# Engineering Acoustics/Clarinet Acoustics

## Introduction

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.

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:

Figure 2: Mouthpiece-Reed system diagram

The lumped reed model is described by [1]:

${\displaystyle m{\frac {d^{2}y}{dt^{2}}}+\mu {\frac {dy}{dt}}+k(\Delta p)y=\Delta p,}$

where ${\displaystyle y}$ is the reed displacement, ${\displaystyle m}$ is the mass, ${\displaystyle \mu }$ is the damping coefficient, ${\displaystyle k}$ is the stiffness and is treated as a function of ${\displaystyle \Delta p}$.

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.

Figure 3: Input air flow vs. pressure difference (${\displaystyle \Delta p=p_{mouth}-p_{chamber}}$)

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 ${\displaystyle U}$ and the pressure difference ${\displaystyle \Delta p}$ that across the reed channel is mathematically described by Bernoulli's Equation:

${\displaystyle U=hw{\sqrt {\frac {2|\Delta p|}{\rho }}}sgn(\Delta p),}$

where ${\displaystyle h}$ is the reed opening, ${\displaystyle w}$ is the channel's width and ${\displaystyle \rho }$ is the fluid density. The reed opening ${\displaystyle h}$ is related to pressure difference ${\displaystyle \Delta p}$. Approximately, increasing ${\displaystyle \Delta p}$ results in decreasing ${\displaystyle h}$ 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.

Figure 4: Pressure and velocity distribution of f0 in a lossless cylindrical closed-open pipe

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.

Figure 5: Pressure and velocity distribution of f1, f3 and f5 in a lossless cylindrical closed-open pipe

#### 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 shape the unique timbre of clarinets. Specifically, the series resonance frequencies of the closed-open pipe with length L are given by [3]:

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

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 ${\displaystyle L_{eff}=L+0.6a}$,[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. [3] 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:

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}P(x,t)}{\partial t^{2}}}={\frac {\partial ^{2}P(x,t)}{\partial x^{2}}},}$

where ${\displaystyle x}$ is the axis along the propagation direction.

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

${\displaystyle P(x,t)=(Ae^{-jkx}+Be^{jkx})e^{j\omega t},}$

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

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

${\displaystyle U(x,t)={\frac {s}{\rho c}}(Ae^{-jkx}-Be^{jkx})e^{j\omega t},}$

### Acoustic Impedance

Figure 6: The input impedance of an ideal losses cylindrical pipe

The acoustic input impedance ${\displaystyle Z_{in}(j\omega )}$ 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 [3]:

${\displaystyle Z_{in}(j\omega )={\frac {P(j\omega )|_{x=0}}{U(j\omega )|_{x=0}}}=Z_{c}{\frac {Z_{L}cos(kL)+jZ_{c}sin(kL)}{jZ_{L}sin(kL)+Z_{c}cos(kL)}},}$

where ${\displaystyle Z_{L}}$ is the load impedance at the open end of the clarinet's bore, and ${\displaystyle Z_{c}=\rho c/s}$ 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 ${\displaystyle L=0.148}$ meters and radius ${\displaystyle r=0.00775}$ meters by Matlab, which is shown in Figure 6.

Figure 7a: Reflectance magnitude and length correction for open end pipes
Figure 7b: Normalized amplitude of radiation impedance for open end pipes

The load impedance at the open end of the cylindrical bore are represented by radiation impedance ${\displaystyle Z_{r}}$. We assumed ${\displaystyle Z_{r}=0}$ 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 ${\displaystyle Z_{in}}$ by:[7] ${\displaystyle Z_{r}=jZ_{c}tan[atan(Z_{in}/jZ_{c})-kL]}$, where ${\displaystyle L}$ is the length of the pipe and ${\displaystyle a}$ is the radius.

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

${\displaystyle Z_{r}={\frac {\rho c}{S}}{\frac {1+R}{1-R}}}$

where ${\displaystyle \rho }$ is the air density, ${\displaystyle c}$ is the sound speed and ${\displaystyle S}$is the cross-section area of the pipe.

Levine and Schwinger [8] gives the theoretical value of ${\displaystyle R}$ of a tube with a finite wall thickness, where ${\displaystyle R}$ is calculated by its modulus ${\displaystyle |R|}$ and the length correction ${\displaystyle l(\omega )}$ as ${\displaystyle R=-|R|e^{-2jkl}}$. The original equation proposed by Levine and Schwinger is rather complicated. To make life easier, as shown in Figure 7a, ${\displaystyle R}$ and the length correction can be approximated by the rational equation given by Norris and Sheng (1989)[9]. The radiation impedance is followed in Figure 7b.

${\displaystyle |R|={\frac {1+0.2ka-0.084(ka)^{2}}{1+0.2ka+0.416(ka)^{2}}}}$
${\displaystyle l/a={\frac {0.6133+0.027(ka)^{2}}{1+0.19(ka)^{2}}}}$

### 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:

Figure 8: A bore section

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

${\displaystyle {\begin{bmatrix}P_{1}\\U_{1}\end{bmatrix}}}$=${\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}P_{2}\\U_{2}\end{bmatrix}}}$

where ${\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$=${\displaystyle {\begin{bmatrix}cos(kL)&jZ_{c}sin(kL)\\{\frac {j}{Z_{c}}}sin(kL)&cos(kL)\end{bmatrix}}}$

Thus, ${\displaystyle Z_{1}}$ is related to ${\displaystyle Z_{2}}$ by: ${\displaystyle Z_{1}={\frac {b+aZ_{2}}{d+cZ_{2}}}}$. 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.

Figure 9: Shunt and series impedances of the tone hole
Figure 10: Combine the tone hole network with the bore network

The shunt impedance ${\displaystyle Z_{s}}$ and series impedance ${\displaystyle Z_{a}}$ of a tone hole of ridius ${\displaystyle b}$ in a main bore of ridius ${\displaystyle a}$ are given by: [10]

${\displaystyle Z_{sc}=(\rho c/\pi a^{2})(a/b)^{2}(-jcotkt),}$
${\displaystyle Z_{so}=(\rho c/\pi a^{2})(a/b)^{2}(jkt_{e}),}$
${\displaystyle Z_{a}=(\rho c/\pi a^{2})(a/b)^{2}(-jkt_{a}),}$

where ${\displaystyle Z_{sc}}$ is the shunt impedance of closed tone hole, ${\displaystyle Z_{so}}$ is the shunt impedance of open tone hole, ${\displaystyle Z_{a}}$ is the series impedance of either closed or open tone hole, the value of ${\displaystyle t_{e}}$ and ${\displaystyle t_{a}}$ 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 ${\displaystyle Z_{r}}$ and ${\displaystyle Z_{rt}}$ 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 ${\displaystyle Z_{in}=P_{in}/U_{in}}$ 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 (${\displaystyle k=\omega /c}$) 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:[10]

${\displaystyle K=\omega /\nu -j\alpha .}$

We notice two interesting differences here. For one, the sound speed ${\displaystyle c}$ is replaced by "phase velocity" ${\displaystyle \nu }$, which is not a constant but rather a function of frequency and geometric radius of the pipe. For two, there is an imaginary term ${\displaystyle j\alpha }$, where ${\displaystyle \alpha }$ 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.

Figure 10: Dissipated power vs. radiated power