# Engineering Acoustics/Clarinet Acoustics

The clarinet is a member of the woodwind instruments family that is
widely played in orchestra bands or jazz bands. There are different
types of clarinets that differ in sizes and pitches: B flat, E flat,
bass, contrabass, etc. A clarinet typically provides a flow of air of
about 3 kPa acoustic pressure or 3% of one atmosphere.

A clarinet consists several acoustical components:

-   a mouthpiece-reed system: like an energy source, it produces air
    flow and pressure oscillating components that fill into the
    instrument.
-   a cylindrical bore: a resonator that forms the air column and
    produces the standing wave.
-   a bell (at the open end of the cylindrical bore) and open tone
    hole(s): act as radiators.

![](Clarinet_sy.png "Clarinet_sy.png"){width="650"}

From the energy\'s point of view, most of the energy injected by the
player compensates the thermal and viscous losses to the wall inside the
cylindrical bore, while only a fractional part of energy is radiated via
bell and open holes and heard by listeners.

### Mouthpiece-Reed System

The reed serves as a spring-like oscillator. It converts steady input
air flow (DC) into acoustically vibrated air flow (AC). However, it is
more than a single-way converter because it also interacts with the
resonance of the air column in the instrument, i.e.:

-   initially, increasing the blowing pressure results in more air
    flowing into the clarinet bore.
-   but too much difference of the blowing pressure and the mouthpiece
    pressure will close the aperture between the reed and the mouthpiece
    and finally result in zero air flow.

This behavior is roughly depicted in Figure 2:

\[\[Image:mouthpiece.png\|left\|thumb\|450px\|

```{=html}
<center>
```
Figure 2: Mouthpiece-Reed system diagram

```{=html}
</center>
```
\]\]

The lumped reed model is described by:[^1]

```{=html}
<center>
```
$m\frac{d^2 y}{dt^2}+ \mu \frac{dy}{dt}+k(\Delta p)y = \Delta p ,$

```{=html}
</center>
```
where $y$ is the reed displacement, $m$ is the mass, $\mu$ is the
damping coefficient, $k$ is the stiffness and is treated as a function
of $\Delta p$.

Let\'s go to a bit more details of the relation between the input air
flow, the air pressure in the player\'s mouth as well as that in the
mouthpiece chamber.

\[\[Image:ReedResistance.png\|right\|thumb\|550px\|

```{=html}
<center>
```
Figure 3: Input air flow vs. pressure difference
($\Delta p = p_{mouth} - p_{chamber}$)

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<center>
```
\]\]

Figure 3 is roughly divided by two parts. The left part shows a
resistance-like feature, i.e., the air flow increases with increasing
difference between the mouth pressure and the mouthpiece pressure. The
right part shows a negative resistance, i.e., the air flow decreases
with increasing pressure difference. The AC oscillating only occurs
within the right part, so the player must play with a mouth pressure
that fall into a certain range. Specifically, the pressure difference
must be larger than the minimum pressure corresponding to the beginning
of the right part, while no more than the maximum pressure that will
shut the reed off.

The relation between the volume flow $U$ and the pressure difference
$\Delta p$ that across the reed channel is mathematically described by
Bernoulli\'s Equation:

```{=html}
<center>
```
$U = hw \sqrt{\frac{2|\Delta p|}{\rho}}sgn(\Delta p) ,$

```{=html}
</center>
```
where $h$ is the reed opening, $w$ is the channel\'s width and $\rho$ is
the fluid density. The reed opening $h$ is related to pressure
difference $\Delta p$. Approximately, increasing $\Delta p$ results in
decreasing $h$ until the mouthpiece channel be closed and no air flow
filled in.

The non-linear behavior of the mouthpiece-reed system is complicated and
is beyond the scope of linear acoustics. Andrey da Silva (2008) in
McGill University simulated the fully coupled fluid-structure
interaction in single reed mouthpieces by using a 2-D lattice Boltzmann
model, where the velocity fields for different instants are visualized
in his PhD
thesis.[^2]

### Cylindrical Bore

If all tone holes are closed, the main bore of a clarinet is
approximately cylindrical and the mouthpiece end can be looked as a
closed end. Therefore, the prime acoustical behavior of the main bore is
similar to a cylindrical closed-open pipe (a pipe with one closed end
and one open end). Also, to further simplify the problem, here we assume
the walls of the pipe are rigid, perfectly smooth and thermally
insulated.

Sound propagation in the bore can be expressed as a sum of numerous
normal modes. These modes are produced by wave motion along all three
axis in a circular cylindrical coordinate system, namely the wave motion
along transverse concentric circles, the wave motion along the
transverse radial plane and the plane wave motions along the principal
axis of the pipe. However, since transverse modes are only weakly
excited in real instruments, we will not discuss them here but only
focus on the longitudinal plane wave.

#### Fundamental frequency

The natural vibrations of the air column confined by the main bore are
supported by a series of standing waves. Even without any mathematical
analysis, by inspecting boundary conditions, we can intuitively learn
some important physical features of these standing waves. There must be
pressure nodes at the open end because the total pressure near the open
end is almost the same as the ambient pressure, that means zero
acoustical pressure at the open end. Then as we look at the closed end
(actually, the end connecting with the mouthpiece chamber is not
completely closed, there is always an opening to let the air fill in,
but we \"pretend\" the end is completely closed to simplify the analysis
at this moment), since the volume velocity of the air flow is almost
zero, the pressure is at its maximum value. The lowest frequency of
these standing waves can then be found from the wave with the longest
wavelength, which is four times of the length of the instrument bore.
Why? Because if we plot one quarter circle of this sinusoid wave and fit
it into a closed-open pipe, that the peak amplitude is located at the
closed end and the zero amplitude is located at the open end, this is a
perfect representation of a pressure standing wave inside a closed-open
pipe. Figure 4 depicts the pressure wave and the velocity wave
corresponding to the lowest pitch (the 1st resonance frequency) in an
ideal closed-open cylindrical pipe.
\[\[Image:c_o\_pipe.png\|left\|thumb\|350px\|

```{=html}
<center>
```
Figure 4: Pressure and velocity distribution of f0 in a lossless
cylindrical closed-open pipe

```{=html}
</center>
```
\]\]

Figure 5 is the normalized pressure and velocity distribution of the
1st, 3rd and 5th resonance frequency of a closed-open pipe of length
148 cm. To simplify the question, the reflectance of the open end is
simplified to -1 and the viscous losses are not accounted.

\[\[Image:c_o\_pipe_pu_f135.png\|right\|thumb\|650px\|

```{=html}
<center>
```
Figure 5: Pressure and velocity distribution of f1, f3 and f5 in a
lossless cylindrical closed-open pipe

```{=html}
</center>
```
\]\]

#### Harmonic series

In the main bore, standing waves of other higher frequencies are also
possible, but their frequencies must be the odd harmonics of the
fundamental frequency due to the closed-open restriction. This is also
an important factor that shapes the unique timbre of clarinets.
Specifically, the series resonance frequencies of the closed-open pipe
with length L are given by:[^3]

$$f_n = \frac{(2n-1)c}{4L}$$, where $n=1,2,...$

For example, for a bore of length of 14.8 cm, the first 5 harmonics are:
0.581, 1.7432, 2.9054, 4.0676, 5.2297 kHz, respectively. The calculation
is based on an ideal cylindrical pipe. For a real clarinet, however, the
resonance frequencies are determined not only by the length of the bore,
but also by the shape of the bore (which is not a perfect cylinder) and
by the fingering of tone holes. Also, due to the end
correction effects caused
by radiation impedance at the open end, the effective length of an
unflanged open pipe is $L_{eff}=L+0.6a$,.[^4] hence the fundamental
frequency and the harmonic series are lowered a bit.

### Tone Holes

The role of tone holes of clarinets can be viewed from two aspects.

Firstly, the open tone holes change the effective length of the main
bore and hence the resonance frequencies of the enclosed air column.
Each discrete note produced by a clarinet is determined by a specific
fingering, i.e., a particular configuration of open and closed tone
holes. By using advance playing techniques, a player can play pitch
bending (a continuous variation of pitch from one note to the next).
These techniques include partially covering a tone hole (for limited
pitch bending of notes from G3/175 Hz to G4/349 Hz and above D5/523 Hz)
and using vocal tract (for substantial pitch bending above
D5/523 Hz).[^5] The beginning bars of Gershwin\'s *Rhapsody in Blue*[^6]
demonstrates a famous example of a large pitch bending over the range up
to 2.5 octave.

Secondly, the sound radiates from both open holes and the bell end, this
makes the clarinets (and other woodwind instruments) have different
directivity patterns comparing to another family of wind instruments,
the brass instruments, which have a similar open bell end but don\'t
have side holes.

We will see how to calculate the acoustic impedance that changed by tone
holes later.

### Bell

The flaring bell of a clarinet is less important than that of a brass
instrument, because open tone holes contribute to sound radiation in
addition to the bell end. The main function of bell is *to form a smooth
impedance transition from the interior of the bore to the surrounding
air*.[^7] A clarinet is still functional for most notes even without a
bell.

### Register Holes

The main purpose of a register hole is to disrupt the fundamental but
keep higher harmonics as much as possible, such that the frequency of
the note will be tripled by opening the register hole.

## Wave Propagation in the Bore

### Wave Equation

The sound waves propagation inside the main bore of a clarinet are
described by the one-dimensional wave equation:

```{=html}
<center>
```
$\frac{1}{c^2}\frac{\partial^2 P(x,t)}{\partial t^2}=\frac{\partial^2 P(x,t)}{\partial x^2},$

```{=html}
</center>
```
where $x$ is the axis along the propagation direction.

The complex solution for the sound pressure wave $P(y,t)$ is:

```{=html}
<center>
```
$P(x,t)=(Ae^{-jkx}+Be^{jkx})e^{j\omega t},$

```{=html}
</center>
```
where $k=\omega /c$ is the wave number, $\omega=2\pi f$, A and B are the
complex amplitudes of the left-,right-going traveling pressure waves,
respectively.

Another interesting physical parameter is the volume velocity $U(x,t)$,
defined as particle velocity $V(x,t)$ times cross-sectional area $s$.
The complex solution for the volume velocity $U(x,t)$ is given by:

```{=html}
<center>
```
$U(x,t)=\frac{s}{\rho c}(Ae^{-jkx}-Be^{jkx})e^{j\omega t},$

```{=html}
</center>
```
### Acoustic Impedance

\[\[Image:InputImpedanceLoseless.png\|right\|thumb\|550px\|

```{=html}
<center>
```
Figure 6: The input impedance of an ideal losses cylindrical pipe

```{=html}
</center>
```
\]\] The acoustic input impedance $Z_{in}(j\omega)$ provides very useful
information about the acoustic behavior of a clarinet in the frequency
domain. The intonation and response can be inferred from the input
impedance, e.g., sharper and stronger peaks indicate frequencies that
are easiest to play.

The input impedance (in the frequency domain) is defined as the ratio of
pressure to volume flow at the input end (x=0) of the pipe:[^8]

```{=html}
<center>
```
$Z_{in}(j\omega) = \frac{P(j\omega)|_{x=0}}{U(j\omega)|_{x=0}} = Z_c\frac{Z_L cos(kL) + jZ_c sin(kL)}{jZ_L sin(kL) + Z_c cos(kL)},$

```{=html}
</center>
```
where $Z_L$ is the load impedance at the open end of the clarinet\'s
bore, and $Z_c=\rho c/s$ is the characteristic impedance.

At this point, if we want to have a \"quick glance\" of the input
impedance of a clarinet bore, we may neglect the radiation losses and
assume zero load impedance at the open end of the main bore to simplify
the problem. We may also neglect the sound absorption due to wall
losses. With these simplifications, we can calculate the theoretical
input impedance of a cylindrical pipe of length $L=0.148$ meters and
radius $r = 0.00775$ meters by Matlab, which is shown in Figure 6.

### Radiation Impedance

\[\[Image:Reflectance.png\|left\|thumb\|550px\|

```{=html}
<center>
```
Figure 7a: Reflectance magnitude and length correction for open end
pipes

```{=html}
</center>
```
\]\] \[\[Image:Radiation impedance chart.png\|left\|thumb\|550px\|

```{=html}
<center>
```
Figure 7b: Normalized amplitude of radiation impedance for open end
pipes

```{=html}
</center>
```
\]\]

The load impedance at the open end of the cylindrical bore are
represented by radiation impedance $Z_r$. We assumed $Z_r=0$ previously
when we discussed the input impedance of an ideal cylindrical pipe.
Although it is very small, the radiation impedance of a real clarinet is
obviously not zero. And not only for the open end of the main bore, each
open tone hole features its own radiation impedance as well.

It is not easy to measure the radiation impedance directly. However, we
can obtain the radiation impedance of a pipe from its input impedance
$Z_{in}$ by:[^9] $Z_r = jZ_c tan[atan(Z_{in}/jZ_c)-kL]$, where $L$ is
the length of the pipe and $a$ is the radius.

Alternatively, we can also calculate the radiation impedance from
reflection coefficient $R$ at the open end by this relation:

```{=html}
<center>
```
$Z_r = \frac{\rho c}{S} \frac{1+R}{1-R}$

```{=html}
</center>
```
where $\rho$ is the air density, $c$ is the sound speed and $S$is the
cross-section area of the pipe.

Levine and Schwinger [^10] gives the theoretical value of $R$ of a tube
with a finite wall thickness, where $R$ is calculated by its modulus
$|R|$ and the length correction $l(\omega)$ as $R = -|R|e^{-2jkl}$. The
original equation proposed by Levine and Schwinger is rather
complicated. To make life easier, as shown in Figure 7a, $R$ and the
length correction can be approximated by the rational equation given by
Norris and Sheng (1989).[^11] The radiation impedance is followed in
Figure 7b.

```{=html}
<center>
```
$|R|=\frac{1+0.2ka-0.084(ka)^2}{1+0.2ka+0.416(ka)^2}$

```{=html}
</center>
```
```{=html}
<center>
```
$l/a=\frac{0.6133+0.027(ka)^2}{1+0.19(ka)^2}$

```{=html}
</center>
```
### Transmission Matrices

#### Bore section

Since the acoustic impedance is so important for the quality and feature
of a clarinet, we somehow are interested in knowing the acoustic
impedance at any place along the main bore. This problem can be solved
by transmission matrices method. We will see the effects of tone holes
can also be incorporated into the acoustic impedance network of the
instrument by introducing extra series and shunt impedances.

The entire bore can be seen as a series cascade cylindrical sections,
each section with an input end and a output end, as shown in Figure
below: \[\[Image:BoreTM.svg\|center\|thumb\|450px\|

```{=html}
<center>
```
Figure 8: A bore section

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</center>
```
\]\]

The pressure and volume velocity at the input end and that at the output
end are related by the associated transmission matrix:

```{=html}
<center>
```
$\begin{bmatrix} P_1 \\ U_1 \end{bmatrix}$=$\begin{bmatrix}  a & b \\ c & d \end{bmatrix}$$\begin{bmatrix} P_2 \\ U_2 \end{bmatrix}$

where
$\begin{bmatrix}  a & b \\ c & d \end{bmatrix}$=$\begin{bmatrix}  cos (k L) & j Z_c sin (k L) \\ \frac{j}{Z_c} sin(kL)  & cos(kL) \end{bmatrix}$

```{=html}
</center>
```
Thus, $Z_1$ is related to $Z_2$ by: $Z_1=\frac{b+aZ_2}{d+cZ_2}$. Given
the input impedance or the load impedance of a cylindrical pipe, we can
calculate the acoustic impedance at any position of the pipe along the
propagation axis.

#### Tonehole section

Now we deal with the tone holes. The influence of an open or closed tone
hole can be represented by a network of shunt and series impedances, as
shown in Figure 9. \[\[Image:Toneholenetwork.svg\|right\|thumb\|550px\|

```{=html}
<center>
```
Figure 9: Shunt and series impedances of the tone hole

```{=html}
</center>
```
\]\] \[\[Image:Tonehole2.svg\|right\|thumb\|550px\|

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<center>
```
Figure 10: Combine the tone hole network with the bore network

```{=html}
</center>
```
\]\]

The shunt impedance $Z_s$ and series impedance $Z_a$ of a tone hole of
ridius $b$ in a main bore of ridius $a$ are given by:[^12]

```{=html}
<center>
```
$Z_{sc}=(\rho c/\pi a^2) (a/b)^2 (-jcotkt),$

```{=html}
</center>
```
```{=html}
<center>
```
$Z_{so}=(\rho c/\pi a^2) (a/b)^2 (jkt_e),$

```{=html}
</center>
```
```{=html}
<center>
```
$Z_{a}=(\rho c/\pi a^2) (a/b)^2 (-jkt_a),$

```{=html}
</center>
```
where $Z_{sc}$ is the shunt impedance of closed tone hole, $Z_{so}$ is
the shunt impedance of open tone hole, $Z_{a}$ is the series impedance
of either closed or open tone hole, the value of $t_e$ and $t_a$ are
related to the geometrical chimney height.

The network of a tone hole can be inserted into the bore section as a
zero-length section, as shown in Figure 10, where $Z_{r}$ and $Z_{rt}$
are radiation impedance of the bore and the open tone hole,
respectively. In the low frequency approximation, a tone hole can be
viewed as a short cylindrical bore, and its radiation impedance can be
calculated in a similar way. The input acoustic impedance of the
combination $Z_{in}=P_{in}/U_{in}$ can be calculated from the entire
network.

#### Wall losses

We assumed a perfect rigid and smooth and thermally insulated wall in
the previous discussions. The bore of a clarinet is of course not that
ideal in the real case, so the losses due to viscous drag and thermal
exchange must be taken into account. The full physical detail of
thermal-viscous losses is complex and tedious and is beyond the scope of
this article. Fortunately, we don\'t have to go to that detail if we
only concerns the final effects, i.e., we just simply replacing the wave
number ($k=\omega /c$) of the transmission matrix coefficients with its
complex brother and bingo, our new transmission matrices take care of
the wall losses effects automatically. This complex version of wave
number is given by:[^13]

```{=html}
<center>
```
$K=\omega /\nu - j\alpha.$

```{=html}
</center>
```
We notice two interesting differences here. For one, the sound speed $c$
is replaced by \"phase velocity\" $\nu$, which is not a constant but
rather a function of frequency and geometric radius of the pipe. For
two, there is an imaginary term $j\alpha$, where $\alpha$ is the
attenuation coefficient per unit length of path, which is also a
function of frequency and geometric radius of the pipe. Both the phase
velocity and the attenuation coefficient are subject to environmental
parameters such as temperature, air viscosity, specific heat and thermal
conductivity of air.

The fact that the phase velocity and the attenuation coefficient are
frequency related suggests that not only the amplitude, but also the
phase of the acoustic impedance is affected by wall losses. In other
words, not only the loudness but also the tonality of the instrument is
affected by the wall losses. That implies when we design a clarinet, if
we calculate the input impedance based on the assumption of an ideal
cylindrical pipe with a perfect rigid and smooth wall, the tone of this
instrument will be problematic! Using a complex wave number that related
to the physics properties of the materials will improve our design, at
least will shorten the gap between the theoretical prediction and the
real results.

The fact that the complex wave number are also influenced by
environmental parameters suggests that the tonality of a woodwind
instrument may change by environmental factors, say, the room
temperature.

It would be interesting to compare the dissipated power due to the
thermal-viscous losses with the radiated power over the clarinet bore.
The comparison over the range from 0 Hz to 2000 Hz is simulated in
Matlab, where the length of the pipe is 0.148 m and the radius is
0.00775 m, the properties of air is chosen at the temperature of 20
degree. We found the dissipated power is much larger than the radiated
power for most frequencies excepts small areas around the resonance
frequencies.

\[\[Image:Prad.png\|center\|thumb\|550px\|

```{=html}
<center>
```
Figure 10: Dissipated power vs. radiated power

```{=html}
</center>
```
\]\]

## Other Useful Links

-   Clarinet Acoustics by
    UNSW: almost
    everything about clarinet acoustics and general musical acoustics.
    This excellent online knowledge base is maintained by The University
    of New South Wales.
-   NICTA-UNSW Clarinet
    Robot: online
    video shows a robot that can play clarinet and help people to better
    understand clarinet playing.
-   Clarinet Acoustics: An
    Introduction.
-   Basic Clarinet
    Acoustics: Yet another
    online article about basic clarinet acoustics.
-   What is Acoustic Impedance and why is it
    important? Explain
    the acoustic impedance in an intuitive way.
-   Physical modeling of woodwind
    instruments.
    The physical behavior of a clarinet can be modeled by digital
    waveguide - a highly efficient time-domain modeling technique.
    Digital waveguide is used in Yamaha\'s virtual instruments, such as
    VL70m.
    See J.O.Smith\'s online book Physical Audio Signal
    Processing for more detail
    about physical modeling technique based on digital waveguide.
-   Physical modeling
    clarinet:,
    Electronic implementation of another physical modeling clarinet.

## References


[^1]:
    
[^2]:
    
[^3]:
    
[^4]:
    
[^5]:
    
[^6]: 1, George Gershwin:
    Rhapsody in Blue - Fantasia 2000

[^7]:

[^8]:

[^9]:
    
[^10]:
    
[^11]:
    
[^12]:
    
[^13]:
