{\Delta}^{+})$ or the like. A direct signal lookup, though requiring much higher resolution, would eliminate the multiplication associated with the scattering coefficient. For example, if $ p_b^{+}$ and $ p_b^{-}$ are 16-bit signal samples, the table would contain on the order of 64K 16-bit $ p_{\Delta}^\ast $ samples. Clearly, some compression of this table would be desirable. Since $ p_{\Delta}^\ast (p_{\Delta}^{+}/2)$ is smoothly varying, significant compression is in fact possible. However, because the table is directly in the signal path, comparatively little compression can be done while maintaining full audio quality (such as 16-bit accuracy).

Figure 9.42: Simple, qualitatively chosen reed table for the digital waveguide clarinet.
\includegraphics[width=4in]{eps/fReedTable}

In the field of computer music, it is customary to use simple piecewise linear functions for functions other than signals at the audio sampling rate, e.g., for amplitude envelopes, FM-index functions, and so on [382,380]. Along these lines, good initial results were obtained [431] using the simplified qualitatively chosen reed table

$\displaystyle \hat\rho (h_{\Delta}^{+}) = \left\{ \begin{array}{ll} 1-m(h_{\Del...
...ent_mark>2515 1, & h_{\Delta}^c\leq h_{\Delta}^{+}\leq 1 \\ \end{array} \right.$ (10.49)

depicted in Fig. 9.42 for $ m=1/(h_{\Delta}^c+1)$. The corner point $ h_{\Delta}^c$ is the smallest pressure difference giving reed closure.10.19 Embouchure and reed stiffness correspond to the choice of offset $ h_{\Delta}^c$ and slope $ m$. Brighter tones are obtained by increasing the curvature of the function as the reed begins to open; for example, one can use $ \hat\rho ^k(h_{\Delta}^{+})$ for increasing $ k\geq1$.

Another variation is to replace the table-lookup contents by a piecewise polynomial approximation. While less general, good results have been obtained in practice [89,91,92]. For example, one of the SynthBuilder [353] clarinet patches employs this technique using a cubic polynomial.

An intermediate approach between table lookups and polynomial approximations is to use interpolated table lookups. Typically, linear interpolation is used, but higher order polynomial interpolation can also be considered (see Chapter 4).


Clarinet Synthesis Implementation Details

To finish off the clarinet example, the remaining details of the SynthBuilder clarinet patch Clarinet2.sb are described.

The input mouth pressure is summed with a small amount of white noise, corresponding to turbulence. For example, 0.1% is generally used as a minimum, and larger amounts are appropriate during the attack of a note. Ideally, the turbulence level should be computed automatically as a function of pressure drop $ p_{\Delta}$ and reed opening geometry [141,530]. It should also be lowpass filtered as predicted by theory.

Referring to Fig. 9.39, the reflection filter is a simple one-pole with transfer function

$\displaystyle H(z) = \frac{1+a_1(t)}{1+a_1(t) z^{-1}}$ (10.50)

where $ a_1(t) = v(t) - 0.642$, $ v(t) = A_v\sin(2\pi f_v t)$, $ A_v$ is vibrato amplitude (e.g., $ 0.03$), and $ f_v$ is vibrato frequency (e.g., $ 5$ Hz). Further loop filtering occurs as a result of using simple linear interpolation of the delay line. (There is only one delay line in the actual implementation since the lower delay line of Fig. 9.39 can be commuted with the reflection filter and combined with the upper delay line, ignoring the path to the output filter since a pure delay of less than a period in the final output sound is inconsequential.) There is no transmission filter or tone-hole modeling.

Legato note transitions are managed using two delay line taps and cross-fading from one to the other during a transition [208,441,405]. In general, legato problems arise when the bore length is changed suddenly while sounding, corresponding to a new fingering. The reason is that really the model itself should be changed during a fingering change from that of a statically terminated bore to that of a bore with a new scattering junction appearing where each ``finger'' is lifting, and with disappearing scattering junctions where tone holes are being covered. In addition, if a hole is covered abruptly (especially when there are large mechanical caps, as in the saxophone), there will also be new signal energy injected in both directions on the bore in superposition with the signal scattering. As a result of this ideal picture, is difficult to get high quality legato performance using only a single delay line.

A reduced-cost, approximate solution for obtaining good sounding note transitions in a single delay-line model was proposed in [208]. In this technique, the bore delay line is ``branched'' during the transition, i.e., a second feedback loop is formed at the new loop delay, thus forming two delay lines sharing the same memory, one corresponding to the old pitch and the other corresponding to the new pitch. A cross-fade from the old-pitch delay to the new-pitch delay sounds good if the cross-fade time and duration are carefully chosen. Another way to look at this algorithm is in terms of ``read pointers'' and ``write pointers.'' A normal delay line consists of a single write pointer followed by a single read pointer, delayed by one period. During a legato transition, we simply cross-fade from a read-pointer at the old-pitch delay to a read-pointer at the new-pitch delay. In this type of implementation, the write-pointer always traverses the full delay memory corresponding to the minimum supported pitch in order that read-pointers may be instantiated at any pitch-period delay at any time. Conceptually, this simplified model of note transitions can be derived from the more rigorous model by replacing the tone-hole scattering junction by a single reflection coefficient.

STK software implementing a model as in Fig.9.39 can be found in the file Clarinet.cpp.


Tonehole Modeling

Toneholes in woodwind instruments are essentially cylindrical holes in the bore. One modeling approach would be to treat the tonehole as a small waveguide which connects to the main bore via one port on a three-port junction. However, since the tonehole length is small compared with the distance sound travels in one sampling instant ( $ cT
= 1125/44100 = 0.3$in, e.g.), it is more straightforward to treat the tonehole as a lumped load along the bore, and most modeling efforts have taken this approach.

The musical acoustics literature contains experimentally verified models of tone-hole acoustics, such as by Keefe [238]. Keefe's tonehole model is formulated as a ``transmission matrix'' description, which we may convert to a traveling-wave formulation by a simple linear transformation (described in §9.5.4 below) [465]. For typical fingerings, the first few open tone holes jointly provide a bore termination [38]. Either the individual tone holes can be modeled as (interpolated) scattering junctions, or the whole ensemble of terminating tone holes can be modeled in aggregate using a single reflection and transmission filter, like the bell model. Since the tone hole diameters are small compared with most audio frequency wavelengths, the reflection and transmission coefficients can be implemented to a reasonable approximation as constants, as opposed to cross-over filters as in the bell. Taking into account the inertance of the air mass in the tone hole, the tone hole can be modeled as a two-port loaded junction having load impedance equal to the air-mass inertance [143,509]. At a higher level of accuracy, adapting transmission-matrix parameters from the existing musical acoustics literature leads to first-order reflection and transmission filters [238,406,403,404,465]. The individual tone-hole models can be simple lossy two-port junctions, modeling only the internal bore loss characteristics, or three-port junctions, modeling also the transmission characteristics to the outside air. Another approach to modeling toneholes is the ``wave digital'' model [527] (see §F.1 for a tutorial introduction to this approach). The subject of tone-hole modeling is elaborated further in [406,502]. For simplest practical implementation, the bell model can be used unchanged for all tunings, as if the bore were being cut to a new length for each note and the same bell were attached. However, for best results in dynamic performance, the tonehole model should additionally include an explicit valve model for physically accurate behavior when slowly opening or closing the tonehole [405].

The Clarinet Tonehole as a Two-Port Junction

Figure 9.43: Lumped-parameter description of the clarinet tonehole.
\includegraphics[scale=0.9]{eps/fFingerHoleKeefe}

The clarinet tonehole model developed by Keefe [240] is parametrized in terms of series and shunt resistance and reactance, as shown in Fig. 9.43. The transmission matrix description of this two-port is given by the product of the transmission matrices for the series impedance $ R_a/2$, shunt impedance $ R_s$, and series impedance $ R_a/2$, respectively:

$\displaystyle \left[\begin{array}{c} P_1 \\ [2pt] U_1 \end{array}\right]$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 1 & R_a/2 \\ [2pt] 0 & 1 \end{array}\righ...
...1 \end{array}\right]
\left[\begin{array}{c} P_2 \\ [2pt] U_2 \end{array}\right]$  
  $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 1+\frac{R_a}{2R_s} & R_a[1+\frac{R_a}{4R_...
...} \end{array}\right]
\left[\begin{array}{c} P_2 \\ [2pt] U_2 \end{array}\right]$  

where all quantities are written in the frequency domain, and the impedance parameters are given by
(open-hole shunt impedance)$\displaystyle \quad R_s^o$ $\displaystyle =$ $\displaystyle R_b (j k t_e + \xi_e)$  
(closed-hole shunt impedance)$\displaystyle \quad R_s^c$ $\displaystyle =$ $\displaystyle -j R_b \cot(k t_h)$ (10.51)
(open-hole series impedance)$\displaystyle \quad R_a^o$ $\displaystyle =$ $\displaystyle -j R_b k t_a^o$  
(closed-hole series impedance)$\displaystyle \quad R_a^c$ $\displaystyle =$ $\displaystyle -j R_b k t_a^c$  

where $ R_b = \rho c / (\pi b^2)$ is the wave impedance of the tonehole entrance, i.e., that of an acoustic tube of cross-sectional area $ \pi b^2$ ($ \rho$ is air density and $ c$ is sound speed as usual), $ b$ is the tonehole radius, $ k = \omega/c = 2\pi/\lambda$ is the wavenumber (radian spatial frequency), $ t_e$ is the open-tonehole effective length (which is slightly greater than its physical length due to the formation of a small air-piston inside the open tonehole), $ \xi_e$ is the ``specific resistance'' of the open tonehole due to air viscosity in and radiation from the hole, $ t_h$ is the closed-tonehole height, defined such that its product times the cross-sectional area of the tonehole exactly equals the geometric volume $ V_h$ of the closed tonehole. Finally, $ t_a^o$ and $ t_a^c$ are the equivalent series lengths of the open and closed tonehole, respectively, and are given by
$\displaystyle t_a^o$ $\displaystyle =$ $\displaystyle \frac{0.47b (b/a)^4}{\tanh(1.84 t_h/b) + 0.62(b/a)^2 + 0.64 (b/a)}$  
$\displaystyle t_a^c$ $\displaystyle =$ $\displaystyle \frac{0.47b (b/a)^4}{\coth(1.84 t_h/b) + 0.62(b/a)^2 + 0.64 (b/a)}$  

where $ a$ is the radius of the main bore. The closed-tonehole height $ V_h/(\pi b^2)$ can be estimated as [240]

$\displaystyle t_h = t_w + \frac{1}{8}\frac{b^2}{a}\left[1+0.172\left(\frac{b}{a}\right)^2\right]
$

where $ t_w$ is the physical tonehole chimney height at its center.

Note that the specific resistance of the open tonehole, $ \xi_e$, is the only real impedance and therefore the only source of wave energy loss at the tonehole. It is given by [240]

$\displaystyle \xi_e = 0.25 (kb)^2 + \alpha t_h + (1/4) k d_v\,$ln$\displaystyle (2b/r_c),
$

where $ r_c$ is the radius of curvature of the tonehole, $ d_v$ is the viscous boundary layer thickness which expressible in terms of the shear viscosity $ \eta$ of air as

$\displaystyle d_v = \sqrt{\frac{2\eta}{\rho\omega}}
$

and $ \alpha$ is the real part of the propagation wavenumber (or minus the imaginary part of complex spatial frequency $ k$). In [239], for the large-tube limit (i.e., when the tube radius is large compared with the viscous boundary layer), $ \alpha$ is given by

$\displaystyle \alpha = \frac{1}{2bc}\left[\,\sqrt{\frac{2\eta\omega}{\rho}}
+ (\gamma-1)\sqrt{\frac{2\kappa\omega}{\rho C_p}}\,\right]
$

where $ \gamma=1.4$ is the adiabatic gas constant for air [318], $ \kappa$ is the thermal conductivity of air, and $ C_p$ is the specific heat of air at constant pressure. In [239], the following values are given for air at $ 300^\circ$ Kelvin ( $ 26.85^\circ$ C), and valid within $ \pm 10$ degrees of that temperature:

\begin{eqnarray*}
\rho &=& 1.1769 \times 10^{-3}(1-0.00335\Delta T)\,\mbox{g}/\m...
... \frac{0.750}{r_v^3} \right) \quad
\mbox{(valid for $r_v > 2$)}
\end{eqnarray*}

where

$\displaystyle r_v = b\sqrt{\frac{\rho\omega}{\eta}} = \sqrt{2}\frac{b}{d_v}
$

can be interpreted as $ \sqrt{2}$ times the ratio of the tonehole radius $ b$ to the viscous boundary layer thickness $ d_v$ [239]. The constant $ \nu^2$ is referred to as the Prandtl number, and $ \eta$ is the shear viscosity coefficient [239]. In [71], it is noted that $ r_v$ is greater than $ 8$ under practical conditions in musical acoustics, and so it is therefore sufficient to keep only the first and second-order terms in the expression above for $ \alpha$.

The open-hole effective length $ t_e$, assuming no pad above the hole, is given in [240] as

$\displaystyle t_e = \frac{(1/k)\tan(kt) + b [1.40 - 0.58(b/a)^2]}{1 - 0.61 kb \tan(kt)}
$

See [240] for the case in which a pad lies above the open hole. In [405], a unified tonehole model is given which supports continuous opening and closing of the tonehole.

For implementation in a digital waveguide model, the lumped parameters above must be converted to scattering parameters. Such formulations of toneholes have appeared in the literature: Vesa Välimäki [509,502] developed tonehole models based on a ``three-port'' digital waveguide junction loaded by an inertance, as described in Fletcher and Rossing [143], and also extended his results to the case of interpolated digital waveguides. It should be noted in this context, however, that in the terminology of Appendix C, Välimäki's tonehole representation is a loaded 2-port junction rather than a three-port junction. (A load can be considered formally equivalent to a ``waveguide'' having wave impedance given by the load impedance.) Scavone and Smith [402] developed digital waveguide tonehole models based on the more rigorous ``symmetric T'' acoustic model of Keefe [240], using general purpose digital filter design techniques to obtain rational approximations to the ideal tonehole frequency response. A detailed treatment appears in Scavone's CCRMA Ph.D. thesis [406]. This section, adapted from [465], considers an exact translation of the Keefe tonehole model, obtaining two one-filter implementations: the ``shared reflectance'' and ``shared transmittance'' forms. These forms are shown to be stable without introducing an approximation which neglects the series inertance terms in the tonehole model.

By substituting $ k=\omega/c$ in (9.53) to convert spatial frequency to temporal frequency, and by substituting

$\displaystyle P_i$ $\displaystyle =$ $\displaystyle P_i^{+}+ P_i^{-}$ (10.52)
$\displaystyle U_i$ $\displaystyle =$ $\displaystyle \frac{P_i^{+}- P_i^{-}}{R_0}$ (10.53)

for $ i=1,2$, into (9.51) to convert physical variables to wave variables, ( $ R_0=\rho c /(\pi a^2)$ is the bore wave impedance), we may solve for the outgoing waves $ P_1^{-}, P_2^{-}$ in terms of the incoming waves $ P_1^{+}, P_2^{+}$. Mathematica code for obtaining the general conversion formula from lumped parameters to scattering parameters is as follows:
        Clear["t*", "p*", "u*", "r*"]
        transmissionMatrix = {{t11, t12}, {t21, t22}};
        leftPort = {{p2p+p2m}, {(p2p-p2m)/r2}};
        rightPort = {{p1p+p1m}, {(p1p-p1m)/r1}};
        Format[t11, TeXForm] := "{T_{11}}"
        Format[p1p, TeXForm] := "{P_1^+}"
        ... (etc. for all variables) ...
        TeXForm[Simplify[Solve[leftPort ==
               transmissionMatrix . rightPort, {p1m, p2p}]]]
The above code produces the following formulas:
$\displaystyle P_1^-$ $\displaystyle =$ $\displaystyle \frac{2 {P_2^-} {R_1} - {P_1^+} {R_1} {T_{11}} -
{P_1^+} {T_{12}}...
...} {T_{22}}}{{R_1} {T_{11}} - {T_{12}} -
{R_1} {R_2} {T_{21}} + {R_2} {T_{22}}},$  
$\displaystyle P_2^+$ $\displaystyle =$ $\displaystyle \frac{{P_2^-} {R_1} {T_{11}} - {P_2^-} {T_{12}} +
{P_2^-} {R_1} {...
...}} {T_{22}}}{{R_1} {T_{11}} - {T_{12}} -
{R_1} {R_2} {T_{21}} + {R_2} {T_{22}}}$  
    $\displaystyle % Get eqn number on next line below
$ (10.54)

Substituting relevant values for Keefe's tonehole model, we obtain, in matrix notation,
$\displaystyle \left[\begin{array}{c} P_1^{-} \\ [2pt] P_2^{+} \end{array}\right]$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} S & T \\ [2pt] T & S \end{array}\right]
\left[\begin{array}{c} P_1^{+} \\ [2pt] P_2^{-} \end{array}\right]$  
  $\displaystyle =$ $\displaystyle \frac{1}{(2R_0+R_a)(2R_0+R_a+4R_s)} \;\times$  
    $\displaystyle \quad
\left[\begin{array}{cc} 4R_aR_s + R_a^2 - 4R_0^2 & 8R_0R_s ...
...rray}\right]
\left[\begin{array}{c} P_1^{+} \\ [2pt] P_2^{-} \end{array}\right]$ (10.55)

We thus obtain the scattering formulation depicted in Fig. 9.44, where

$\displaystyle S(\omega) = \frac{4R_aR_s + R_a^2 - 4R_0^2}{(2R_0+ R_a)(2R_0+ R_a + 4R_s)} \approx - \frac{R_0}{R_0+ 2R_s}$ (10.56)

is the reflectance of the tonehole (the same from either direction), and

$\displaystyle T(\omega) = \frac{8R_0R_s}{(2R_0+ R_a)(2R_0+ R_a + 4R_s)} \approx \frac{2R_s}{R_0+ 2R_s}$ (10.57)

is the transmittance of the tonehole (also the same from either direction). The notation ``$ S$'' for reflectance is chosen because every reflectance is a Schur function (stable and not exceeding unit magnitude on the unit circle in the $ z$ plane) [428, p. 221].

Figure 9.44: Frequency-domain, traveling-wave description of the clarinet tonehole.
\includegraphics[scale=0.9]{eps/fFingerHoleScat}

The approximate forms in (9.57) and (9.58) are obtained by neglecting the negative series inertance $ R_a$ which serves to adjust the effective length of the bore, and which therefore can be implemented elsewhere in the interpolated delay-line calculation as discussed further below. The open and closed tonehole cases are obtained by substituting $ \{R_a = R_a^o,
R_s =
R_s^o\}$ and $ \{R_a = R_a^c, R_s =
R_s^c\}$, respectively, from (9.53).

In a manner analogous to converting the four-multiply Kelly-Lochbaum (KL) scattering junction [245] into a one-multiply form (cf. (C.60) and (C.62) on page [*]), we may pursue a ``one-filter'' form of the waveguide tonehole model. However, the series inertance gives some initial trouble, since

$\displaystyle [1+S(\omega)] - T(\omega) = \frac{2R_a}{2R_0+ R_a} \isdef L(\omega)
$

instead of zero as in the KL junction. In the scattering formulas (C.100) and (C.101) on page [*] for the general loaded waveguide junction, the reflectance seen on any branch is always the transmittance from that branch to any other branch minus $ 1$. I.e., if $ \alpha_i$ denotes the transmittance from branch $ i$ to all other branches meeting at the junction, then $ \alpha_i-1$ is the reflectance seen on branch $ i$. Substituting

$\displaystyle T= 1 + S- L
$

into the basic scattering relations (9.56), and factoring out $ S$, we obtain, in the frequency domain,
$\displaystyle P_1^{-}(\omega)$ $\displaystyle =$ $\displaystyle SP_1^{+}+ TP_2^{+}$  
  $\displaystyle =$ $\displaystyle SP_1^{+}+ [1 + S- L] P_2^{+}$  
  $\displaystyle =$ $\displaystyle S[P_1^{+}+ P_2^{+}] + [1 - L] P_2^{+}$  
  $\displaystyle \isdef$ $\displaystyle S[P_1^{+}+ P_2^{+}] + AP_2^{+}$ (10.58)

and, similarly,
$\displaystyle P_2^{-}(\omega)$ $\displaystyle =$ $\displaystyle S[P_1^{+}+ P_2^{+}] + AP_1^{+}$ (10.59)

The resulting tonehole implementation is shown in Fig. 9.45. We call this the ``shared reflectance'' form of the tonehole junction.

In the same way, an alternate form is obtained from the substitution

$\displaystyle S= T- 1 + L
$

which yields the ``shared transmittance'' form:
$\displaystyle P_1^{-}$ $\displaystyle =$ $\displaystyle T[P_1^{+}+ P_2^{+}] - AP_1^{+}$ (10.60)
$\displaystyle P_2^{-}$ $\displaystyle =$ $\displaystyle T[P_1^{+}+ P_2^{+}] - AP_2^{+}$ (10.61)

shown in Fig. 9.46.

Figure 9.45: ``Shared-reflectance'' implementation of the clarinet tonehole model.
\includegraphics[scale=0.9]{eps/fFingerHoleOneMul}

Figure 9.46: ``Shared-transmittance'' implementation of the clarinet tonehole model.
\includegraphics[scale=0.9]{eps/fFingerHoleOneMulAlt}

Figure 9.47: ``Shared-reflectance'' tonehole model with unstable allpasses pulled out to the inputs.
\includegraphics[scale=0.9]{eps/fFingerHoleOneMulCommuted}

Figure 9.48: ``Shared-transmittance'' tonehole model with unstable allpasses pulled out to inputs.
\includegraphics[width=\twidth]{eps/fFingerHoleOneMulAltCommuted}

Since $ L(\omega)\approx 0$, it can be neglected to first order, and $ A(\omega)\approx 1$, reducing both of the above forms to an approximate ``one-filter'' tonehole implementation.

Since $ R_a = -jR_b \omega t_a/c$ is a pure negative reactance, we have

$\displaystyle A(\omega) = 1 - L(\omega) = \frac{R_0- R_a/2}{R_0+ R_a/2} = \frac{p+j\omega}{p-j\omega}, \quad p=\frac{R_0c}{R_b t_a}$ (10.62)

In this form, it is clear that $ A(\omega)$ is a first-order allpass filter with a single pole-zero pair near infinity. Unfortunately, the pole is in the right-half-plane and hence unstable. We cannot therefore implement it as shown in Fig. 9.45 or Fig. 9.46. Using elementary manipulations, the unstable allpasses in Figs. 9.45 and Fig. 9.46 can be moved to the configuration shown in Figs. 9.47 and 9.48, respectively. Note that $ T(\omega)/A(\omega)$ is stable whenever $ T$ is stable. The unstable allpasses now operate only on the two incoming wave variables, and they can be implemented implicitly by slightly reducing the (interpolated) delay-lines leading to the junction from either side. The tonehole then requires only one filter $ S/A$ or $ T/A$.

We now see precisely how the negative series inertance $ R_a$ provides a negative, frequency-dependent, length correction for the bore. From (9.63), the phase delay of $ A(\omega)$ can be computed as

$\displaystyle D_A(\omega) \isdef -\frac{\angle A(\omega)}{\omega}
= -2\tan^{-1}(\omega/p) = -2\tan^{-1}(k t_a R_b / R_0)
$

Thus, the negative delay correction goes to zero with frequency $ k=\omega/c$, series tonehole length $ t_a$, tonehole impedance $ R_b$, or main bore admittance $ \Gamma _0= 1/R_0$.

In practice, it is common to combine all delay corrections into a single ``tuning allpass filter'' for the whole bore [428,207]. Whenever the desired allpass delay goes negative, we simply add a sample of delay to the desired allpass phase-delay and subtract it from the nearest delay. In other words, negative delays have to be ``pulled out'' of the allpass and used to shorten an adjacent interpolated delay line. Such delay lines are normally available in practical modeling situations.


Tonehole Filter Design

The tone-hole reflectance and transmittance must be converted to discrete-time form for implementation in a digital waveguide model. Figure 9.49 plots the responses of second-order discrete-time filters designed to approximate the continuous-time magnitude and phase characteristics of the reflectances for closed and open toneholes, as carried out in [403,406]. These filter designs assumed a tonehole of radius $ b = 4.765$ mm, minimum tonehole height $ t_{w}
= 3.4$ mm, tonehole radius of curvature $ r_{c} = 0.5$ mm, and air column radius $ a = 9.45$ mm. Since the measurements of Keefe do not extend to 5 kHz, the continuous-time responses in the figures are extrapolated above this limit. Correspondingly, the filter designs were weighted to produce best results below 5 kHz.

The closed-hole filter design was carried out using weighted $ L2$ equation-error minimization [428, p. 47], i.e., by minimizing $ \vert\vert\,W(e^{j\omega})[{\hat A}(e^{j\omega})H(e^{j\omega}) - {\hat B}(e^{j\omega})]\,\vert\vert _2$, where $ W$ is the weighting function, $ H(e^{j\omega})$ is the desired frequency response, $ \Omega$ denotes discrete-time radian frequency, and the designed filter response is $ {\hat H}(e^{j\omega}) = {\hat B}(e^{j\omega})/{\hat A}(e^{j\omega})$. Note that both phase and magnitude are matched by equation-error minimization, and this error criterion is used extensively in the field of system identification [288] due to its ability to design optimal IIR filters via quadratic minimization. In the spirit of the well-known Steiglitz-McBride algorithm [287], equation-error minimization can be iterated, setting the weighting function at iteration $ i+1$ to the inverse of the inherent weighting $ {\hat A}_i$ of the previous iteration, i.e., $ W_{i+1}(e^{j\omega})
= 1/{\hat A}_i(e^{j\omega})$. However, for this study, the weighting was used only to increase accuracy at low frequencies relative to high frequencies. Weighted equation-error minimization is implemented in the matlab function invfreqz()8.6.4).

The open-hole discrete-time filter was designed using Kopec's method [297], [428, p. 46] in conjunction with weighted equation-error minimization. Kopec's method is based on linear prediction:

  • Given a desired complex frequency response $ H(e^{j\omega})$, compute an allpole model $ 1/{\hat A}(z)$ using linear prediction
  • Compute the error spectrum $ \hat E(e^{j\omega})\isdef {\hat A}(e^{j\omega})H(e^{j\omega})$.
  • Compute an allpole model $ 1/{\hat B}(z)$ for $ \hat E^{-1}(e^{j\omega})$ by minimizing

    $\displaystyle \left\Vert\,{\hat B}(e^{j\omega})\hat E^{-1}(e^{j\omega})\,\right...
...at B}(e^{j\omega})}{{\hat A}(e^{j\omega})}H^{-1}(e^{j\omega})\,\right\Vert _2.
$

Use of linear prediction is equivalent to minimizing the $ L2$ ratio error

$\displaystyle \left\Vert\,\hat E(e^{j\omega})\,\right\Vert _2 = \left\Vert\,{\hat A}(e^{j\omega})H(e^{j\omega})\,\right\Vert _2.
$

This optimization criterion causes the filter to fit the upper spectral envelope of the desired frequency-response. Since the first step of Kopec's method captures the upper spectral envelope, the ``nulls'' and ``valleys'' are largely ``saved'' for the next step which computes zeros. When computing the zeros, the spectral ``dips'' become ``peaks,'' thereby receiving more weight under the $ L2$ ratio-error norm. Thus, in Kopec's method, the poles model the upper spectral envelope, while the zeros model the lower spectral envelope. To apply Kopec's method to the design of an open-tonehole filter, a one-pole model $ \hat{H}_{1}(z)$ was first fit to the continuous-time response, $ H(e^{j\Omega}).$ Subsequently, the inverse error spectrum, $ \hat{H}_{1}(e^{j\Omega})/H(e^{j\Omega})$ was modeled with a two-pole digital filter, $ \hat{H}_{2}(z).$ The discrete-time approximation to $ H(e^{j\Omega})$ was then given by $ \hat{H}_{1}(z)/\hat{H}_{2}(z).$

Figure 9.49: Two-port tonehole junction closed-hole and open-hole reflectances based on Keefe's acoustic measurements (dashed) versus second-order digital filter approximations (solid). Top: Reflectance magnitude; Bottom: Reflectance phase. The closed tonehole has one resonance in the audio band just above $ 16$ kHz. The open tonehole has one anti-resonance in the audio band near $ 10$ kHz. At dc, the open tonehole fully reflects, while the closed tonehole reflects close to nothing (from [403]).
\includegraphics[width=\twidth]{eps/twoptfilts}

The reasonably close match in both phase and magnitude by second-order filters indicates that there is in fact only one important tonehole resonance and/or anti-resonance within the audio band, and that the measured frequency responses can be modeled with very high audio accuracy using only second-order filters.

Figure 9.50 plots the reflection function calculated for a six-hole flute bore, as described in [240].

Figure 9.50: Reflection functions for note $ G$ (three finger holes closed, three finger holes open) on a simple flute (from [403]). (top) Transmission-line calculation; (bottom) Digital waveguide two-port tonehole implementation.
\includegraphics[width=\twidth]{eps/gtwoport}
The upper plot was calculated using Keefe's frequency-domain transmission matrices, such that the reflection function was determined as the inverse Fourier transform of the corresponding reflection coefficient. This response is equivalent to that provided by [240], though scale factor discrepancies exist due to differences in open-end reflection models and lowpass filter responses. The lower plot was calculated from a digital waveguide model using two-port tonehole scattering junctions. Differences between the continuous- and discrete-time results are most apparent in early, high-frequency, closed-hole reflections. The continuous-time reflection function was low-pass filtered to remove time-domain aliasing effects incurred by the inverse Fourier transform operation and to better correspond with the plots of [240]. By trial and error, a lowpass filter with a cutoff frequency around 4 kHz was found to produce the best match to Keefe's results. The digital waveguide result was obtained at a sampling rate of 44.1 kHz and then lowpass filtered to a 10 kHz bandwidth, corresponding to that of [240]. Further lowpass filtering is inherent from the first-order Lagrangian, delay-line length interpolation technique used in this model [502]. Because such filtering is applied at different locations along the ``bore,'' a cumulative effect is difficult to accurately determine. The first tonehole reflection is affected by only two interpolation filters, while the second tonehole reflection is affected by four of these filtering operations. This effect is most responsible for the minor discrepancies apparent in the plots.


The Tonehole as a Two-Port Loaded Junction

It seems reasonable to expect that the tonehole should be representable as a load along a waveguide bore model, thus creating a loaded two-port junction with two identical bore ports on either side of the tonehole. From the relations for the loaded parallel junction (C.101), in the two-port case with $ R_1=R_2=R_0$, and considering pressure waves rather than force waves, we have

$\displaystyle P_J(s)$ $\displaystyle =$ $\displaystyle \alpha P_1^{+}+ \alpha P_2^{+}, \quad \alpha = 2\Gamma _0/[G_J(s)+2\Gamma _0]$ (10.63)
$\displaystyle P_1^{-}(s)$ $\displaystyle =$ $\displaystyle P_J(s) - P_1^{+}
= (\alpha-1)P_1^{+}+ \alpha P_2^{+}= \alpha(P_1^{+}+P_2^{+})-P_1^{+}$ (10.64)
$\displaystyle P_2^{-}(s)$ $\displaystyle =$ $\displaystyle P_J(s) - P_2^{+}= \alpha P_1^{+}+ (\alpha-1)P_2^{+}
= \alpha(P_1^{+}+P_2^{+})-P_2^{+}$ (10.65)

Thus, the loaded two-port junction can be implemented in ``one-filter form'' as shown in Fig. 9.48 with $ A(\omega)=1$ ( $ L(\omega)=0$) and

$\displaystyle T(\omega)=\alpha = \frac{2\Gamma _0}{2\Gamma _0+ G_J(s)} = \frac{2R_J(s)}{2R_J(s)+R_0}
$

Comparing with (9.58), we see that the simplified Keefe tonehole model with the negative series inertance removed ($ R_a=0$), is equivalent to a loaded two-port waveguide junction with $ R_J=R_s$, i.e., the parallel load impedance is simply the shunt impedance in the tonehole model.

Each series impedance $ R_a/2$ in the split-T model of Fig. 9.43 can be modeled as a series waveguide junction with a load of $ R_a/2$. To see this, set the transmission matrix parameters in (9.55) to the values $ T_{11} = T_{22} = 1$, $ T_{12} = R_a/2$, and $ T_{21}=0$ from (9.51) to get

$\displaystyle P_1^-$ $\displaystyle =$ $\displaystyle (1-\alpha) P_1^+ + \alpha P_2^-$  
$\displaystyle P_2^+$ $\displaystyle =$ $\displaystyle \alpha P_1^+ + (1-\alpha) P_2^-$ (10.66)

where $ \alpha = 2R_0/(2R_0+R_a/2)$ is the alpha parameter for a series loaded waveguide junction involving two impedance $ R_0$ waveguides joined in series with each other and with a load impedance of $ R_a/2$, as can be seen from (C.99). To obtain exactly the loaded series scattering relations (C.100), we first switch to the more general convention in which the ``$ +$'' superscript denotes waves traveling into a junction of any number of waveguides. This exchanges ``$ +$'' with ``$ -$'' at port 2 to yield
$\displaystyle P_1^-$ $\displaystyle =$ $\displaystyle (1-\alpha) P_1^+ + \alpha P_2^+$  
$\displaystyle P_2^-$ $\displaystyle =$ $\displaystyle \alpha P_1^+ + (1-\alpha) P_2^+$ (10.67)

Next we convert pressure to velocity using $ P_i^+ = R_0U_i^+$ and $ P_i^- = -R_0U_i^-$ to obtain
$\displaystyle U_1^-$ $\displaystyle =$ $\displaystyle (\alpha-1) U_1^+ - \alpha U_2^+$  
$\displaystyle U_2^-$ $\displaystyle =$ $\displaystyle -\alpha U_1^+ + (\alpha-1) U_2^+$ (10.68)

Finally, we toggle the reference direction of port 2 (the ``current'' arrow for $ u_2$ on port 2 in Fig. 9.43) so that velocity is positive flowing into the junction on both ports (which is the convention used to derive (C.100) and which is typically followed in circuit theory). This amounts to negating $ U_2^{\pm}$, giving
$\displaystyle U_1^-$ $\displaystyle =$ $\displaystyle U_J - U_1^+$  
$\displaystyle U_2^-$ $\displaystyle =$ $\displaystyle U_J - U_2^+$ (10.69)

where $ U_J \isdef (\alpha U_1^+ + \alpha U_2^+)$. This is then the canonical form (C.100).


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