# Acoustics/Flow-induced Oscillations of a Helmholtz Resonator

![](Acoustics_flow_induced.JPG "Acoustics_flow_induced.JPG")

## Introduction

The importance of flow excited acoustic resonance lies in the large
number of applications in which it occurs. Sound production in organ
pipes, compressors, transonic wind tunnels, and open sunroofs are only a
few examples of the many applications in which flow excited resonance of
Helmholtz resonators can be found.\[4\] An instability of the fluid
motion coupled with an acoustic resonance of the cavity produce large
pressure fluctuations that are felt as increased sound pressure levels.

Passengers of road vehicles with open sunroofs often experience
discomfort, fatigue, and dizziness from self-sustained oscillations
inside the car cabin. This phenomenon is caused by the coupling of
acoustic and hydrodynamic flow inside a cavity which creates strong
pressure oscillations in the passenger compartment in the 10 to 50 Hz
frequency range. Some effects experienced by vehicles with open sunroofs
when buffeting include: dizziness, temporary hearing reduction,
discomfort, driver fatigue, and in extreme cases nausea. The importance
of reducing interior noise levels inside the car cabin relies primarily
in reducing driver fatigue and improving sound transmission from
entertainment and communication devices.

This Wikibook page aims to theoretically and graphically explain the
mechanisms involved in the flow-excited acoustic resonance of Helmholtz
resonators. The interaction between fluid motion and acoustic resonance
will be explained to provide a thorough explanation of the behavior of
self-oscillatory Helmholtz resonator systems. As an application example,
a description of the mechanisms involved in sunroof buffeting phenomena
will be developed at the end of the page.

## Feedback loop analysis

As mentioned before, the self-sustained oscillations of a Helmholtz
resonator in many cases is a continuous interaction of hydrodynamic and
acoustic mechanisms. In the frequency domain, the flow excitation and
the acoustic behavior can be represented as transfer functions. The flow
can be decomposed into two volume velocities.

:   qr: flow associated with acoustic response of cavity
:   qo: flow associated with excitation

## Acoustical characteristics of the resonator

### Lumped parameter model

The lumped parameter model of a Helmholtz resonator consists of a
rigid-walled volume open to the environment through a small opening at
one end. The dimensions of the resonator in this model are much less
than the acoustic wavelength, in this way allowing us to model the
system as a lumped system.

Figure 2 shows a sketch of a Helmholtz resonator on the left, the
mechanical analog on the middle section, and the electric-circuit analog
on the right hand side. As shown in the Helmholtz resonator drawing, the
air mass flowing through an inflow of volume velocity includes the mass
inside the neck (Mo) and an end-correction mass (Mend). Viscous losses
at the edges of the neck length are included as well as the radiation
resistance of the tube. The electric-circuit analog shows the resonator
modeled as a forced harmonic oscillator. \[1\] \[2\]\[3\]

```{=html}
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***Figure 2***

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V: cavity volume

$\rho$: ambient density

c: speed of sound

S: cross-section area of orifice

K: stiffness

$M_a$: acoustic mass

$C_a$: acoustic compliance

The equivalent stiffness K is related to the potential energy of the
flow compressed inside the cavity. For a rigid wall cavity it is
approximately:

```{=html}
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```
$K = \left(\frac{\rho c^2}{V}\right)S^2$

```{=html}
</center>
```
The equation that describes the Helmholtz resonator is the following:

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```
$S \hat{P}_e =\frac{\hat{q}_e}{j\omega S}(-\omega ^2 M + j\omega R + K)$

```{=html}
</center>
```
$\hat{P}_e$: excitation pressure

M: total mass (mass inside neck Mo plus end correction, Mend)

R: total resistance (radiation loss plus viscous loss)

From the electrical-circuit we know the following:

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```
$M_a = \frac{L \rho}{S}$

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```
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$C_a = \frac{V}{\rho c^2}$

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```
```{=html}
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```
$L ' = \ L + \ 1.7 \ re$

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```
The main cavity resonance parameters are resonance frequency and quality
factor which can be estimated using the parameters explained above
(assuming free field radiation, no viscous losses and leaks, and
negligible wall compliance effects)

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$\omega_r^2 = \frac{1}{M_a C_a}$

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```{=html}
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$f_r = \frac{c}{2 \pi} \sqrt{\frac{S}{L' V}}$

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```
The sharpness of the resonance peak is measured by the quality factor Q
of the Helmholtz resonator as follows:

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```
$Q = 2 \pi \sqrt{V (\frac{L'} {S})^3}$

```{=html}
</center>
```
$f_r$: resonance frequency in Hz

$\omega_r$: resonance frequency in radians

:   L: length of neck
:   L\': corrected length of neck

From the equations above, the following can be deduced:

-   The greater the volume of the resonator, the lower the resonance
    frequencies.
-   If the length of the neck is increased, the resonance frequency
    decreases.

### Production of self-sustained oscillations

The acoustic field interacts with the unstable hydrodynamic flow above
the open section of the cavity, where the grazing flow is continuous.
The flow in this section separates from the wall at a point where the
acoustic and hydrodynamic flows are strongly coupled. \[5\]

The separation of the boundary layer at the leading edge of the cavity
(front part of opening from incoming flow) produces strong vortices in
the main stream. As observed in Figure 3, a shear layer crosses the
cavity orifice and vortices start to form due to instabilities in the
layer at the leading edge.

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***Figure 3***

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From Figure 3, L is the length of the inner cavity region, d denotes the
diameter or length of the cavity length, D represents the height of the
cavity, and $\delta$ describes the gradient length in the grazing
velocity profile (boundary layer thickness).

The velocity in this region is characterized to be unsteady and the
perturbations in this region will lead to self-sustained oscillations
inside the cavity. Vortices will continually form in the opening region
due to the instability of the shear layer at the leading edge of the
opening.

## Applications to Sunroof Buffeting

### How are vortices formed during buffeting?

In order to understand the generation and convection of vortices from
the shear layer along the sunroof opening, the animation below has been
developed. At a certain range of flow velocities, self-sustained
oscillations inside the open cavity (sunroof) will be predominant.
During this period of time, vortices are shed at the trailing edge of
the opening and continue to be convected along the length of the cavity
opening as pressure inside the cabin decreases and increases. Flow
visualization experimentation is one method that helps obtain a
qualitative understanding of vortex formation and conduction.

The animation below shows, in the middle, a side view of a car cabin
with the sunroof open. As the air starts to flow at a certain mean
velocity Uo, air mass will enter and leave the cabin as the pressure
decreases and increases again. At the right hand side of the animation,
a legend shows a range of colors to determine the pressure magnitude
inside the car cabin. At the top of the animation, a plot of circulation
and acoustic cavity pressure versus time for one period of oscillation
is shown. The symbol x moving along the acoustic cavity pressure plot is
synchronized with pressure fluctuations inside the car cabin and with
the legend on the right. For example, whenever the x symbol is located
at the point where t=0 (when the acoustic cavity pressure is minimum)
the color of the car cabin will match that of the minimum pressure in
the legend (blue).

```{=html}
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```
![](theplot.gif "theplot.gif")

```{=html}
</center>
```
The perturbations in the shear layer propagate with a velocity of the
order of 1/2Uo which is half the mean inflow velocity. \[5\] After the
pressure inside the cavity reaches a minimum (blue color) the air mass
position in the neck of the cavity reaches its maximum outward position.
At this point, a vortex is shed at the leading edge of the sunroof
opening (front part of sunroof in the direction of inflow velocity). As
the pressure inside the cavity increases (progressively to red color)
and the air mass at the cavity entrance is moved inwards, the vortex is
displaced into the neck of the cavity. The maximum downward displacement
of the vortex is achieved when the pressure inside the cabin is also
maximum and the air mass in the neck of the Helmholtz resonator (sunroof
opening) reaches its maximum downward displacement. For the rest of the
remaining half cycle, the pressure cavity falls and the air below the
neck of the resonator is moved upwards. The vortex continues displacing
towards the downstream edge of the sunroof where it is convected upwards
and outside the neck of the resonator. At this point the air below the
neck reaches its maximum upwards displacement.\[4\] And the process
starts once again.

### How to identify buffeting

Flow induced tests performed over a range of flow velocities are helpful
to determine the change in sound pressure levels (SPL) inside the car
cabin as inflow velocity is increased. The following animation shows
typical auto spectra results from a car cabin with the sunroof open at
various inflow velocities. At the top right hand corner of the
animation, it is possible to see the inflow velocity and resonance
frequency corresponding to the plot shown at that instant of time.

```{=html}
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```
![](curve.gif "curve.gif")

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</center>
```
It is observed in the animation that the SPL increases gradually with
increasing inflow velocity. Initially, the levels are below 80 dB and no
major peaks are observed. As velocity is increased, the SPL increases
throughout the frequency range until a definite peak is observed around
a 100 Hz and 120 dB of amplitude. This is the resonance frequency of the
cavity at which buffeting occurs. As it is observed in the animation, as
velocity is further increased, the peak decreases and disappears.

In this way, sound pressure level plots versus frequency are helpful in
determining increased sound pressure levels inside the car cabin to find
ways to minimize them. Some of the methods used to minimize the
increased SPL levels achieved by buffeting include: notched deflectors,
mass injection, and spoilers.

# Useful websites

This link: 1 takes you to the website of EXA
Corporation, a developer of PowerFlow for Computational Fluid Dynamics
(CFD) analysis.

This link:
2 is a
small news article about the current use of(CFD) software to model
sunroof buffeting.

This link:
3
is a small industry brochure that shows the current use of CFD for
sunroof buffeting.

# References

1.  Acoustics: An introduction to its Physical Principles and
    Applications ; Pierce, Allan D., Acoustical Society of America,
    1989.
2.  Prediction and Control of the Interior Pressure Fluctuations in a
    Flow-excited Helmholtz resonator ; Mongeau, Luc, and Hyungseok
    Kook., Ray W. Herrick Laboratories, Purdue University, 1997.
3.  Influence of leakage on the flow-induced response of vehicles with
    open sunroofs ; Mongeau, Luc, and Jin-Seok Hong., Ray W. Herrick
    Laboratories, Purdue University.
4.  Fluid dynamics of a flow excited resonance, part I: Experiment ;
    P.A. Nelson, Halliwell and Doak.; 1991.
5.  An Introduction to Acoustics ; Rienstra, S.W., A. Hirschberg.,
    Report IWDE 99--02, Eindhoven University of Technology, 1999.


# Acoustics/Active Control

![](Acoustics_active_control.JPG "Acoustics_active_control.JPG")

## Introduction

The principle of active control of noise, is to create destructive
interferences using a secondary source of noise. Thus, any noise can
theoretically disappear. But as we will see in the following sections,
only low frequencies noises can be reduced for usual applications, since
the amount of secondary sources required increases very quickly with
frequency. Moreover, predictable noises are much easier to control than
unpredictable ones. The reduction can reach up to 20 dB for the best
cases. But since good reduction can only be reached for low frequencies,
the perception we have of the resulting sound is not necessarily as good
as the theoretical reduction. This is due to psychoacoustics
considerations, which will be discussed later on.

## Fundamentals of active control of noise

### Control of a monopole by another monopole

Even for the free space propagation of an acoustic wave created by a
punctual source it is difficult to reduce noise in a large area, using
active noise control, as we will see in the section.

In the case of an acoustic wave created by a monopolar source, the
Helmholtz equation becomes:

```{=html}
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```
$\Delta p + k^2 p =  - j\omega \rho _0 q$

```{=html}
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```
where q is the flow of the noise sources.

The solution for this equation at any M point is:

```{=html}
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```
$p_p (M) = \frac{{j\omega \rho _0 q_p }}{{4\pi }}\frac{{e^{ - jkr_p } }}{{r_p }}$

```{=html}
</center>
```
where the p mark refers to the primary source.

Let us introduce a secondary source in order to perform active control
of noise. The acoustic pressure at that same M point is now:

```{=html}
<center>
```
${\rm{p(M) = }}\frac{{{\rm{j}}\omega \rho _{\rm{0}} q_p }}{{4\pi }}\frac{{e^{ - jkr_p } }}{{r_p }} + \frac{{{\rm{j}}\omega \rho _{\rm{0}} q_s }}{{4\pi }}\frac{{e^{ - jkr_s } }}{{r_s }}$

```{=html}
</center>
```
It is now obvious that if we chose
$q_s  =  - q_p \frac{{r_s }}{{r_p }}e^{ - jk(r_p  - r_s )}$ there is no
more noise at the M point. This is the most simple example of active
control of noise. But it is also obvious that if the pressure is zero in
M, there is no reason why it should also be zero at any other N point.
This solution only allows to reduce noise in one very small area.

However, it is possible to reduce noise in a larger area far from the
source, as we will see in this section. In fact the expression for
acoustic pressure far from the primary source can be approximated by:

```{=html}
<center>
```
$p(M) = \frac{{j\omega \rho _0 }}{{4\pi }}\frac{{e^{ - jkr_p } }}{{r_p }}(q_p  + q_s e^{ - jkD\cos \theta } )$

```{=html}
</center>
```
```{=html}
<center>
```
!Control of a monopole by another
monopole

```{=html}
</center>
```
As shown in the previous section we can adjust the secondary source in
order to get no noise in M. In that case, the acoustic pressure in any
other N point of the space remains low if the primary and secondary
sources are close enough. More precisely, it is possible to have a
pressure close to zero in the whole space if the M point is equally
distant from the two sources and if: $D < \lambda /6$ where D is the
distance between the primary and secondary sources. As we will see later
on, it is easier to perform active control of noise with more than on
source controlling the primary source, but it is of course much more
expensive.

A commonly admitted estimation of the number of secondary sources which
are necessary to reduce noise in an R radius sphere, at a frequency f
is:

```{=html}
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```
$N = \frac{{36\pi R^2 f^2 }}{{c^2 }}$

```{=html}
</center>
```
This means that if you want no noise in a one meter diameter sphere at a
frequency below 340 Hz, you will need 30 secondary sources. This is the
reason why active control of noise works better at low frequencies.

### Active control for waves propagation in ducts and enclosures

This section requires from the reader to know the basis of modal
propagation theory, which will not be explained in this article.

#### Ducts

For an infinite and straight duct with a constant section, the pressure
in areas without sources can be written as an infinite sum of
propagation modes:

```{=html}
<center>
```
$p(x,y,z,\omega ) = \sum\limits_{n = 1}^N {a_n (\omega )\phi _n (x,y)e^{ - jk_n z} }$

```{=html}
</center>
```
where $\phi$ are the eigen functions of the Helmoltz equation and a
represent the amplitudes of the modes.

The eigen functions can either be obtained analytically, for some
specific shapes of the duct, or numerically. By putting pressure sensors
in the duct and using the previous equation, we get a relation between
the pressure matrix P (pressure for the various frequencies) and the A
matrix of the amplitudes of the modes. Furthermore, for linear sources,
there is a relation between the A matrix and the U matrix of the signal
sent to the secondary sources: $A_s  = KU$ and hence:
$A = A_p  + A_s  = A_p  + KU$.

Our purpose is to get: A=0, which means: $A_p  + KU = 0$. This is
possible every time the rank of the K matrix is bigger than the number
of the propagation modes in the duct.

Thus, it is theoretically possible to have no noise in the duct in a
very large area not too close from the primary sources if the there are
more secondary sources than propagation modes in the duct. Therefore, it
is obvious that active noise control is more appropriate for low
frequencies. In fact the more the frequency is low, the less propagation
modes there will be in the duct. Experiences show that it is in fact
possible to reduce the noise from over 60 dB.

#### Enclosures

The principle is rather similar to the one described above, except the
resonance phenomenon has a major influence on acoustic pressure in the
cavity. In fact, every mode that is not resonant in the considered
frequency range can be neglected. In a cavity or enclosure, the number
of these modes rise very quickly as frequency rises, so once again, low
frequencies are more appropriate. Above a critical frequency, the
acoustic field can be considered as diffuse. In that case, active
control of noise is still possible, but it is theoretically much more
complicated to set up.

### Active control and psychoacoustics

As we have seen, it is possible to reduce noise with a finite number of
secondary sources. Unfortunately, the perception of sound of our ears
does not only depend on the acoustic pressure (or the decibels). In
fact, it sometimes happen that even though the number of decibels has
been reduced, the perception that we have is not really better than
without active control.

## Active control systems

Since the noise that has to be reduced can never be predicted exactly, a
system for active control of noise requires an auto adaptable algorithm.
We have to consider two different ways of setting up the system for
active control of noise depending on whether it is possible or not to
detect the noise from the primary source before it reaches the secondary
sources. If this is possible, a feed forward technique will be used
(aircraft engine for example). If not a feed back technique will be
preferred.

### Feedforward

In the case of a feed forward, two sensors and one secondary source are
required. The sensors measure the sound pressure at the primary source
(detector) and at the place we want noise to be reduced (control
sensor). Furthermore, we should have an idea of what the noise from the
primary source will become as he reaches the control sensor. Thus we
approximately know what correction should be made, before the sound wave
reaches the control sensor (forward). The control sensor will only
correct an eventual or residual error. The feedforward technique allows
to reduce one specific noise (aircraft engine for example) without
reducing every other sound (conversations, ...). The main issue for this
technique is that the location of the primary source has to be known,
and we have to be sure that this sound will be detected beforehand.
Therefore portative systems based on feed forward are impossible since
it would require having sensors all around the head.

```{=html}
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!Feedforward System

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</center>
```
### Feedback

In that case, we do not exactly know where the sound comes from; hence
there is only one sensor. The sensor and the secondary source are very
close from each other and the correction is done in real time: as soon
as the sensor gets the information the signal is treated by a filter
which sends the corrected signal to the secondary source. The main issue
with feedback is that every noise is reduced and it is even
theoretically impossible to have a standard conversation.

```{=html}
<center>
```
!Feedback System

```{=html}
</center>
```
## Applications

### Noise cancelling headphone

Usual headphones become useless when the frequency gets too low. As we
have just seen active noise cancelling headphones require the feedback
technique since the primary sources can be located all around the head.
This active control of noise is not really efficient at high frequencies
since it is limited by the Larsen effect. Noise can be reduced up to
30 dB at a frequency range between 30 Hz and 500 Hz.

### Active control for cars

Noise reduction inside cars can have a significant impact on the comfort
of the driver. There are three major sources of noise in a car: the
motor, the contact of tires on the road, and the aerodynamic noise
created by the air flow around the car. In this section, active control
for each of those sources will be briefly discussed.

#### Motor noise

This noise is rather predictable since it a consequence of the rotation
of the pistons in the motor. Its frequency is not exactly the motor's
rotational speed though. However, the frequency of this noise is in
between 20 Hz and 200 Hz, which means that an active control is
theoretically possible. The following pictures show the result of an
active control, both for low and high regime.

```{=html}
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!Low regime

```{=html}
</center>
```
Even though these results show a significant reduction of the acoustic
pressure, the perception inside the car is not really better with this
active control system, mainly for psychoacoustics reasons which were
mentioned above. Moreover such a system is rather expensive and thus are
not used in commercial cars.

#### Tires noise

This noise is created by the contact between the tires and the road. It
is a broadband noise which is rather unpredictable since the mechanisms
are very complex. For example, the different types of roads can have a
significant impact on the resulting noise. Furthermore, there is a
cavity around the tires, which generate a resonance phenomenon. The
first frequency is usually around 200 Hz. Considering the multiple
causes for that noise and its unpredictability, even low frequencies
become hard to reduce. But since this noise is broadband, reducing low
frequencies is not enough to reduce the overall noise. In fact an active
control system would mainly be useful in the case of an unfortunate
amplification of a specific mode.

#### Aerodynamic noise

This noise is a consequence of the interaction between the air flow
around the car and the different appendixes such as the rear views for
example. Once again, it is an unpredictable broadband noise, which makes
it difficult to reduce with an active control system. However, this
solution can become interesting in the case an annoying predictable
resonance would appear.

### Active control for aeronautics

The noise of aircraft propellers is highly predictable since the
frequency is quite exactly the rotational frequency multiplied by the
number of blades. Usually this frequency is around some hundreds of Hz.
Hence, an active control system using the feedforward technique provides
very satisfying noise reductions. The main issues are the cost and the
weigh of such a system. The fan noise on aircraft engines can be reduced
in the same manner.

## Further reading

-   \"Active Noise
    Control\"
    at Dirac delta.


# Acoustics/Rotor Stator interactions

![](Acoustics_RotorStatorInteractions_2.JPG "Acoustics_RotorStatorInteractions_2.JPG")

An important issue for the aeronautical industry is the reduction of
aircraft noise. The characteristics of the turbomachinery noise are to
be studied. The rotor/stator interaction is a predominant part of the
noise emission. We will present an introduction to these interaction
theory, whose applications are numerous. For example, the conception of
air-conditioning ventilators requires a full understanding of this
interaction.

## Noise emission of a Rotor-Stator mechanism

A Rotor wake induces on the downstream Stator blades a fluctuating vane
loading, which is directly linked to the noise emission.

We consider a B blades Rotor (at a rotation speed of $\Omega$) and a V
blades stator, in a unique Rotor/Stator configuration. The source
frequencies are multiples of $B \Omega$, that is to say $mB \Omega$. For
the moment we don't have access to the source levels $F_{m}$. The noise
frequencies are also $mB \Omega$, not depending on the number of blades
of the stator. Nevertheless, this number V has a predominant role in the
noise levels ($P_{m}$) and directivity, as it will be discussed later.

*Example*

*For an airplane air-conditioning ventilator, reasonable data are :*

*$B=13$ and $\Omega = 12000$ rnd/min*

*The blade passing frequency is 2600 Hz, so we only have to include the
first two multiples (2600 Hz and 5200 Hz), because of the human ear
high-sensibility limit. We have to study the frequencies m=1 and m=2.*

## Optimization of the number of blades

As the source levels can\'t be easily modified, we have to focus on the
interaction between those levels and the noise levels.

The transfer function ${{F_m } \over {P_m }}$ contains the following
part :

```{=html}
<center>
```
$\sum\limits_{s =  - \infty }^{s =  + \infty } {e^{ - {{i(mB - sV)\pi } \over 2}} J_{mB - sV} } (mBM)$

```{=html}
</center>
```
Where m is the Mach number and $J_{mB - sV}$ the Bessel function of
mB-sV order. In order to minimize the influence of the transfer
function, the goal is to reduce the value of this Bessel function. To do
so, the argument must be smaller than the order of the Bessel function.

*Back to the example :*

*For m=1, with a Mach number M=0.3, the argument of the Bessel function
is about 4. We have to avoid having mB-sV inferior than 4. If V=10, we
have 13-1x10=3, so there will be a noisy mode. If V=19, the minimum of
mB-sV is 6, and the noise emission will be limited.*

*Remark :*

*The case that is to be strictly avoided is when mB-sV can be nul, which
causes the order of the Bessel function to be 0. As a consequence, we
have to take care having B and V prime numbers.*

## Determination of source levels

The minimization of the transfer function ${{F_m } \over {P_m }}$ is a
great step in the process of reducing the noise emission. Nevertheless,
to be highly efficient, we also have to predict the source levels
$F_{m}$. This will lead us to choose to minimize the Bessel functions
for the most significant values of m. For example, if the source level
for m=1 is very higher than for m=2, we will not consider the Bessel
functions of order 2B-sV. The determination of the source levels is
given by the Sears theory, which will not be discussed here.

## Directivity

All this study was made for a specific direction : the axis of the
Rotor/Stator. All the results are acceptable when the noise reduction is
ought to be in this direction. In the case where the noise to reduce is
perpendicular to the axis, the results are very different, as those
figures shown :

For B=13 and V=13, which is the worst case, we see that the sound level
is very high on the axis (for $\theta = 0$)

```{=html}
<center>
```
![](Acoustics_1313.JPG "Acoustics_1313.JPG")

```{=html}
</center>
```
For B=13 and V=19, the sound level is very low on the axis but high
perpendicularly to the axis (for $\theta = Pi/2$)

```{=html}
<center>
```
![](Acoustics_1319.jpg "Acoustics_1319.jpg")

```{=html}
</center>
```
## Further reading

This module discusses rotor/stator interaction, the predominant part of
the noise emission of turbomachinery. See Acoustics/Noise from Cooling
Fans for a discussion of
other noise sources.

## External references

-   Prediction of rotor wake-stator interaction noise by P. Sijtsma and
    J.B.H.M.
    Schulten


# Acoustics/Car Mufflers

![](Acoustics_car_muflers.JPG "Acoustics_car_muflers.JPG")

## Introduction

A car muffler is a component of the exhaust system of a car. The exhaust
system has mainly 3 functions:

1.  Getting the hot and noxious gas from the engine away from the
    vehicle
2.  Reduce exhaust emission
3.  Attenuating the noise output from the engine

The last specified function is the function of the car muffler. It is
necessary because the gas coming from the combustion in the pistons of
the engine would generate an extremely loud noise if it were sent
directly in the ambient air surrounding engine through the exhaust
valves. There are 2 techniques used to dampen the noise: absorption and
reflection. Each technique has its advantages and disadvantages.

!Muffler type \"Cherry
bomb\"\|right{width="150"}

## The absorber muffler

The muffler is composed of a tube covered by sound absorbing material.
The tube is perforated so that some part of the sound wave goes through
the perforation to the absorbing material. The absorbing material is
usually made of fiberglass or steel wool. The dampening material is
protected from the surrounding by a supplementary coat made of a bend
metal sheet.

The advantage of this method is low back pressure with a relatively
simple design. The inconvenience of this method is low sound damping
ability compared to the other techniques, especially at low frequency.

The mufflers using the absorption technique are usually sports vehicle
because they increase the performances of the engine because of their
low back pressure. A trick to improve their
muffling ability consists of lining up several \"straight\" mufflers.

## The reflector muffler

Principle: Sound wave reflection is used to create a maximum amount of
destructive interferences

!Destructive
interference\|right{width="400"}

### Definition of destructive interferences

Let\'s consider the noise a person would hear when a car drives past.
This sound would physically correspond to the pressure variation of the
air which would make his ear-drum vibrate. The curve A1 of the graph 1
could represent this sound. The pressure amplitude is a function of the
time at a certain fixed place. If another sound wave A2 is produced at
the same time, the pressure of the two waves will add. If the amplitude
of A1 is exactly the opposite of the amplitude A2, then the sum will be
zero, which corresponds physically to the atmospheric pressure. The
listener would thus hear nothing although there are two radiating sound
sources. A2 is called the destructive interference.

!Wave
reflection\|right{width="250"}

### Definition of the reflection

The sound is a traveling wave i.e. its position changes in function of
the time. As long as the wave travels in the same medium, there is no
change of speed and amplitude. When the wave reaches a frontier between
two mediums which have different impedances, the speed, and the pressure
amplitude change (and so does the angle if the wave does not propagate
perpendicularly to the frontier). The figure 1 shows two medium A and B
and the 3 waves: incident transmitted and reflected.

### Example

If plane sound waves are propagating across a tube and the section of
the tube changes, the impedance of the tube will change. Part of the
incident waves will be transmitted through the discontinuity and the
other part will be reflected.

Animation

Mufflers using the reflection technique are the most commonly used
because they dampen the noise much better than the absorber muffler.
However they induce a higher back pressure,
lowering the performance of the engine (an engine would be most
efficient or powerful without the use of a muffler).

!Schema{width="400"}

The upper right image represents a car muffler\'s typical architecture.
It is composed of 3 tubes. There are 3 areas separated by plates, the
part of the tubes located in the middle area are perforated. A small
quantity of pressure \"escapes\" from the tubes through the perforation
and cancel one another.

Some mufflers using the reflection principle may incorporate cavities
which dampen noise. These cavities are called Helmholtz
Resonators
in acoustics. This feature is usually only available for up market class
mufflers.

![](Muffler_resonator.png "Muffler_resonator.png"){width="300"}

## Back pressure

Car engines are 4 stroke cycle engines. Out of these 4 strokes, only one
produces the power, this is when the explosion occurs and pushes the
pistons back. The other 3 strokes are necessary evil that don\'t produce
energy. They on the contrary consume energy. During the exhaust stroke,
the remaining gas from the explosion is expelled from the cylinder. The
higher the pressure behind the exhaust valves (i.e. back pressure), the
higher the effort necessary to expel the gas out of the cylinder. So, a
low back pressure is preferable in order to have a higher engine
horsepower.

## Muffler modeling by transfer matrix method

This method is easy to use on computer to obtain theoretical values for
the transmission loss of a muffler. The transmission loss gives a value
in dB that correspond to the ability of the muffler to dampen the noise.

### Example

!Muffler working with waves
reflections{width="500"}

P stands for Pressure \[Pa\] and U stand for volume velocity \[m3/s\]

$\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=$\begin{bmatrix}  T1 \end{bmatrix}
\begin{bmatrix} P2 \\ U2 \end{bmatrix}$ and
$\begin{bmatrix} P2 \\ U2 \end{bmatrix}$=$\begin{bmatrix}  T2 \end{bmatrix}
\begin{bmatrix} P3 \\ U3 \end{bmatrix}$ and
$\begin{bmatrix} P3 \\ U3 \end{bmatrix}$=$\begin{bmatrix}  T3 \end{bmatrix}
\begin{bmatrix} P4 \\ U4 \end{bmatrix}$

So, finally: $\begin{bmatrix} P1 \\ U1 \end{bmatrix}$=
$\begin{bmatrix} T1 \end{bmatrix}
\begin{bmatrix} T2 \end{bmatrix}
\begin{bmatrix} T3 \end{bmatrix}
\begin{bmatrix} P4 \\ U4 \end{bmatrix}$

with

$\begin{bmatrix} T_i \end{bmatrix}$=$\begin{bmatrix}  cos (k L_i) & j sin (k L_i) \frac{\rho c}{S_i}  \\ j sin (k L_i) \frac{S_i}{\rho c} & cos (k L_i) \end{bmatrix}$

Si stands for the cross section area

k is the wave
number

$\ \rho$ is the medium density

c is the speed of sound of the medium

### Results

!Schema{width="400"}

https://commons.wikimedia.org/wiki/File:Transmission_loss.png#Source_code
Matlab
code
of the graph above.

### Comments

The higher the value of the transmission loss and the better the
muffler.

The transmission loss depends on the frequency. The sound frequency of a
car engine is approximately between 50 and 3000 Hz. At resonance
frequencies, the transmission loss is zero. These frequencies correspond
to the lower peaks on the graph.

The transmission loss is independent of the applied pressure or velocity
at the input.

The temperature (about 600 Fahrenheit) has an impact on the air
properties : the speed of sound is higher and the mass density is lower.

The elementary transfer matrix depends on the element which is modelled.
For instance the transfer matrix of a Helmotz Resonator is
$\begin{bmatrix} 1 & 0 \\ \frac{1}{Z} & 1 \end{bmatrix}$ with
$\ Z = j \rho ( \frac{\omega L_i}{S_i} - \frac{c^2}{\omega V})$

The transmission loss and the insertion loss are different terms. The
transmission loss is 10 times the logarithm of the ratio output/input.
The insertion loss is 10 times the logarithm of the ratio of the
radiated sound power with and without muffler.

## Links

-   General Information about Filter Design &
    Implementation
-   More information about the Transfer Matrix
    Method
-   General information about car
    mufflers


# Acoustics/Sonic Boom

![](Acoustics_sonic_boom.JPG "Acoustics_sonic_boom.JPG")

!Warplane passing the sound
barrier.

A **sonic boom** is the audible component of a shock wave in air. The
term is commonly used to refer to the air shocks caused by the
supersonic flight of military aircraft or passenger transports such as
Concorde (Mach 2.2, no longer flying) and the Space Shuttle (Mach 27).
Sonic booms generate enormous amounts of sound energy, sounding much
like an explosion; typically the shock front may approach 100 megawatts
per square meter, and may exceed 200 decibels.

!When an aircraft is near the sound barrier, an unusual cloud sometimes
forms in its wake. A Prandtl-Glauert Singularity results from a drop in
pressure, due to shock wave formation. This pressure change causes a
sharp drop in temperature, which in humid conditions leads the water
vapor in the air to condense into droplets and form the
cloud._-_filtered.jpg "When an aircraft is near the sound barrier, an unusual cloud sometimes forms in its wake. A Prandtl-Glauert Singularity results from a drop in pressure, due to shock wave formation. This pressure change causes a sharp drop in temperature, which in humid conditions leads the water vapor in the air to condense into droplets and form the cloud."){width="200"}

## Cause of sonic booms

As an object moves through the air it creates a series of pressure waves
in front and behind it, similar to the bow and stern waves created by a
boat. These waves travel at the speed of sound, and as the speed of the
aircraft increases the waves are forced together or \'compressed\'
because they cannot \"get out of the way\" of each other, eventually
merging into a single shock wave at the speed of sound. This critical
speed is known as Mach 1 and is 1,225 km/h (761 mph) at sea level.

In smooth flight, the shock wave starts at the nose of the aircraft and
ends at the tail. There is a sudden increase in pressure at the nose,
decreasing steadily to a negative pressure at the tail, where it
suddenly returns to normal. This \"overpressure profile\" is known as
the N-wave due to its shape. We experience the \"boom\" when there is a
sudden increase in pressure, so the N-wave causes two booms, one when
the initial pressure rise from the nose hits, and another when the tail
passes and the pressure suddenly returns to normal. This leads to a
distinctive \"double boom\" from supersonic aircraft. When maneuvering
the pressure; distribution changes into different forms, with a
characteristic U-wave shape. Since the boom is being generated
continually as long as the aircraft is supersonic, it traces out a path
on the ground following the aircraft\'s flight path, known as the **boom
carpet**. frame\|A cage around the engine reflects any shock waves. A
spike behind the engine converts them into
thrust. frame\|To
generate lift a supersonic airplane has to produce at least two shock
waves: One over-pressure downwards wave, and one under-pressure upwards
wave. Whitcomb area rule states, we can reuse air displacement without
generating additional shock waves. In this case the fuselage reuses some
displacement of the
wings.

A sonic boom or \"tunnel boom\" can also be caused by high-speed trains
in tunnels (e.g. the Japanese Shinkansen). In order to reduce the sonic
boom effect, a special shape of the train car and a widened opening of
the tunnel entrance is necessary. When a high speed train enters a
tunnel, the sonic boom effect occurs at the tunnel exit. In contrast to
the (super)sonic boom of an aircraft, this \"tunnel boom\" is caused by
a rapid change of subsonic flow (due to the sudden narrowing of the
surrounding space) rather than by a shock wave. In close range to the
tunnel exit this phenomenon can causes disturbances to residents.

## Characteristics

The power, or volume, of the shock wave is dependent on the quantity of
air that is being accelerated, and thus the size and weight of the
aircraft. As the aircraft increases speed the shocks grow \"tighter\"
around the craft, and do not become much \"louder\". At very high speeds
and altitudes the cone does not intersect the ground, and no boom will
be heard. The \"length\" of the boom from front to back is dependent on
the length of the aircraft, although to a factor of 3:2 not 1:1. Longer
aircraft therefore \"spread out\" their booms more than smaller ones,
which leads to a less powerful boom.

The nose shockwave compresses and pulls the air along with the aircraft
so that the aircraft behind its shockwave sees subsonic airflow.

However, this means that several smaller shock waves can, and usually
do, form at other points on the aircraft, primarily any convex points or
curves, the leading wing edge and especially the inlet to engines. These
secondary shockwaves are caused by the subsonic air behind the main
shockwave being forced to go supersonic again by the shape of the
aircraft (for example due to the air\'s acceleration over the top of a
curved wing).

The later shock waves are somehow faster than the first one, travel
faster and add to the main shockwave at some distance away from the
aircraft to create a much more defined N-wave shape. This maximizes both
the magnitude and the \"rise time\" of the shock, which makes it seem
louder. On most designs the characteristic distance is about 40,000 ft,
meaning that below this altitude the sonic boom will be \"softer\".
However the drag at this altitude or below makes supersonic travel
particularly inefficient, which poses a serious problem.

## Abatement

In the late 1950s when SST designs were being actively pursued, it was
thought that although the boom would be very large, they could avoid
problems by flying higher. This premise was proven false when the North
American B-70 *Valkyrie* started flying and it was found that the boom
was a very real problem even at 70,000 ft (21,000m). It was during these
tests that the N-wave was first characterized.

Richard Seebass and his colleague Albert George at Cornell University
studied the problem extensively, and eventually defined a \"figure of
merit\", **FM**, to characterize the sonic boom levels of different
aircraft. FM is proportional to the aircraft weight divided by the
three-halves of the aircraft length, FM = W/(3/2·L) = 2W/3L. The lower
this value, the less boom the aircraft generates, with figures of about
1 or lower being considered acceptable. Using this calculation they
found FM\'s of about 1.4 for Concorde, and 1.9 for the Boeing 2707. This
eventually doomed most SST projects as public resentment, somewhat blown
out of proportion, mixed with politics eventually resulted in laws that
made any such aircraft impractical (flying only over water for
instance).

Seebass-George also worked on the problem from another angle, examining
ways to reduce the \"peaks\" of the N-wave and therefore smooth out the
shock into something less annoying. Their theory suggested that body
shaping might be able to use the secondary shocks to either \"spread
out\" the N-wave, or interfere with each other to the same end. Ideally
this would raise the characteristic altitude from 40,000 ft to 60,000,
which is where most SST designs fly. The design required some fairly
sophisticated shaping in order to achieve the dual needs of reducing the
shock and still leaving an aerodynamically efficient shape, and
therefore had to wait for the advent of computer-aided design before
being able to be built.

This remained untested for decades, until DARPA started the **Quiet
Supersonic Platform** project and funded the **Shaped Sonic Boom
Demonstration** aircraft to test it. SSBD used a F-5 Freedom Fighter
modified with a new body shape, and was tested over a two year period in
what has become the most extensive study on the sonic boom to date.
After measuring the 1,300 recordings, some taken inside the shock wave
by a chase plane, the SSBD demonstrated a reduction in boom by about
one-third. Although one-third is not a huge reduction, it could reduce
Concorde below the FM = 1 limit for instance.

There are theoretical designs that do not appear to create sonic booms
at all, such as the Busemann\'s Biplane. Nobody has been able to suggest
a practical implementation of this concept, as yet.

## Perception and noise

The sound of a sonic boom depends largely on the distance between the
observer and the aircraft producing the sonic boom. A sonic boom is
usually heard as a deep double \"boom\" as the aircraft is usually some
distance away. However, as those who have witnessed landings of space
shuttles have heard, when the aircraft is nearby the sonic boom is a
sharper \"bang\" or \"crack\". The sound is much like the \"aerial
bombs\" used at firework displays.

In 1964, NASA and the FAA began the Oklahoma City sonic boom tests,
which caused eight sonic booms per day over a period of six months.
Valuable data was gathered from the experiment, but 15,000 complaints
were generated and ultimately entangled the government in a class action
lawsuit, which it lost on appeal in 1969.

In late October 2005, Israel began using nighttime sonic boom raids
against civilian populations in the Gaza Strip
1 as a
method of psychological warfare. The practice was condemned by the
United Nations. A senior Israeli army intelligence source said the
tactic was intended to break civilian support for armed Palestinian
groups.

## Media

These videos include jets achieving supersonic speeds.

First supersonic
flight
(info)

:   Chuck Yeager broke the sound barrier on October 14, 1947 in the Bell
    X-1.

F-14 Tomcat sonic boom flyby (with
audio)
(info)

:   F-14 Tomcat flies at Mach 1 over the water, creating a sonic boom as
    it passes.

F-14A Tomcat supersonic
flyby
(info)

:   Supersonic F-14A Tomcat flying by the USS Theodore Roosevelt CVN-71
    in 1986 for the tiger cruise.

Shuttle passes sound
barrier
(info)

:   Space shuttle Columbia crosses the sound barrier at 45 seconds after
    liftoff.

## External links

-   NASA opens new chapter in supersonic
    flight
-   \"Sonic
    Boom,\" a
    tutorial from the \"Sonic Boom, Sound Barrier, and Condensation
    Clouds\"
    (or \"Sonic Boom, Sound Barrier, and Prandtl-Glauert Condensation
    Clouds\") collection of tutorials by Mark S. Cramer, Ph.D. at
    <http://FluidMech.net> (Tutorials, Sound Barrier).
-   decibel chart including sonic
    booms


# Acoustics/Sonar

**SONAR** (**so**und **n**avigation **a**nd **r**anging) is a technique
that uses sound propagation under water to navigate or to detect other
vessels. There are two kinds of sonar: active and passive.

## History

The French physicist Paul Langevin, working with a Russian émigré
electrical engineer, Constantin Chilowski, invented the first active
sonar-type device for detecting submarines in 1915. Although
piezoelectric transducers later superseded the electrostatic transducers
they used, their work influenced the future of sonar designs. In 1916,
under the British Board of Inventions and Research, Canadian physicist
Robert Boyle took on the project, which subsequently passed to the
**Anti-** (or **Allied**) **Submarine Detection Investigation
Committee**, producing a prototype for testing in mid-1917, hence the
British acronym **ASDIC**.

By 1918, both the U.S. and Britain had built active systems. The UK
tested what they still called ASDIC on *HMS Antrim* in 1920, and started
production of units in 1922. The 6th Destroyer Flotilla had
ASDIC-equipped vessels in 1923. An anti-submarine school, *HMS Osprey*,
and a training flotilla of four vessels were established on Portland in
1924.

The U.S. Sonar QB set arrived in 1931. By the outbreak of World War II,
the Royal Navy had five sets for different surface ship classes, and
others for submarines. The greatest advantage came when it was linked to
the Squid anti-submarine weapon.

## Active sonar

Active sonar creates a pulse of sound, often called a \"ping\", and then
listens for reflections of the pulse. To measure the distance to an
object, one measures the time from emission of a pulse to reception. To
measure the bearing, one uses several hydrophones, and measures the
relative arrival time to each in a process called beamforming.

The pulse may be at constant frequency or a chirp of changing frequency.
For a chirp, the receiver correlates the frequency of the reflections to
the known chirp. The resultant processing gain allows the receiver to
derive the same information as if a much shorter pulse of the same total
energy were emitted. In practice, the chirp signal is sent over a longer
time interval; therefore the instantaneous emitted power will be
reduced, which simplifies the design of the transmitter. In general,
long-distance active sonars use lower frequencies. The lowest have a
bass \"BAH-WONG\" sound.

The most useful small sonar looks roughly like a waterproof flashlight.
One points the head into the water, presses a button, and reads a
distance. Another variant is a \"fishfinder\" that shows a small display
with shoals of fish. Some civilian sonars approach active military
sonars in capability, with quite exotic three-dimensional displays of
the area near the boat. However, these sonars are not designed for
stealth.

When active sonar is used to measure the distance to the bottom, it is
known as echo sounding.

Active sonar is also used to measure distance through water between two
sonar transponders. A transponder is a device that can transmit and
receive signals but when it receives a specific interrogation signal it
responds by transmitting a specific reply signal. To measure distance,
one transponder transmits an interrogation signal and measures the time
between this transmission and the receipt of the other transponder\'s
reply. The time difference, scaled by the speed of sound through water
and divided by two, is the distance between the two transponders. This
technique, when used with multiple transponders, can calculate the
relative positions of static and moving objects in water.

### Analysis of active sonar data

Active sonar data is obtained by measuring detected sound for a short
period of time after the issuing of a ping; this time period is selected
so as to ensure that the ping\'s reflection will be detected. The
distance to the seabed (or other acoustically reflective object) can be
calculated from the elapsed time between the ping and the detection of
its reflection. Other properties can also be detected from the shape of
the ping\'s reflection:

-   When collecting data on the seabed, some of the reflected sound will
    typically reflect off the air-water interface, and then reflect off
    the seabed a second time. The size of this second echo provides
    information about the acoustic hardness of the seabed.
-   The roughness of a seabed affects the variance in reflection time.
    For a smooth seabed, all of the reflected sound will take much the
    same path, resulting in a sharp spike in the data. For a rougher
    seabed, sound will be reflected back over a larger area of seabed,
    and some sound may bounce between seabed features before reflecting
    to the surface. A less sharp spike in the data therefore indicates a
    rougher seabed.

### Sonar and marine animals

Some marine animals, such as whales and dolphins, use echolocation
systems similar to active sonar to locate predators and prey. It is
feared that sonar transmitters could confuse these animals and cause
them to lose their way, perhaps preventing them from feeding and mating.
A recent article on the BBC Web site (see below) reports findings
published in the journal *Nature* to the effect that military sonar may
be inducing some whales to experience decompression sickness (and
resultant beachings).

High-powered sonar transmitters may indirectly harm marine animals,
although scientific evidence suggests that a confluence of factors must
first be present. In the Bahamas in 2000, a trial by the United States
Navy of a 230 decibel transmitter in the frequency range 3 -- 7 kHz
resulted in the beaching of sixteen whales, seven of which were found
dead. The Navy accepted blame in a report published in the Boston Globe
on 1 January 2002. However, at low powers, sonar can protect marine
mammals against collisions with ships.

A kind of sonar called mid-frequency sonar has been correlated with mass
cetacean strandings throughout the world's oceans, and has therefore
been singled out by environmentalists as causing the death of marine
mammals. International press coverage of these events can be found at
this active sonar news clipping Web
site. A lawsuit was filed in Santa Monica, California on 19 October 2005
contending that the U.S. Navy has conducted sonar exercises in violation
of several environmental laws, including the National Environmental
Policy Act, the Marine Mammal Protection Act, and the Endangered Species
Act.

## Passive sonar

Passive sonar listens without transmitting. It is usually employed in
military settings, although a few are used in science applications.

### Speed of sound

Sonar operation is affected by sound speed. Sound speed is slower in
fresh water than in sea water. In all water sound velocity is affected
by density (or the mass per unit of volume). Density is affected by
temperature, dissolved molecules (usually salinity), and pressure. The
speed of sound (in feet per second) is approximately equal to 4388 +
(11.25 × temperature (in °F)) + (0.0182 × depth (in feet) + salinity (in
parts-per-thousand)). This is an empirically derived approximation
equation that is reasonably accurate for normal temperatures,
concentrations of salinity and the range of most ocean depths. Ocean
temperature varies with depth, but at between 30 and 100 metres there is
often a marked change, called the thermocline, dividing the warmer
surface water from the cold, still waters that make up the rest of the
ocean. This can frustrate sonar, for a sound originating on one side of
the thermocline tends to be bent, or refracted, off the thermocline. The
thermocline may be present in shallower coastal waters, however, wave
action will often mix the water column and eliminate the thermocline.
Water pressure also affects sound propagation. Increased pressure
increases the density of the water and raises the sound velocity.
Increases in sound velocity cause the sound waves to refract away from
the area of higher velocity. The mathematical model of refraction is
called Snell\'s law.

Sound waves that are radiated down into the ocean bend back up to the
surface in great arcs due to the effect of pressure on sound. The ocean
must be at least 6000 feet (1850 meters) deep, or the sound waves will
echo off the bottom instead of refracting back upwards. Under the right
conditions these waves will then be focused near the surface and
refracted back down and repeat another arc. Each arc is called a
convergence zone. Where an arc intersects the surface a CZ annulus is
formed. The diameter of the CZ depends on the temperature and salinity
of the water. In the North Atlantic, for example, CZs are found
approximately every 33 nautical miles (61 km), depending on the season,
forming a pattern of concentric circles around the sound source. Sounds
that can be detected for only a few miles in a direct line can therefore
also be detected hundreds of miles away. Typically the first, second and
third CZ are fairly useful; further out than that the signal is too
weak, and thermal conditions are too unstable, reducing the reliability
of the signals. The signal is naturally attenuated by distance, but
modern sonar systems are very sensitive.

### Identifying sound sources

Military sonar has a wide variety of techniques for identifying a
detected sound. For example, U.S. vessels usually operate 60 Hz
alternating current power systems. If transformers are mounted without
proper vibration insulation from the hull, or flooded, the 60 Hz sound
from the windings and generators can be emitted from the submarine or
ship, helping to identify its nationality. In contrast, most European
submarines have 50 Hz power systems. Intermittent noises (such as a
wrench being dropped) may also be detectable to sonar.

Passive sonar systems may have large sonic databases, however most
classification is performed manually by the sonar operator. A computer
system frequently uses these databases to identify classes of ships,
actions (i.e., the speed of a ship, or the type of weapon released), and
even particular ships. Publications for classification of sounds are
provided by and continually updated by the U.S. Office of Naval
Intelligence.

### Sonar in warfare

Modern naval warfare makes extensive use of sonar. The two types
described before are both used, but from different platforms, i.e.,
types of water-borne vessels.

Active sonar is extremely useful, since it gives the exact position of
an object. Active sonar works the same way as radar: a signal is
emitted. The sound wave then travels in many directions from the
emitting object. When it hits an object, the sound wave is then
reflected in many other directions. Some of the energy will travel back
to the emitting source. The echo will enable the sonar system or
technician to calculate, with many factors such as the frequency, the
energy of the received signal, the depth, the water temperature, etc.,
the position of the reflecting object. Using active sonar is somewhat
hazardous however, since it does not allow the sonar to identify the
target, and any vessel around the emitting sonar will detect the
emission. Having heard the signal, it is easy to identify the type of
sonar (usually with its frequency) and its position (with the sound
wave\'s energy). Moreover, active sonar, similar to radar, allows the
user to detect objects at a certain range but also enables other
platforms to detect the active sonar at a far greater range.

Since active sonar does not allow an exact identification and is very
noisy, this type of detection is used by fast platforms (planes,
helicopters) and by noisy platforms (most surface ships) but rarely by
submarines. When active sonar is used by surface ships or submarines, it
is typically activated very briefly at intermittent periods, to reduce
the risk of detection by an enemy\'s passive sonar. As such, active
sonar is normally considered a backup to passive sonar. In aircraft,
active sonar is used in the form of disposable sonobuoys that are
dropped in the aircraft\'s patrol area or in the vicinity of possible
enemy sonar contacts.

Passive sonar has fewer drawbacks. Most importantly, it is silent.
Generally, it has a much greater range than active sonar, and allows an
identification of the target. Since any motorized object makes some
noise, it may be detected eventually. It simply depends on the amount of
noise emitted and the amount of noise in the area, as well as the
technology used. To simplify, passive sonar \"sees\" around the ship
using it. On a submarine, the nose mounted passive sonar detects in
directions of about 270°, centered on the ship\'s alignment, the
hull-mounted array of about 160° on each side, and the towed array of a
full 360°. The no-see areas are due to the ship\'s own interference.
Once a signal is detected in a certain direction (which means that
something makes sound in that direction, this is called broadband
detection) it is possible to zoom in and analyze the signal received
(narrowband analysis). This is generally done using a Fourier transform
to show the different frequencies making up the sound. Since every
engine makes a specific noise, it is easy to identify the object.

Another use of the passive sonar is to determine the target\'s
trajectory. This process is called Target Motion Analysis (TMA), and the
resultant \"solution\" is the target\'s range, course, and speed. TMA is
done by marking from which direction the sound comes at different times,
and comparing the motion with that of the operator\'s own ship. Changes
in relative motion are analyzed using standard geometrical techniques
along with some assumptions about limiting cases.

Passive sonar is stealthy and very useful. However, it requires
high-tech components (band pass filters, receivers) and is costly. It is
generally deployed on expensive ships in the form of arrays to enhance
the detection. Surface ships use it to good effect; it is even better
used by submarines, and it is also used by airplanes and helicopters,
mostly to a \"surprise effect\", since submarines can hide under thermal
layers. If a submarine captain believes he is alone, he may bring his
boat closer to the surface and be easier to detect, or go deeper and
faster, and thus make more sound.

In the United States Navy, a special badge known as the Integrated
Undersea Surveillance System Badge is awarded to those who have been
trained and qualified in sonar operation and warfare.

In World War II, the Americans used the term **SONAR** for their system.
The British still called their system **ASDIC**. In 1948, with the
formation of NATO, standardization of signals led to the dropping of
ASDIC in favor of sonar.


# Acoustics/Interior Sound Transmission

## Introduction to NVH

Noise is characterized by frequency (20--20 kHz), level (dB) and
quality. Noise may be undesirable in some cases, i.e. road NVH yet may
be desirable in other cases, i.e. powerful sounding engine.

Vibration is defined as the motion sensed by the body, mainly in
0.5 Hz - 50 Hz range. It is characterized by frequency, level and
direction.

Harshness is defined as rough, grating or discordant sensation.

Sound quality is defined according to Oxford English Dictionary, \"That
distinctive Quality of a Sound other than it\'s Pitch or Loudness\"

In generally, ground vehicle NVH and sound quality design are attributed
by the following categories:

1.) Powertrain NVH and SQ: Interior Idle NVH, Acceleration NVH,
Deceleration NVH, Cruising NVH, Sound Quality Character, Diesel
Combusition Noise, Engine Start-Up/Shut-Down

2.) Wind Noise: Motorway Speed Wind Noise (80-130kph), High Speed Wind
Noise (\>130kph), Open Glazing Wind Noise

3.) Road NVH: Road Noise, Road Vibration, Impact Noise

4.) Operational Sound Quality: Closure Open/Shut Sound Quality, Customer
Operated Feature Sound Quality, Audible Warning Sounds

5.) Squeaks and Rattles

**== Noise generation ==**

## Structural vibration response

## Structural acoustic response

## Sound propagation

## Applications to vehicle interior sound transmission

## Useful websites

## References


# Acoustics/Anechoic and reverberation rooms

![](Acoustics_anechoic_reverberation.JPG "Acoustics_anechoic_reverberation.JPG")

## Introduction

Acoustic experiments often require to realise measurements in rooms with
special characteristics. Two types of rooms can be distinguished:
anechoic rooms and reverberation rooms.

## Anechoic room

The principle of this room is to simulate a free field. In a free space,
the acoustic waves are propagated from the source to infinity. In a
room, the reflections of the sound on the walls produce a wave which is
propagated in the opposite direction and comes back to the source. In
anechoic rooms, the walls are very absorbent in order to eliminate these
reflections. The sound seems to die down rapidly. The materials used on
the walls are rockwool, glasswool or foams, which are materials that
absorb sound in relatively wide frequency bands. Cavities are dug in the
wool so that the large wavelength corresponding to bass frequencies are
absorbed too. Ideally the sound pressure level of a punctual sound
source decreases about 6 dB per a distance doubling.

Anechoic rooms are used in the following experiments:

```{=html}
<center>
```
Intensimetry: measurement of the acoustic power of a source.

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</center>
```
```{=html}
<center>
```
Study of the source directivity.

```{=html}
</center>
```
## Reverberation room

The walls of a reverberation room mostly consist of concrete and are
covered with reflecting paint. Alternative design consist of sandwich
panels with metal surface. The sound reflects off the walls many times
before dying down. It gives a similar impression of a sound in a
cathedral. Ideally all sound energy is absorbed by air. Because of all
these reflections, a lot of plane waves with different directions of
propagation interfere in each point of the room. Considering all the
waves is very complicated so the acoustic field is simplified by the
diffuse field hypothesis: the field is homogeneous and isotropic. Then
the pressure level is uniform in the room. The truth of this thesis
increases with ascending frequency, resulting in a lower limiting
frequency for each reverberation room, where the density of standing
waves is sufficient.

Several conditions are required for this approximation: The absorption
coefficient of the walls must be very low (α\<0.2) The room must have
geometrical irregularities (non-parallel walls, diffusor objects) to
avoid nodes of pressure of the resonance modes.

With this hypothesis, the theory of Sabine can be applied. It deals with
the reverberation time which is the time required to the sound level to
decrease of 60 dB. T depends on the volume of the room V, the absorption
coefficient αi and the area Si of the different materials in the room :

Reverberation rooms are used in the following experiments:

```{=html}
<center>
```
measurement of the ability of a material to absorb a sound

```{=html}
</center>
```
```{=html}
<center>
```
measurement of the ability of a partition to transmit a sound

```{=html}
</center>
```
```{=html}
<center>
```
Intensimetry

```{=html}
</center>
```
```{=html}
<center>
```
measurement of sound power

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


# Acoustics/Basic Room Acoustic Treatments

![](Acoustics_basic_room_acoustic_treatments.JPG "Acoustics_basic_room_acoustic_treatments.JPG")

## Introduction

Many people use one or two rooms in their living space as \"theatrical\"
rooms where theater or music room activities commence. It is a common
misconception that adding speakers to the room will enhance the quality
of the room acoustics. There are other simple things that can be done to
increase the room\'s acoustics to produce sound that is similar to
\"theater\" sound. This site will take you through some simple
background knowledge on acoustics and then explain some solutions that
will help improve sound quality in a room.

## Room sound combinations

The sound you hear in a room is a combination of direct sound and
indirect sound. Direct sound will come directly from your speakers while
the other sound you hear is reflected off of various objects in the
room.

![](sound_lady.jpg "sound_lady.jpg")

The Direct sound is coming right out of the TV to the listener, as you
can see with the heavy black arrow. All of the other sound is reflected
off surfaces before they reach the listener.

## Good and bad reflected sound

Have you ever listened to speakers outside? You might have noticed that
the sound is thin and dull. This occurs because when sound is reflected,
it is fuller and louder than it would if it were in an open space. So
when sound is reflected, it can add a fullness, or spaciousness. The bad
part of reflected sound occurs when the reflections amplify some notes,
while cancelling out others, making the sound distorted. It can also
affect tonal quality and create an echo-like effect. There are three
types of reflected sound, pure reflection, absorption, and diffusion.
Each reflection type is important in creating a \"theater\" type
acoustic room.

![](sound.jpg "sound.jpg")

### Reflected sound

Reflected sound waves, good and bad, affect the sound you hear, where it
comes from, and the quality of the sound when it gets to you. The bad
news when it comes to reflected sound is standing waves.

These waves are created when sound is reflected back and forth between
any two parallel surfaces in your room, ceiling and floor or wall to
wall.

Standing waves can distort noises 300 Hz and down. These noises include
the lower mid frequency and bass ranges. Standing waves tend to collect
near the walls and in corners of a room, these collecting standing waves
are called room resonance modes.

#### Finding your room resonance modes

First, specify room dimensions (length, width, and height). **Then
follow this example:**

![](equationandexample.jpg "equationandexample.jpg")![](Resmodepic.jpg "Resmodepic.jpg")![](exampletable.jpg "exampletable.jpg")

#### Working with room resonance modes to increase sound quality

##### There are some room dimensions that produce the largest amount of standing waves.

1.  Cube
2.  Room with 2 out of the three dimensions equal
3.  Rooms with dimensions that are multiples of each other

##### Move chairs or sofas away from the walls or corners to reduce standing wave effects

### Absorbed

The sound that humans hear is actually a form of acoustic energy.
Different materials absorb different amounts of this energy at different
frequencies. When considering room acoustics, there should be a good mix
of high frequency absorbing materials and low frequency absorbing
materials. A table including information on how different common
household absorb sound can be found
here.

### Diffused sound

Using devices that diffuse sound is a fairly new way of increasing
acoustic performance in a room. It is a means to create sound that
appears to be \"live\". They can replace echo-like reflections without
absorbing too much sound.

Some ways of determining where diffusive items should be placed were
found on this
\[<http://www.crutchfieldadvisor.com/S-hpU9sw2hgbG/learningcenter/home/speakers_roomacoustics.html?page=4>:
website\].

1.  If you have carpet or drapes already in your room, use diffusion to
    control side wall reflections.
2.  A bookcase filled with odd-sized books makes an effective diffuser.
3.  Use absorptive material on room surfaces between your listening
    position and your front speakers, and treat the back wall with
    diffusive material to re-distribute the reflections.

## How to find overall trouble spots in a room

Every surface in a room does not have to be treated in order to have
good room acoustics. Here is a simple method of finding trouble spots in
a room.

1.  Grab a friend to hold a mirror along the wall near a certain speaker
    at speaker height.
2.  The listener sits in a spot of normal viewing.
3.  The friend then moves slowly toward the listening position (stay
    along the wall).
4.  Mark each spot on the wall where the listener can see any of the
    room speakers in the mirror.
5.  Congratulations! These are the trouble spots in the room that need
    an absorptive material in place. Don\'t forget that diffusive
    material can also be placed in those positions.

## References

-   Acoustic Room Treatment
    Articles
-   Room Acoustics: Acoustic
    Treatments
-   Home Improvement: Acoustic
    Treatments
-   Crutchfield
    Advisor


# Acoustics/Human Vocal Fold

![](Acoustics_human_vocal_fold.JPG "Acoustics_human_vocal_fold.JPG")

## Physiology of vocal fold

The human vocal fold is a set of lip-like tissues located inside the
larynx, and is the source of
sound for humans and many animals.

The larynx is located at the top of the trachea. It is mainly composed
of cartilages and muscles, and the largest cartilage, thyroid, is well
known as the \"Adam\'s Apple.\"

The organ has two main functions; to act as the last protector of the
airway, and to act as a sound source for voice. This page focuses on the
latter function.

Links on Physiology:

-   Discover The
    Larynx

## Voice production

Although the science behind sound production for a vocal fold is
complex, it can be thought of as similar to a brass player\'s lips, or a
whistle made out of grass. Basically, vocal folds (or lips or a pair of
grass) make a constriction to the airflow, and as the air is forced
through the narrow opening, the vocal folds oscillate. This causes a
periodical change in the air pressure, which is perceived as sound.

Vocal Folds Video

When the airflow is introduced to the vocal folds, it forces open the
two vocal folds which are nearly closed initially. Due to the stiffness
of the folds, they will then try to close the opening again. And now the
airflow will try to force the folds open etc\... This creates an
oscillation of the vocal folds, which in turn, as I stated above,
creates sound. However, this is a damped oscillation, meaning it will
eventually achieve an equilibrium position and stop oscillating. So how
are we able to \"sustain\" sound?

As it will be shown later, the answer seems to be in the changing shape
of vocal folds. In the opening and the closing stages of the
oscillation, the vocal folds have different shapes. This affects the
pressure in the opening, and creates the extra pressure needed to push
the vocal folds open and sustain oscillation. This part is explained in
more detail in the \"Model\" section.

This flow-induced oscillation, as with many fluid mechanics problems, is
not an easy problem to model. Numerous attempts to model the oscillation
of vocal folds have been made, ranging from a single mass-spring-damper
system to finite element models. In this page I would like to use my
single-mass model to explain the basic physics behind the oscillation of
a vocal fold.

Information on vocal fold models: National Center for Voice and
Speech

## Model

thumb\|Figure 1:
Schematics.png "wikilink")

The most simple way of simulating the motion of vocal folds is to use a
single mass-spring-damper system as shown above. The mass represents one
vocal fold, and the second vocal fold is assumed to be symmetry about
the axis of symmetry. Position 3 represents a location immediately past
the exit (end of the mass), and position 2 represents the glottis (the
region between the two vocal folds).

### The pressure force

The major driving force behind the oscillation of vocal folds is the
pressure in the glottis. The Bernoulli\'s equation from fluid mechanics
states that:

$P_1 + \frac{1}{2}\rho U^2 + \rho gh = Constant$ \-\-\-\--EQN 1

Neglecting potential difference and applying EQN 1 to positions 2 and 3
of Figure 1,

$P_2 + \frac{1}{2}\rho U_2^2 = P_3 + \frac{1}{2}\rho U_3^2$ \-\-\-\--EQN
2

Note that the pressure and the velocity at position 3 cannot change.
This makes the right hand side of EQN 2 constant. Observation of EQN 2
reveals that in order to have oscillating pressure at 2, we must have
oscillation velocity at 2. The flow velocity inside the glottis can be
studied through the theories of the orifice flow.

The constriction of airflow at the vocal folds is much like an orifice
flow with one major difference: with vocal folds, the orifice profile is
continuously changing. The orifice profile for the vocal folds can open
or close, as well as change the shape of the opening. In Figure 1, the
profile is converging, but in another stage of oscillation it takes a
diverging shape.

The orifice flow is described by Blevins as:

$U = C\frac{2(P_1 - P_3)}{\rho}$ \-\-\-\--EQN 3

Where the constant C is the orifice coefficient, governed by the shape
and the opening size of the orifice. This number is determined
experimentally, and it changes throughout the different stages of
oscillation.

Solving equations 2 and 3, the pressure force throughout the glottal
region can be determined.

### The Collision Force

As the video of the vocal folds shows, vocal folds can completely close
during oscillation. When this happens, the Bernoulli equation fails.
Instead, the collision force becomes the dominating force. For this
analysis, Hertz collision model was applied.

$F_H = k_H delta^{3/2} (1 + b_H delta')$ \-\-\-\--EQN 4

where

$k_H = \frac{4}{3} \frac{E}{1 - \mu_H^2} \sqrt{r}$

Here delta is the penetration distance of the vocal fold past the line
of symmetry.

## Simulation of the model

The pressure and the collision forces were inserted into the equation of
motion, and the result was simulated.

thumb\|Figure 2: Area Opening and Volumetric Flow
Rate

Figure 2 shows that an oscillating volumetric flow rate was achieved by
passing a constant airflow through the vocal folds. When simulating the
oscillation, it was found that the collision force limits the amplitude
of oscillation rather than drive the oscillation. Which tells us that
the pressure force is what allows the sustained oscillation to occur.

## The acoustic output

This model showed that the changing profile of glottal opening causes an
oscillating volumetric flow rate through the vocal folds. This will in
turn cause an oscillating pressure past the vocal folds. This method of
producing sound is unusual, because in most other means of sound
production, air is compressed periodically by a solid such as a speaker
cone.

Past the vocal folds, the produced sound enters the vocal tract.
Basically this is the cavity in the mouth as well as the nasal cavity.
These cavities act as acoustic filters, modifying the character of the
sound. These are the characters that define the unique voice each person
produces.

## Related links

-   FEA
    Model
-   Two Mass
    Model

## References

1.  Fundamentals of Acoustics; Kinsler et al, John Wiley & Sons, 2000
2.  Acoustics: An introduction to its Physical Principles and
    Applications; Pierce, Allan D., Acoustical Society of America, 1989.
3.  Blevins, R.D. (1984). Applied Fluid Dynamics Handbook. Van Nostrand
    Reinhold Co. 81-82.
4.  Titze, I. R. (1994). Principles of Voice Production. Prentice-Hall,
    Englewood Cliffs, NJ.
5.  Lucero, J. C., and Koenig, L. L. (2005). Simulations of temporal
    patterns of oral airflow in men and women using a two-mass model of
    the vocal folds under dynamic control, Journal of the Acostical
    Society of America 117, 1362-1372.
6.  Titze, I.R. (1988). The physics of small-amplitude oscillation of
    the vocal folds. Journal of the Acoustical Society of America 83,
    1536--1552


# Acoustics/How an Acoustic Guitar Works

![](Acoustics_how_an_acoustic_guitar_works.JPG "Acoustics_how_an_acoustic_guitar_works.JPG")

What are sound vibrations that contribute to sound production? First of
all, there are the strings. Any string that is under tension will
vibrate at a certain frequency. The weight and length of the string, the
tension in the string, and the compliance of the string determine the
frequency at which it vibrates. The guitar controls the length and
tension of six differently weighted strings to cover a very wide range
of frequencies. Second, there is the body of the guitar. The guitar body
is very important for the lower frequencies of the guitar. The air mass
just inside the sound hole oscillates, compressing and decompressing the
compliant air inside the body. In practice this concept is called a
Helmholtz resonator. Without this, it would be difficult to produce the
wonderful timbre of the guitar.

![](Acoustic_guitar-en.svg "Acoustic_guitar-en.svg"){width="320"}

## The strings

The strings of the guitar vary in linear density, length, and tension.
This gives the guitar a wide range of attainable frequencies. The larger
the linear density is, the slower the string vibrates. The same goes for
the length; the longer the string is the slower it vibrates. This causes
a low frequency. Inversely, if the strings are less dense and/or shorter
they create a higher frequency. The resonance frequencies of the strings
can be calculated by

$$f_1 = \frac{\sqrt{\frac{T}{\rho_l}}}{2 L}\quad \text{with}\quad T = \text{string tension},\ \rho_l = \text{linear density of string},\ L = \text{string length}.$$

The string length, $L$, in the equation is what changes when a player
presses on a string at a certain fret. This will shorten the string
which in turn increases the frequency it produces when plucked. The
spacing of these frets is important. The length from the nut to bridge
determines how much space goes between each fret. If the length is 25
inches, then the position of the first fret should be located
(25/17.817) inches from the nut. Then the second fret should be located
 inches from the
first fret. This results in the equation

$$d = \frac{L}{17.817} \quad \text{with}\quad d = \text{spacing between frets}, L = \text{length from previous fret to bridge}.$$

When a string is plucked, a disturbance is formed and travels in both
directions away from point where the string was plucked. These \"waves\"
travel at a speed that is related to the tension and linear density and
can be calculated by

$$c = \sqrt{\frac{T}{\rho_l}} \quad \text{with}\quad c = \text{wave speed},\ T = \text{string tension},\ \rho_l = \text{linear density}.$$

The waves travel until they reach the boundaries on each end where they
are reflected back. The link below displays how the waves propagate in a
string.

Plucked String @
www.phys.unsw.edu

The strings themselves do not produce very much sound because they are
so thin. They can\'t \"push\" the air that surrounds them very
effectively. This is why they are connected to the top plate of the
guitar body. They need to transfer the frequencies they are producing to
a large surface area which can create more intense pressure
disturbances.

## The body

The body of the guitar transfers the vibrations of the bridge to the air
that surrounds it. The top plate contributes to most of the pressure
disturbances, because the player dampens the back plate and the sides
are relatively stiff. This is why it is important to make the top plate
out of a light springy wood, like spruce. The more the top plate can
vibrate, the louder the sound it produces will be. It is also important
to keep the top plate flat, so a series of braces are located on the
inside to strengthen it. Without these braces the top plate would bend
and crack under the large stress created by the tension in the strings.
This would also affect the magnitude of the sound being transmitted. The
warped plate would not be able to \"push\" air very efficiently. A good
experiment to try, in order to see how important this part of the guitar
is in the amplification process, is as follows:

1.  Start with an ordinary rubber band, a large bowl, adhesive tape, and
    plastic wrap.
2.  Stretch the rubber band and pluck it a few times to get a good sense
    for how loud it is.
3.  Stretch the plastic wrap over the bowl to form a sort of drum.
4.  Tape down one end of the rubber band to the plastic wrap.
5.  Stretch the rubber band and pluck it a few times.
6.  The sound should be much louder than before.

## The air

The final part of the guitar is the air inside the body. This is very
important for the lower range of the instrument. The air just inside the
soundhole oscillates compressing and expanding the air inside the body.
This is just like blowing across the top of a bottle and listening to
the tone it produces. This forms what is called a Helmholtz resonator.
For more information on Helmholtz resonators go to Helmholtz
Resonance. This link
also shows the correlation to acoustic guitars in great detail. The
acoustic guitar makers often tune these resonators to have a resonance
frequency between F#2 and A2 (92.5 to 110.0 Hz). Having such a low
resonance frequency is what aids the amplification of the lower
frequency strings. To demonstrate the importance of the air in the
cavity, simply play an open A on the guitar (the fifth string - second
lowest note). Now, as the string is vibrating, place a piece of
cardboard over the soundhole. The sound level is reduced dramatically.
This is because you\'ve stopped the vibration of the air mass just
inside the soundhole, causing only the top plate to vibrate. Although
the top plate still vibrates a transmits sound, it isn\'t as effective
at transmitting lower frequency waves, thus the need for the Helmholtz
resonator.


# Acoustics/Basic Acoustics of the Marimba

![](Acoustics_basic_acoustics_marimba.JPG "Acoustics_basic_acoustics_marimba.JPG")

## Introduction

Like a xylophone, a marimba has octaves of wooden bars that are struck
with mallets to produce tones. Unlike the harsh sound of a xylophone, a
marimba produces a deep, rich tone. Marimbas are not uncommon and are
played in most high school bands.

Now, while all the trumpet and flute and clarinet players are busy
tuning up their instruments, the marimba player is back in the
percussion section with her feet up just relaxing. This is a bit
surprising, however, since the marimba is a melodic instrument that
needs to be in tune to sound good. So what gives? Why is the marimba
never tuned? How would you even go about tuning a marimba? To answer
these questions, the acoustics behind (or within) a marimba must be
understood.

## Components of sound

What gives the marimba its unique sound? It can be boiled down to two
components: the bars and the resonators. Typically, the bars are made of
rosewood (or some synthetic version of wood). They are cut to size
depending on what note is desired, then the tuning is refined by shaving
wood from the underside of the bar.

### Example

Rosewood bar, middle C, 1 cm thick

The equation that relates the length of the bar with the desired
frequency comes from the theory of modeling a bar that is free at both
ends. This theory yields the following equation:

$Length = \sqrt{\frac{3.011^2\cdot \pi \cdot t \cdot c}{8 \cdot \sqrt{12}\cdot f}}$

where t is the thickness of the bar, c is the speed of sound in the bar,
and f is the frequency of the note.

For rosewood, c = 5217 m/s. For middle C, f=262 Hz.

Therefore, to make a middle C key for a rosewood marimba, cut the bar to
be:

$Length = \sqrt{\frac{3.011^2\cdot \pi \cdot .01 \cdot 5217}{8 \cdot \sqrt{12}\cdot 262}}= .45 m = 45 cm$
\*\*\*

The resonators are made from metal (usually aluminum) and their lengths
also differ depending on the desired note. It is important to know that
each resonator is open at the top but closed by a stopper at the bottom
end.

### Example

Aluminum resonator, middle C

The equation that relates the length of the resonator with the desired
frequency comes from modeling the resonator as a pipe that is driven at
one end and closed at the other end. A \"driven\" pipe is one that has a
source of excitation (in this case, the vibrating key) at one end. This
model yields the following:

$Length = \frac {c}{4\cdot f}$

where c is the speed of sound in air and f is the frequency of the note.

For air, c = 343 m/s. For middle C, f = 262 Hz.

Therefore, to make a resonator for the middle C key, the resonator
length should be:

$Length = \frac {343}{4 \cdot 262} = .327m = 32.7 cm$

### Resonator shape

The shape of the resonator is an important factor in determining the
quality of sound that can be produced. The ideal shape is a sphere. This
is modeled by the Helmholtz resonator. However, mounting big, round,
beach ball-like resonators under the keys is typically impractical. The
worst choices for resonators are square or oval tubes. These shapes
amplify the non-harmonic pitches sometimes referred to as "junk
pitches". The round tube is typically chosen because it does the best
job (aside from the sphere) at amplifying the desired harmonic and not
much else.

As mentioned in the second example above, the resonator on a marimba can
be modeled by a closed pipe. This model can be used to predict what type
of sound (full and rich vs dull) the marimba will produce. Each pipe is
a \"quarter wave resonator\" that amplifies the sound waves produced by
of the bar. This means that in order to produce a full, rich sound, the
length of the resonator must exactly match one-quarter of the
wavelength. If the length is off, the marimba will produce a dull or
off-key sound for that note.

## Why would the marimba need tuning?

In the theoretical world where it is always 72 degrees with low
humidity, a marimba would not need tuning. But, since weather can be a
factor (especially for the marching band) marimbas do not always perform
the same way. Hot and cold weather can wreak havoc on all kinds of
percussion instruments, and the marimba is no exception. On hot days,
the marimba tends to be sharp and for cold days it tends to be flat.
This is the exact opposite of what happens to string instruments. Why?
The tone of a string instrument depends mainly on the tension in the
string, which decreases as the string expands with heat. The decrease in
tension leads to a flat note. Marimbas on the other hand produce sound
by moving air through the resonators. The speed at which this air is
moved is the speed of sound, which varies proportionately with
temperature! So, as the temperature increases, so does the speed of
sound. From the equation given in example 2 from above, you can see that
an increase in the speed of sound (c) means a longer pipe is needed to
resonate the same note. If the length of the resonator is not increased,
the note will sound sharp. Now, the heat can also cause the wooden bars
to expand, but the effect of this expansion is insignificant compared to
the effect of the change in the speed of sound.

## Tuning myths

It is a common myth among percussionists that the marimba can be tuned
by simply moving the resonators up or down (while the bars remain in the
same position.) The thought behind this is that by moving the resonators
down, for example, you are in effect lengthening them. While this may
sound like sound reasoning, it actually does not hold true in practice.
Judging by how the marimba is constructed (cutting bars and resonators
to specific lengths), it seems that there are really two options to
consider when looking to tune a marimba: shave some wood off the
underside of the bars, or change the length of the resonator. For
obvious reasons, shaving wood off the keys every time the weather
changes is not a practical solution. Therefore, the only option left is
to change the length of the resonator.

As mentioned above, each resonator is plugged by a stopper at the bottom
end. So, by simply shoving the stopper farther up the pipe, you can
shorten the resonator and sharpen the note. Conversely, pushing the
stopper down the pipe can flatten the note. Most marimbas do not come
with tunable resonators, so this process can be a little challenging.
(Broomsticks and hammers are common tools of the trade.)

### Example

Middle C Resonator lengthened by 1 cm

For ideal conditions, the length of the middle C (262 Hz) resonator
should be 32.7 cm as shown in example 2. Therefore, the change in
frequency for this resonator due to a change in length is given by:

$\Delta Frequency = 262 Hz - \frac {c}{4\cdot (.327 + \Delta L)}$

If the length is increased by 1 cm, the change in frequency will be:

$\Delta Frequency = \frac {343}{4\cdot (.327 + .01)} - 262 Hz = 7.5 Hz$

The acoustics behind the tuning a marimba go back to the design that
each resonator is to be ¼ of the total wavelength of the desired note.
When marimbas get out of tune, this length is no longer exactly equal to
¼ the wavelength due to the lengthening or shortening of the resonator
as described above. Because the length has changed, resonance is no
longer achieved, and the tone can become muffled or off-key.

## Conclusions

Some marimba builders are now changing their designs to include tunable
resonators. Since any leak in the end-seal will cause major loss of
volume and richness of the tone, this is proving to be a very difficult
task. At least now, though, armed with the acoustic background of their
instruments, percussionists everywhere will now have something to do
when the conductor says, "tune up!"

## Links and References

1.  <http://www.gppercussion.com/html/resonators.html>
2.  <http://www.mostlymarimba.com/>
3.  <http://www.craftymusicteachers.com/bassmarimba/>


# Acoustics/Bessel Functions and the Kettledrum

![](Acoustics_bessel_ketteldrum.JPG "Acoustics_bessel_ketteldrum.JPG")

## Introduction

In class, we have begun to discuss the solutions of multidimensional
wave equations. A particularly interesting aspect of these
multidimensional solutions are those of bessel functions for circular
boundary conditions. The practical application of these solutions is the
kettledrum. This page will explore in qualitative and quantitative terms
how the kettledrum works. More specifically, the kettledrum will be
introduced as a circular membrane and its solution will be discussed
with visuals (e.g. visualization of bessel functions, video of
kettledrums and audio forms (wav files of kettledrums playing. In
addition, links to more information about this material, including
references will be included.

## The math behind the kettledrum: the brief version

When one looks at how a kettledrum produces sound, one should look no
farther than the drum head. The vibration of this circular membrane (and
the air in the drum enclosure) is what produces the sound in this
instrument. The mathematics behind this vibrating drum are relatively
simple. If one looks at a small element of the drum head, it looks
exactly like the mathematical model for a vibrating string (see:). The
only difference is that there are two dimensions where there are forces
on the element, the two dimensions that are planar to the drum. As this
is the same situation, we have the same equation, except with another
spatial term in the other planar dimension. This allows us to model the
drum head using a helmholtz equation. The next step (solved in detail
below) is to assume that the displacement of the drum head (in polar
coordinates) is a product of two separate functions for theta and r.
This allows us to turn the PDE into two ODES which are readily solved
and applied to the situation of the kettledrum head. For more info, see
below.

## The math behind the kettledrum: the derivation

So starting with the trusty general Helmholtz equation:

$$\nabla^2\Psi+k^2\Psi=0.$$

Where $k$ is the wave number, the frequency of the forced oscillations
divided by the speed of sound in the membrane.

Since we are dealing with a circular object, it makes sense to work in
polar coordinates (in terms of radius and angle) instead of rectangular
coordinates. For polar coordinates the Laplacian term of the Helmholtz
relation ($\nabla^2$) becomes
$\frac{\partial^2 \Psi}{\partial r^2} + \frac{1}{r} \frac{\partial\Psi}{\partial r} +\frac{1}{r^2} \frac{\partial^2 \Psi}{\partial \theta^2}$

Using the method of separation of variables (see Reference 3 for more
info), we will assume a solution of the form

$$\Psi (r,\theta) = R(r) \Theta(\theta).$$

Substituting this result back into our trusty Helmholtz equation, then
multiplying through by $r^2/(R\Theta)$ gives

$$\frac{1}{R} \left(r^2\frac{d^2 R}{dr^2} + r \frac{dR}{dr}\right) + k^2 r^2 = -\frac{1}{\Theta} \frac{d^2 \Theta}{d\theta^2},$$

where we moved the $\theta$-dependent terms to the right hand side.
Since we separated the variables of the solution into two
one-dimensional functions, the partial derivatives become ordinary
derivatives. In order for the above equality to hold regardless of
changes in $r$ and $\theta$, both sides must be equal to some constant.
For simplicity, I will use $\lambda^2$ as this constant. This results in
the following two equations:

$$\frac{d^2 \Theta}{d\theta^2} = -\lambda^2 \Theta,$$

$$r^2\frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (k^2 r^2 - \lambda^2) R = 0.$$

The first of these equations readily seen as the standard second order
ordinary differential equation which has a harmonic solution of sines
and cosines with the frequency based on $\lambda$. The second equation
is what is known as Bessel\'s Equation. The solution to this equation is
cryptically called Bessel functions of order $\lambda$ of the first and
second kind. These functions, while sounding very intimidating, are
simply oscillatory functions of the radius times the wave number. Both
sets of functions diminish as $kr$ becomes large, but are unbounded as
$kr$ goes to zero for the Bessel functions of the second kind. !Bessel
functions of the first
kind..svg "Bessel functions of the first kind."){width="320"}

Now that we have the general solution to this equation, we can now model
a infinite radius kettledrum head. However, since i have yet to see an
infinite kettle drum, we need to constrain this solution of a vibrating
membrane to a finite radius. We can do this by applying what we know
about our circular membrane: along the edges of the kettledrum, the drum
head is attached to the drum. This means that there can be no
displacement of the membrane at the termination at the radius of the
kettle drum. This boundary condition can be mathematically described as
the following:

$$R(a) = 0$$

Where a is the arbitrary radius of the kettledrum. In addition to this
boundary condition, the displacement of the drum head at the center must
be finite. This second boundary condition removes the bessel function of
the second kind from the solution. This reduces the $R$ part of our
solution to:

$$R(r) = AJ_{\lambda}(kr)$$

Where $J_{\lambda}$ is a bessel function of the first kind of order
$\lambda$. Apply our other boundary condition at the radius of the drum
requires that the wave number $k$ must have discrete values,
($j_{mn}/a$) which can be looked up. Combining all of these gives us our
solution to how a drum head behaves (which is the real part of the
following):

$$y_{\lambda n}(r,\theta,t) = A_{\lambda n} J_{\lambda n}(k_{\lambda n} r)e^{j \lambda \theta+j w_{\lambda n} t}$$

## The math behind the kettledrum: the entire drum

The above derivation is just for the drum head. An actual kettledrum has
one side of this circular membrane surrounded by an enclosed cavity.
This means that air is compressed in the cavity when the membrane is
vibrating, adding more complications to the solution. In mathematical
terms, this makes the partial differential equation non-homogeneous or
in simpler terms, the right side of the Helmholtz equation does not
equal zero. This result requires significantly more derivation, and will
not be done here. If the reader cares to know more, these results are
discussed in the two books under references 6 and 7.

## Sites of interest

As one can see from the derivation above, the kettledrum is very
interesting mathematically. However, it also has a rich historical music
tradition in various places of the world. As this page\'s emphasis is on
math, there are few links provided below that reference this rich
history.

-   A discussion of Persian kettledrums: Kettle drums of Iran and other
    countries
-   A discussion of kettledrums in classical music: Kettle drum
    Lit.
-   A massive resource for kettledrum history, construction and
    technique\" Vienna Symphonic
    Library

## References

1.  Eric W. Weisstein. \"Bessel Function of the First Kind.\" From
    MathWorld---A Wolfram Web Resource.
    <http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html>
2.  Eric W. Weisstein. \"Bessel Function of the Second Kind.\" From
    MathWorld---A Wolfram Web Resource.
    <http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html>
3.  Eric W. Weisstein. \"Bessel Function.\" From MathWorld---A Wolfram
    Web Resource. <http://mathworld.wolfram.com/BesselFunction.html>
4.  Eric W. Weisstein et al. \"Separation of Variables.\" From
    MathWorld---A Wolfram Web Resource.
    <http://mathworld.wolfram.com/SeparationofVariables.html>
5.  Eric W. Weisstein. \"Bessel Differential Equation.\" From
    MathWorld---A Wolfram Web Resource.
    <http://mathworld.wolfram.com/BesselDifferentialEquation.html>
6.  Kinsler and Frey, \"Fundamentals of Acoustics\", fourth edition,
    Wiley & Sons
7.  Haberman, \"Applied Partial Differential Equations\", fourth
    edition, Prentice Hall Press