# Engineering Acoustics/Basic Acoustics of the Marimba

## Introduction

!Marimba Band \"La Gloria Antigueña\", Antigua Guatemala,
1979

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:                |
|                                                                      |
| $Length = \sqrt                                                      |
| {\frac{3.011^2\cdot \pi \cdot t \cdot v}{8 \cdot \sqrt{12}\cdot f}}$ |
|                                                                      |
| where t is the thickness of the bar, v is the speed of sound in the  |
| bar, and f is the frequency of the note. For rosewood, v = 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:                                 |
|                                                                      |
| $Length = \sqrt{\frac{3.011^2\cdot \p                                |
| i \cdot .01 \cdot 5217}{8 \cdot \sqrt{12}\cdot 262}}= .45 m = 45 cm$ |
+----------------------------------------------------------------------+

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. (For more see Helmholtz
Resonator
page)
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: |
|                                                                      |
| $\D                                                                  |
| elta Frequency = \frac {343}{4\cdot (.327 + .01)} - 262 Hz = 7.5 Hz$ |
+----------------------------------------------------------------------+

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. There are in fact several marimba companies that have had
tunable resonators for decades. However, only a few offer full range
tuning. 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/>

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# Engineering Acoustics/How an Acoustic Guitar works

## Introduction

There are three main parts of the guitar that contribute to sound
production.

First of all, there are strings. Any string that is under tension will
vibrate at a certain frequency. The tension and gauge in 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 of all, there is the body of the guitar. The guitar body is
connected directly to one end of each of the strings. The body receives
the vibrations of the strings and transmits them to the air around the
body. It is the body's large surface area that allows it to "push" a lot
more air than a string.

Finally, there is the air inside the body. This is very important for
the lower frequencies of the guitar. The air mass just inside the
soundhole oscillates, compressing and decompressing the compliant air
inside the body. In practice, this concept is called a Helmholtz
resonator. Without this, it would difficult to produce the wonderful
timbre of the guitar.

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

## 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 lowest resonance frequencies of each
string can be calculated by

$f_1 = \frac{1}{2 L} \sqrt{\frac{T}{\rho_1}}$ where $T$= string tension,
$\rho_1$=linear density, $L$ = 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
(25-(25/17.817))/17.817 inches from the first fret. This results in the
equation

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

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

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

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. 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
sound hole 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)(Hz stands for Hertz).
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 second string).
Now, as the string is vibrating, place a piece of cardboard over the
sound hole. The sound level is reduced dramatically. This is because
you\'ve stopped the vibration of the air mass just inside the sound
hole, causing only the top plate to vibrate. Although the top plate
still vibrates and transmits sound, it isn\'t as effective at
transmitting lower frequency waves, thus the need for the Helmholtz
resonator.

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# Engineering Acoustics/Bessel Functions and the Kettledrum

# Abstract

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 of the kettledrum works. More specifically, the kettledrum will
be introduced as a circular membrane and its solution will be discussed
in visual (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.

# What is a kettledrum

A kettledrum is a percussion instrument with a circular drumhead mounted
on a \"kettle-like\" enclosure. When one strikes the drumhead with a
mallet, it vibrates which produces its sound. The pitch of this sound is
determined by the tension of the drumhead, which is precisely tuned
before playing. The sound of the kettldrum (called the Timpani in
classical music) is present in many forms of music from many difference
places of the world.

![](myKettledrum.jpeg "myKettledrum.jpeg")

# The math behind the kettledrum:the brief version

When one looks at how a kettledrum produces sound, one should look no
farther than the drumhead. 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 situation for the 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
drumhead using a helmholtz equation. The next step (solved in detail
below) is to assume that the displacement of the drumhead (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 make 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
$\partial^2 \Psi/ \partial r^2 + 1/r \partial^2\Psi/ \partial r +1/r^2 \partial^2 \Psi /\partial \theta^2$

Now lets assume that$$\Psi (r,\theta) = R(r) \Theta(\theta)$$

This assumption follows the method of separation of variables. (see
Reference 3 for more info) Substituting this result back into our trusty
Helmholtz equation gives the following:

$r^2 / R (d^2 R/dr^2 + 1/r dR/dr) + k^2 r^2 = -1/\Theta d^2 \Theta /d\theta^2$

Since we separated the variables of the solution into two
one-dimensional functions, the partial derivatives become ordinary
derivatives. Both sides of this result must equal the same constant. For
simplicity, I will use $\lambda$ as this constant. This results in the
following two equations:

$d^2 \Theta / d\theta^2 = -\lambda^2 \Theta$

$d^2 R / dr^2 + 1/r dR/dr + (k^2 - \lambda^2 / r^2) R = 0$

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 that
are unbounded at when kr (for the function of the second kind)
approaches zero and diminish as kr get larger. (For more information on
what these functions look like see References 1,2, and 3)

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 drumhead 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 differencial 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

Wikibooks sister cite, references under Timpani: Wikipedia
reference

# 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

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# Engineering Acoustics/Acoustics in Violins

# Acoustics of the Violin

For detail anatomy of violin, please refer to Atelierla
Bussiere.

![](Violin_front_view.jpg "Violin_front_view.jpg")![](backview.jpg "backview.jpg")

## How Does A Violin Make Sound?

### General Concept

When a violinist bows a string, which can produce vibrations with
abundant harmonics. The vibrations of the strings are structurally
transmitted to the bridge and the body of the instrument through the
bridge. The bridge transmits the vibrational energy produced by the
strings to the body through its feet, further triggering the vibration
of body. The vibration of the body determines sound radiation and sound
quality, along with the resonance of the cavity.

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

### String

The vibration pattern of the strings can be easily be observed. To the
naked eye, the string appears to move back and forth in a parabolic
shape (see figure), which resembles the first mode of free vibration of
a stretched string. The vibration of strings was first investigated by
Hermann Von
Helmholtz,
the famous mathematician and physicist in 19th century. A surprising
scenario was discovered that the string actually moves in an inverse "V"
shape rather than parabolas (see figure). What we see is just an
envelope of the motion of the string. To honor his findings, the motion
of bowed strings had been called "Helmholtz motion."

![](String.jpg "String.jpg")
![](Helmholtzmotion.jpg "Helmholtzmotion.jpg")

### Bridge

The primary role of the bridge is to transform the motion of vibrating
strings into periodic driving forces by its feet to the top plate of the
violin body. The configuration of the bridge can be referred to the
figure. The bridge stands on the belly between f holes, which have two
primary functions: One is to connect the air inside the body with
outside air, and the other one is to make the belly between f holes move
more easily than other parts of the body. The fundamental frequency of a
violin bridge was found to be around 3000 Hz when it is on a rigid
support, and it is an effective energy-transmitting medium to transmit
the energy from the string to body at frequencies from 1 kHz to 4 kHz,
which is in the range of keen sensitivity of human hearing. If a
violinist desires a darker sound from the violin, he or she may attach a
mute to the top of the bridge. The mute is actually an additional mass
which reduces the fundamental frequency of the bridge. As a result, the
sound at higher frequencies is diminished since the force transferred to
the body has been decreased. On the other hand, the fundamental
frequency of the bridge can be raised by attaching an additional
stiffness in the form of tiny wedges, and the sound at higher
frequencies will be amplified accordingly.

The sound post connects the flexible belly to the much stiffer back
plate. The sound post can prevent the collapse of the belly due to high
tension force in the string, and, at the same time, couples the
vibration of the plate. The bass bar under the belly extends beyond the
f holes and transmits the force of the bridge to a larger area of the
belly. As can be seen in the figure, the motion of the treble foot is
restricted by the sound post, while, conversely, the foot over bass bar
can move up and down more easily. As a result, the bridge tends to move
up and down, pivoting about the treble foot. The forces appearing at the
two feet remain equal and opposite up to 1 kHz. At higher frequencies,
the forces become uneven. The force on the soundpost foot predominates
at some frequencies, while it is the bass bar foot at some.

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

### Body

The body includes top plate, back plate, the sides, and the air inside,
all of which serve to transmit the vibration of the bridge into the
vibration of air surrounding the violin. For this reason, the violin
needs a relatively large surface area to push enough amount of air back
and forth. Thus, the top and back plates play important roles in the
mechanism. Violin makers have traditionally pay much attention on the
vibration of the top and back plates of the violin by listening to the
tap tones, or, recently, by observing the vibration mode shapes of the
body plates. The vibration modes of an assembled violin are, however,
much more complicated.

The vibration modes of top and back plates can be easily observed in a
similar technique first performed by Ernest Florens Friedrich Chaldni
(1756--1827), who is often respectfully referred "the father of
acoustics." First, the fine sand is uniformly sprinkled on the plate.
Then, the plate can be resonated, either by a powerful sound wave tuned
to the desired frequencies, by being bowed by a violin bow, or by being
excited mechanically or electromechanically at desired frequencies.
Consequently, the sand disperses randomly due to the vibration of plate.
Some of the sand falls outside the region of plate, while some of the
sand is collected by the nodal regions, which have relatively small
movement, of the plate. Hence, the mode shapes of the plate can be
visualized in this manner, which can be referred to the figures in the
reference site, Violin
Acoustics. The first
seven modes of the top and back plates of violin are presented, with
nodal lines depicted by using black sands.

The air inside the body is also important, especially in the range of
lower frequencies. It is like the air inside a bottle when you blow into
the neck, or, as known as Helmholtz
resonance,
which has its own modes of vibration. The air inside the body can
communicate with air outside through the f holes, and the outside air
serves as medium carrying waves from the violin.

see www.violinbridges.co.uk for more articles on bridges and acoustics.

### Sound Radiation

A complete description of sound radiation of a violin should include the
information about radiation intensity as functions both of frequency and
location. The sound radiation can be measured by a microphone connected
to a pressure level meter which is rotatably supported on a stand arm
around the violin, while the violin is fastened at the neck by a clip.
The force is introduced into the violin by using a miniature impact
hammer at the upper edge of the bridge in the direction of bowing. The
detail can be referred to Martin Schleske, master studio for
violinmaking.
The radiation intensity of different frequencies at different locations
can be represented by directional characteristics, or acoustic maps. The
directional characteristics of a violin can be shown in the figure in
the website of Martin
Schleske, where
the radial distance from the center point represents the absolute value
of the sound level (re 1Pa/N) in dB, and the angular coordinate of the
full circle indicates the measurement point around the instrument.
According to the directional characteristics of violins, the principal
radiation directions for the violin in the horizontal plane can be
established. For more detail about the principal radiation direction for
violins at different frequencies, please refer to reference (Meyer
1972).

## References And Other Links

-   Violin Acoustics
-   Paul Galluzzo\'s
    Homepage
-   Martin Schleske, master studio for
    violinmaking
-   Atelierla Bussiere
-   Fletcher, N. H., and Rossing, T. D., *The physics of musical
    instrument*, Springer-Verlag, 1991
-   Meyer, J., \"Directivity of bowed stringed instruments and its
    effect on orchestral sound in concert halls\", J. Acoustic. Soc.
    Am., 51, 1972, pp. 1994--2009

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# Engineering Acoustics/Clarinet Acoustics

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

A clarinet consists several acoustical components:

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

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

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

### Mouthpiece-Reed System

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

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

This behavior is roughly depicted in Figure 2:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Cylindrical Bore

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

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

#### Fundamental frequency

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

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

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

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

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

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

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

#### Harmonic series

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

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

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

### Tone Holes

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

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

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

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

### Bell

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

### Register Holes

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

## Wave Propagation in the Bore

### Wave Equation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Radiation Impedance

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

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

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

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

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

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

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

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

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

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

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

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

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

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

#### Bore section

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

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

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

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

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

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

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

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

#### Tonehole section

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

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

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

```{=html}
<center>
```
Figure 10: Combine the tone hole network with the bore network

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

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

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

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

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

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

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

#### Wall losses

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

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

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

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

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

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

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

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

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

## Other Useful Links

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

## References

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]: 1, George Gershwin:
    Rhapsody in Blue - Fantasia 2000

[^7]:

[^8]:

[^9]:

[^10]:

[^11]:

[^12]:

[^13]:


# Engineering Acoustics/Acoustic Guitars

I plan to discuss the workings of an acoustic guitar, and how the topics
that we have studied apply. This will largely be vibrations of strings
and vibrations of cavities.

## Introduction

The acoustic guitar is one of the most well known musical instruments.
Although precise dates are not known, the acoustic guitar is generally
thought to have originated sometime during the Renaissance in the form
of a lute, a smaller fretless form of what is known today. After
evolving over the course of about 500 years, the guitar today consists
of a few major components: the strings and neck, the bridge, soundboard,
head, and internal cavity.

## Strings, Neck, and Head

The strings are what actually create vibration on the guitar. On a
standard acoustic, there are six strings, each with a different constant
linear density. Strings run along the length of the neck, and are wound
around adjustable tuning pegs located on the head. These tuning pegs can
be turned to adjust the tension in the string. This allows a
modification of the wave speed, governed by the equation

$c^2=T/\rho$

where c is the wave speed \[m/s\] as a function of tension \[N\], T, and
rho is the linear density \[kg/m\^3\]. The string is assumed to fixed at
the head (x=0) and mass loaded at the bridge (x=L).

To determine the vibrating frequency of an open string, a general
harmonic solution (GHS) is assumed,
$y(x,t)=Aexp(j(wt-kx))+Bexp(j(wt-kx))$

To solve for coefficients A and B, boundary conditions at x=0 and x=L
are evaluated. At x=0, string velocity (dy/dx) must be zero at all times
because that end is assumed to be fixed. Applying this knowledge to the
GHS produces

$y(x,t)=-2jAsin(kx)*exp(jwt)$

Alternatively, at the bridge (a.k.a the mass load at x=L), the bridge
and soundboard (along with any other piece that may vibrate) is assumed
to be a lumped element of mass m. The overall goal with this boundary
condition is to determine the velocity of the mass. From Newton\'s
second law (F=ma), the only force involved is the tension force in the
string. The y-component of this force divided by mass m equals the
acceleration. Knowing that acceleration equals velocity times jw
(a=jwu),

$u(L,t)=-T/(j*w*m)*(dy/dx)$

evaluated at x=L. Combining the two boundary equations and simplifying,
a final equation can be obtained

$cot(kL)=(m/ms)kL$

where k is the wavenumber (w/c), L is the string length, m is the lumped
mass of the guitar body, ms is the total mass of the string (linear
density times length), w is the frequency, and c is the wave speed. If
the ratio of m/ms is large (which in a guitar\'s case, it is), these
frequencies are designated by kL=n\*pi. Simplified, the fundamental
frequency can be given by

$f=sqrt(T/rho) / 2L$

Therefore to adjust the resonance frequency of the string, either change
the tension (turn the tuning knob), change the linear density (play a
different string), or adjust the length (use the fretboard).

To determine the location of the frets, musical notes must be
considered. In the musical world, it is common practice to use a
tempered scale. In this scale, an A note is set at 440 Hz. To get the
next note in the scale, multiply that frequency by the 12th root of 2
(approximately 1.059), and an A-sharp will be produced. Multiply by the
same factor for the next note, and so on. With this in mind, to increase
f by a factor of 1.059, a corresponding factor should be applied to L.
That factor is 1/17.817, with L in inches. For example, consider an open
A string, vibrating at 440 Hz. For a 26 inch string, the position of the
first fret is (26/17.817=1.459) inches from the head. The second fret
will be ((26-1.459)/17.817) inches from the first, and so on.

## Bridge

The bridge is the connection point between the strings and the
soundboard. The vibration of the string moves the assumed mass load of
the bridge, which vibrates the soundboard, described next.

## Soundboard

The soundboard increases the surface area of vibration, increasing the
initial intensity of the note, and is assisted by the internal cavity.

## Internal Cavity

The internal cavity acts as a Helmholtz resonator, and helps to amplify
the sound. As the sound board vibrates, the sound wave is able to
resonate inside.


# Engineering Acoustics/Room Acoustics and Concert Halls

# Room Acoustics and Concert Halls

## Introduction

From performing on many different rooms and stages all over the United
States, I thought it would be nice to have a better understanding and
source about the room acoustics. This Wikibook page is intended to help
the user with basic technical questions/answers about room acoustics.
Main topics that will be covered are: what really makes a room sound
*good* or *bad*, *alive* or *dead*. This will lead into absorption and
transmission coefficients, decay of sound in the room, and
reverberation. Different use of materials in rooms will be mentioned
also. There is no intention of taking work from another. This page is a
switchboard source to help the user find information about room
acoustics.

## Sound Fields

Two types of sound fields are involved in room acoustics: Direct Sound
and Reverberant Sound.

### Direct Sound

The component of the sound field in a room that involves only a direct
path between the source and the receiver, before any reflections off
walls and other surfaces.

### Reverberant Sound

The component of the sound field in a room that involves the direct path
and the path after it reflects off of walls or any other surfaces. How
the waves deflect off of the mediums all depends on the absorption and
transmission coefficients.

Good example pictures are shown at Crutchfield
Advisor,
a Physics Site from MTSU,
and Voiceteacher.com

## Room Coefficients

In a perfect world, if there is a sound shot right at a wall, the sound
should come right back. But because sounds hit different materials types
of walls, the sound does not have perfect reflection. From 1, these are
explained as follows:

### Absorption & Transmission Coefficients

The best way to explain how sound reacts to different mediums is through
acoustical energy. When sound impacts on a wall, acoustical energy will
be reflected, absorbed, or transmitted through the wall.

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

------------------------------------------------------------------------

Absorption Coefficient:
![](Absorption_Coefficient.jpg "Absorption_Coefficient.jpg") NB: this
chemical structure is unrelated to the acoustics being discussed.

------------------------------------------------------------------------

Transmission Coefficient:
![](Transmission_Coefficient.jpg "Transmission_Coefficient.jpg")

------------------------------------------------------------------------

If all of the acoustic energy hits the wall and none is reflected, the
alpha would equal 1. The energy had zero reflection and was absorbed or
transmitted. This would be an example of a *dead* or *soft* wall because
it takes in everything and doesn\'t reflect anything back. Rooms that
are like this are called Anechoic Rooms which looks like this from
Axiomaudio.

If all of the acoustic energy hits the wall and all reflects back, the
alpha would equal 0. This would be an example of a *live* or *hard* wall
because the sound bounces right back and does not go through the wall.
Rooms that are like this are called Reverberant Rooms like this
McIntosh room. Look how the
walls have nothing attached to them. More room for the sound waves to
bounce off the walls.

### Room Averaged Sound Absorption Coefficient

Not all rooms have the same walls on all sides. The room averaged sound
absorption coefficient can be used to have different types of materials
and areas of walls averaged together.

------------------------------------------------------------------------

RASAC:    ![](Ralpha.jpg "Ralpha.jpg")
![](Sequals.jpg "Sequals.jpg")

------------------------------------------------------------------------

### Absorption Coefficients for Specific Materials

Basic sound absorption Coefficients are shown here at Acoustical
Surfaces.

Brick, unglazed, painted alpha \~ .01 - .03 -\> Sound reflects back

An open door alpha equals 1 -\> Sound goes through

Units are in
Sabins.

## Sound Decay and Reverberation Time

In a large reverberant room, a sound can still propagate after the sound
source has been turned off. This time when the sound intensity level has
decay 60 dB is called the reverberation time of the room.

------------------------------------------------------------------------

![](Revtime.jpg "Revtime.jpg")           ![](Reverberation_Time.jpg "Reverberation_Time.jpg")

------------------------------------------------------------------------

Great Reverberation
Source

## Great Halls in the World

Foellinger Great
Hall

Japan

Budapest

Carnegie Hall in New
York

Carnegie
Hall

Pick Staiger at Northwestern
U

------------------------------------------------------------------------

Concert Hall
Acoustics

------------------------------------------------------------------------

## References

\[1\] Lord, Gatley, Evensen; *Noise Control for Engineers*, Krieger
Publishing, 435 pgs

Back to Engineering
Acoustics

Created by Kevin Baldwin


# Engineering Acoustics/Basic Room Acoustic Treatments

## 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
misconseption 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 acoustics of the room 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.

sound travels in the ground faster than in the air because the particles
in the solid are closer so they vibrate faster and have a quicker
reaction.

## 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, changing the character of the sound. 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

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

a\. Cube

b\. Room with 2 out of the three dimensions equal

c\. Rooms with dimensions that are multiples of each other

##### 2. Move the chair or sofa 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 surfaces absorb sound can be found on the website
<http://www.crutchfieldadvisor.com/learningcenter/home/speakers_roomacoustics.html?page=2#materials_table>

### 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
<http://www.crutchfieldadvisor.com/S-hpU9sw2hgbG/learningcenter/home/speakers_roomacoustics.html?page=4>:

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 diffusor.

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 Sound

<http://www.ecoustics.com/Home/Accessories/Acoustic_Room_Treatments/Acoustic_Room_Treatment_Articles/>

<http://www.audioholics.com/techtips/roomacoustics/roomacoustictreatments.php>

<http://www.diynetwork.com/diy/hi_family_room/article/0,2037,DIY_13912_3471072,00.html>

<http://www.crutchfieldadvisor.com/S-hpU9sw2hgbG/learningcenter/home/speakers_roomacoustics.html?page=1>

Back to the main page


# Engineering Acoustics/Electro Acoustic Installations in Room

As the development of new and sophisticated technology is advancing at
an exponential rate, it has become quite evident that the study of room
acoustics would be rather incomplete without the analysis of the effect
of electroacoustic devices. Whether the Canadian Prime Minister is
addressing the members of parliament, a university professor is giving a
lecture in a large auditorium, the CEO of one of the largest tech
companies holds a press conference to unveil their company's technology,
or thousands of heavy metal fans gather to see Metallica perform live,
loudspeakers and microphones are frequently used as a means of speech
amplification. This addition to the Engineering Acoustics Wikibook will
aim to aid the reader in strategic placement of electroacoustic devices
in any room. It will also serve as an extension to the previous Wikibook
entry, entitled Room Acoustics and Concert Halls.

# Loudspeaker Directivity

Although, a common sound amplification setup consists of a microphone,
amplifier and a loudspeaker, the loudspeaker is perhaps the crucial
component of the arrangement. Loudspeakers must be designed in order to
withstand high power while not causing any distortions in the radiated
sound.\[Kuttruff\]

First, a description of the radiation characteristics of a loudspeaker
is of interest. !Motion of piston creating soundwaves at an angle theta
from the
horizontal
When waves having high frequencies begin propagating due to the motion
of the piston, the radius of the piston can no longer be considered
small relative to the wavelength of the radiated sound. Thus, a
significant phase difference occurs between the radiated wave motion and
the motion of the piston. This leads to various degrees of interference
between both elements, which may lead to total cancellation of all
sound. Thus, it is useful to define a directivity function as follows:

$\delta(\theta)=\frac{2J_1(kasin(\theta))}{kasin(\theta)}$

`J1 is the first order Bessel Function`\
`θ is the angle of radiation`\
`k is the wave number, which is the ratio of angular frequency to the speed of sound `\
`a is the radius of the piston`\
`ka is referred to as the Helmholtz number`

A low Helmholtz number to a minimum, as this leads to a uniform sound
directivity. A large Helmholtz number results in a very directive sound
radiation as shown in the figures below.

Next, let us consider the horn loudspeaker, which provides a more
practical model of the loudspeaker. The main advantage of the horn model
is that it provides a broader directivity than the narrow scope provided
by the piston loudspeaker. It is often desirable to combine several horn
loudspeakers. The horn loudspeaker increases radiation resistance. Since
the directional characteristics of horn loudspeakers depend on both the
size and shape of the opening, as well as on the entire shape of the
horn, an expression for the directivity becomes complex.

Multiple loudspeakers will often be found in a room. Each loudspeaker
may be modeled as a point source. A directivity function for the
ensemble may be defined similarly as for a single piston loudspeaker.

$\delta_a (\theta)=\frac{sin(0.5*Nkdsin(\theta))}{Nsin(0.5kdsin(\theta))}$

Here, the angle of radiation is taken along the normal direction of the
array containing N point sources, spaced along a straight line at an
equal distance, d, from each other.

# Acoustical Feedback

If both the loudspeakers and microphone of an electroacoustic setup are
in the same room, the microphone will pick up sound propagating from the
loudspeakers as well as from the original source. This occurrence leads
to a phenomenon known as acoustical feedback
1. This phenomenon often
leads to a disruption in the entire electroacoustic setup and one may
hear loud howling or whistling sounds. As already discussed, loudspeaker
positioning plays a vital role in the minimization of acoustic feedback.
In order to effectively analyze the effects of acoustical feedback in a
room, it is wise to represent the propagation of sound waves from a
source such as a loudspeaker by means of a block diagram. !Block
Diagram Representation of Acoustical feedback in a room of sound
radiating from a
source

$\frac{Y(\omega)}{S(\omega)}=O(\omega)=\frac{KG'(\omega)}{1-G(\omega)K(\omega)}$

$Y(\omega)$` is the complex amplitude spectrum of the signal at the listener's seat`\
$S(\omega)$` is the sound input (into a microphone)`\
$K$` is the amplifier gain`\
$G'(\omega)$` is a complex transfer function representing the path by which the sound will reach the listener `\
$G(\omega)$` is the transmission function representing the path taken by sound radiating from the loudspeaker back into the microphone`

The closed loop transfer function at right shows how acoustical feedback
is caused by the original source signal passing continuously through the
loop, ending up back at the microphone. The term in the denominator of
complex transfer function, $KG(\omega)$, represents the open-loop gain
of the system. The magnitude of this quantity greatly impacts the
amplitude of the signal outputted to the listener, $Y(\omega)$. This
outputted variable represents the sound waves that actually propagate
toward the listener while the rest enter the feedback loop where they
make there way back to the microphone.

!Open Loop Block Diagram of Sound travelling through a room. This
diagram idealizes the actual closed loop system by neglecting the
presence of any acoustical
feedback

By observing the closed loop transfer function defined by $O(\omega)$,
one can see that the poles of the function for which the denominator
vanishes are of immediate interest. If the value of the open loop gain
equals unity, the entire system becomes unstable. If the open loop gain
equals unity, the closed loop transfer function tends to infinity,
producing a large scale instability. Similarly, if the open loop gain is
than unity, the value of the closed loop transfer function becomes
increasingly large as the value of $KG(\omega)$ tends to one. At such
large values of the open loop transfer function, the sound heard by the
listener will be distorted. This effect is better explained when an
impulsive signal is applied at the source, in which case ringing effects
are heard as the sound waves reach the listener. It is of the highest
importance to allow the system to operate within an appropriate range of
amplifier gain values. Through experimentation (See reference 1,) the
amplifier gain must be set in such a way that the following equations is
satisfied:

$20log(\frac{K}{K_o})<-12 dB$, where K_o is the critical amplifier gain
value for which $O(\omega)$ grows without bound.

Naturally, it is not in one\'s best interest to reduce the amplifier
gain to infinitessimily small values. This would simply render the
amplifier useless and the radiated sound signal would be weak. A wiser
strategy in reducing the effects of acoustical feedback involves in
attempting to reduce G(ω), while increasing G\'(ω). Although this may
prove to be difficult, the meticulous selection of the loudspeaker
directivity, with the main lobe pointing towards the listeners, while
the microphone is positioned away from the main lobes, in a position of
weak radiation, will greatly reduce undesirable feedback effects. Such
unidirectional microphones as Cardioid microphones are commonly used as
they effectively reject sounds propagating from other directions other
than those originating from the source (i.e from the speaker.) The
stability of closed loop feedback systems may also be analysed via such
tools as root locus plots or by using the Routh-Hurwitz criterion. In
traditional control theory, lead or lag compensators are frequently used
to improve the stability of the system
2.

# Loudspeaker Positioning

When deciding on an appropriate location for each loudspeaker in a room,
it is necessary to take into account certain factors. As seen in the
previous section, loudspeaker directivity can be manipulated depending
on the amount of loudspeakers present and if they are arranged linearly
in a group. Regardless, loudspeakers should be placed in such a manner
that a uniform sound energy is supplied to all listeners in the room.
Also, it is necessary to establish an adequate speech intelligibility,
which is measured in accordance with the speech intelligibility index
3.

Loudspeaker location must be chosen in such a way that the audience is
supplied direct sound that is as uniform direct as possible. One way of
achieving this is by mounting the loudspeaker at a higher altitude than
the sound source (i.e. microphone), which also limits feedback. This
also ensures that the direct sound will also arrive to the listener with
a relatively uniform directivity in the horizontal plane. Usually, the
human ear is not sensitive to sound variations in the vertical plane,
thereby making the elevation of loudspeakers a wise chose.

It is also of interest to derive a quantity known as the radius of
reverberation, R~o~. Consider sound leaving the loudspeaker with
intensity, I~o~. This intensity then makes it way through the closed
loop system, arriving back at the microphone input. It is then
re-amplified by a gain A to a value of AI~o~ as it passes through the
amplifier and makes its way back to the source. The intensity near the
loudspeaker will then decay as a function of the radial distance,R, from
the loudspeaker according to 1/R^2^. The intensity will decay until a
distance of R~O~ is reached. This radius of reverberation is then
defined as:

$R_O=0.057sqrt{\frac{V}{T}}$ where

`V is the volume of the room in m`^`3`^\
`T is the reverberation time in seconds`

## Loudspeaker Positioning for the Improvement of the reverberation of a Room using Direct Feedback

Since modern amphitheatres and stadiums serve as multipurpose venues,
controlling the reverberation time of the room to suit a specific event
is of importance. For example, Montreal, Quebec\'s Bell Centre serves as
the home of the Montreal Canadiens hockey team while also hosting a
multitude of live concerts and shows every year. The reverberation time
of such multipurpose venues can be modified with the aide of rotating or
removable walls or ceilings or even thick curtains. Such movable items
can play a role in altering the room\'s absorbtion and reflection
properties. However, the implementation of such devices may prove to be
quite costly. Thus, one\'s attention should be turned to the strategic
placement of loudspeakers in order to increase the reverberation time of
a room. A popular method has been discussed in a 1971 paper by Guelke
and Broadhurst.

In order to increase the reverberation time in a room by means of direct
feedback, it should be noted that the total sound intensity in a room is
the sum of the intensity of the source along with contributions from the
pressure from each of the room modes. Both contributions will decay
exponentially according to the relation:

$I_\mathrm{Tot}=I_\mathrm{Room}e^\mathrm{-kt}+I_\mathrm{sys}e^\mathrm{-Lt}$

$I_\mathrm{Tot}$` is the total sound intensity radiated in the room`\
$I_\mathrm{sys}$` is the sound intensity prom the acoustical system comprising of a microphone, amplifier and loudspeaker`\
$I_\mathrm{Room}$` is the sound intensity due to the pressure modes of the room`\
`k is the room damping coefficient`\
`L is the system damping coefficient`\
$T_\mathrm{60}$` is the reverberation time or more specifically, `\
`the time it takes for the energy density or sound pressure level in the room to decay by 60 dB, once the source is abruptly shut off.`

Furthermore, the system damping coefficient as well as the system sound
intensity may be related to the room\'s damping coefficient and radiated
sound intensity by constants n and m respectively, such that:

$L=nK$

$I_\mathrm{sys}=mI_\mathrm{Room}$

Under initial conditions at time, t=0, the total sound intensity in the
room is:

$I_\mathrm{Tot}=I_\mathrm{room}+mI_\mathrm{room}$

Next, the reverberation time is defined to be the point where the sound
intensity reaches 10^−6^ of its original value. Thus, the above equation
becomes:

$10^\mathrm{-6}(1+m)=e^\mathrm{-kt}+me^\mathrm{-nkt}$

Solving for the reverberation time,$T_\mathrm{60}$, yields:

$T_\mathrm{60}=\frac{-1}{nk}log_{e}(\frac{10^\mathrm{-6}(1+m)}{m})$

If the reverberation time of an isolated room is defined as:

$\frac{log_{e}10^6}{k}=\frac{13.8}{k}$

then the total reverberation time, t, may be written as follows:

$t=\frac{t_\mathrm{room}f(m)}{n}$ where f(m) is
$\frac{log_{e}(\frac{10^\mathrm{-6}(1+m)}{m})}{13.8})$

At the microphone, the magnitude of the sound intensity, I~o~ becomes
$\frac{AI_o}{R_o^2}$, so that A = R~o~^2^. Consequently, the larger the
the radial distance, the larger the average sound intensity in the room.
The directivity factor of the loudspeakers may help to create a more
diffusive sound field. Hence, the radial distance, R, is increased by an
increase in the directivity factor. Also it has been should that delay
times greater than 75 ms produce undesirable whining noises. It is
unwise to increase the reverberation time, T, of such a simple feedback
system beyond 1.5 s as the system very near the point of instability has
at least one frequency with a very long reverberation time.

By taking a value of 75 ms for $\tau$ in the following equation, the
reverberation time, t, in open space is defined by the relation:

$t=\frac{60\tau}{20log_{10}(A)}$, where $\tau = \frac{d}{c}$ c is the
speed of in air, 344 m/s and d is the distance between microphone and
loudspeaker

It is noteworthy to mention the works of M.R. Schroeder presented in his
1964 paper on \"Acoustic-Feedback Stability by Frequency Shifting\" In
essence,in order to avoid any ringing or howling noises of any kind in
the feedback loop, Schroeder suggested a frequency shift of 4 Hz into
the feedback loop. This was shown to greatly improve the stability of
the system. However,the method produces undesirable sidebands Because of
this phenomenon, the method is undesirable for concert halls. To improve
on Schroeder\'s method, Guelke and Broadhurst suggest the introduction
of phase modulation into the system in order to achieve longer
reverberation times and increased system stability.

By definition, if a linear, time invariant (LTI) system, such as the one
described by the closed feedback loop of a microphone, amplifier and
loudspeaker, is unstable, radiated sound will increase exponentially and
without bound at certain frequencies. By the implementation of a phase
reversal switch, sounds at the initially unstable frequency will
stabilize. However, a new set of frequencies that were once stable will
begin to grow without bound. Nonetheless, such a consequence of the
utilization of phase reversal is minor as it may be accounted for by
ensuring that the switch is activated a couple of times a second. This
method allows for the reverberation time to be held constant while
assuring that all strange ringing or howling noises vanish.

Such rapid triggering of the switch does induce another concern in the
continued presence of transient effects. Luckily, such effects become
negligible if the phase is varied sinusoidally.

Ultimately, the modulating frequency must be chosen such that:

`1. The change in phase should be enough to stabilize the system `\
`2. The sidebands produced are so small that they be neglected`

The first criteria may be satisfied if the modulation frequency is set
to a value less than the reciprocal of the delay time, τ. The second
criterion may be satisfied by ensuring the the modulation frequency is
below one hertz. A modulation frequency of 1 Hz is so negligible that it
will not be picked up by the human ear. Briefly, based on the definition
of τ increase the distance between microphone and loudspeaker to achieve
a minimal modulation frequency, which will stabilize the system and be
practically non-existent to the human ear.

For further reading on ensuring that the phase modulation is sinusoidal,
it is advised to consult reference \[2\] or a textbook on control
systems, if desired. Introducing a sinusoidal phase modulator of the
form:

$\phi(t)=ksin(w_mt)$

Introducing a modulator that is purely sinusoidal in nature greatly
simplifies the usually non-linear nature of a real modulator, which
greatly simplifies the problem at hand.

# References

\[1\] Kuttruff, Heinrich; *Room Acoustics, Fourth Edition*, Spon Press
Publishing, 342 pgs

\[2\] Guelke, R. W. and A. D. Broadhurst (1971). \"Reverberation Time
Control by Direct Feedback.\" Acta Acustica united with Acustica 24(1):
33-41.

\[3\] Schroeder, M. R. (1964). \"Improvement of Acoustic‐Feedback
Stability by Frequency Shifting.\" The Journal of the Acoustical Society
of America 36(9): 1718-1724.


# Elements of Political Communication/References

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\_\_NOTOC\_\_


# Communication Theory/Nonverbal Communication

Scholars in this field usually use a strict sense of the term
\"verbal\", meaning \"of or concerned with words,\" and do not use
\"verbal communication\" as a synonym for oral or spoken communication.
Thus, sign languages and
writing are generally understood as forms of
verbal communication, as both make use of words --- although like
speech, both may contain paralinguistic elements and often occur
alongside nonverbal messages. Nonverbal communication can occur through
any sensory
channel "wikilink") ---
sight, sound,
smell,
touch or
taste. Nonverbal communication is also
distinguished from unconscious
communication, which may be
verbal or non-verbal.

## Voluntary

This is less commonly discussed because it seems unproblematic, refers
to movement, gestures and poses intentionally made by the person:
smiling, moving hands, imitating actions, and generally making movements
with full or partial intention of making them and a realisation of what
they communicate. It can apply to many types of soundless communication,
for example, formalized gestures.

## Involuntary

This applies to involuntary movements that may give observers cues about
what one is really thinking or feeling. The ability to interpret such
movements may itself be unconscious, at least for untrained observers.

Many elements of involuntary body language can easily be understood, and
tested, simply by knowing about them. For example, the tendency for
people to raise their eyebrows as one approaches them fact-to-face is
usually indicative of esteem. If you walk down the street and encounter
someone you do not know then the chances are that neither of you will
raise your eyebrows. If you recognize each other, however, even if you
do not greet each another, then eyebrows will likely raise and lower. Of
particular interest here in a work context is that if one is not rated
highly by the other person then that person will not raise their
eyebrows, even though one is
recognised.1.

It is widely believed that involuntary body language is the most
accurate way into a person\'s subconscious. In principle, if people do
not realize what they are doing or why they are doing it, it should be
possible for a trained observer to understand more of what they are
thinking or feeling than they intend - or even more than they realize
themselves. Interrogators, customs examiners, and others who have to
seek information that people do not necessarily want to give have always
relied on explicit or implicit hypotheses about body language. However,
this is a field that is fraught with risk of error, and it has also been
plagued with plausible but superficial or just plain erroneous popular
psychology: just because someone has
their legs crossed toward you, it does not mean that they want to have
sex with you; it could just mean that they are comfortable with you, but
it could also be how they always sit regardless of where you are.
Furthermore, it is not possible to tell reliably whether body language
has been emitted voluntarily or involuntarily, so to rely too heavily on
it is to run the risk of being bluffed.

Research conducted by Paul Ekman at the end
of the 20th Century resolved an old debate about how facial
expressions vary between cultures. He
was interested in whether, for instance smiling,
was a universal phenomenon, or whether there are cultures in which its
expression varies. Ekman found that there were several fundamental sets
of involuntary facial muscle movements relating to the experience of a
corresponding set of emotions: grief, anger, fear, enjoyment and
disgust. He also indicates that, whilst the furrowing of the eyebrows
when experiencing grief is difficult to perform
voluntarily, such expressions can be learnt through practice. Ekman\'s
ideas are described and photographically illustrated in his book
*Emotions
Revealed*2.

The use of video recording has led to important discoveries in the
interpretation of micro-expressions,
facial movements which last a few milliseconds. In particular, it is
claimed that one can detect whether a person is lying by interpreting
micro-expressions correctly. Oliver Sacks,
in his paper *The President\'s Speech*, indicated how people who are
unable to understand speech because of brain damage are nevertheless
able to assess sincerity accurately. He even suggests that such
abilities in interpreting human behavior may be shared by animals such
as domestic dogs.

A recent empirical study of people\'s ability to detect whether another
was lying established that some people can detect dishonesty
consistently reliably. This study showed that certain convicts, American
secret service agents and a Buddhist monk were better at detecting lying
in others than most people, and it is postulated that this ability is
learned by becoming observant of particular facial micro-expressions.

Body language is a product of both genetic and environmental influences.
Blind children will smile and laugh even though
they have never seen a smile. The ethologist
Iraneus Eibl-Eibesfeldt claimed
that a number of basic elements of body language were universal across
cultures and must therefore be fixed action
patterns under
instinctive control. Some forms of human body
language show continuities with communicative gestures of other
apes, though often with changes in meaning - the
human smile, for example, seems to be related to the open-mouthed threat
response seen in most other primates. More
refined gestures, which vary between cultures (for example the gestures
to indicate \"yes\" and \"no\"), must obviously be learned or modified
through learning, usually by unconscious observation of the environment.

## Reference

Argyle, M. (1975). Bodily communication.
New York: International Universities Press.


# This Quantum World/Cover

## About this book

Since 1999 I have taught an introductory course of contemporary physics
to high-school students (grades 11-12) and undergraduates at the Sri
Aurobindo International Centre of
Education
(SAICE) in Pondicherry "wikilink"),
India. The SAICE is not your typical high school or college. While the
students enjoy an exceptional freedom to choose their projects and
courses, the teachers are free to offer subjects of their own choosing
and are encouraged to explore new methods of teaching. My course is
therefore optional. It is suitable for anyone with an interest in what
contemporary physics is trying to tell us about the \"nature of
Nature.\" To students of physics it offers a perspective that is
complementary to those of many excellent textbooks.

Every year at the beginning of the new term I revise and try to improve
the material I hand out to my students. These revisions are to a
considerable extent based on student feedback. I am presently preparing
the handouts for the next term and intend to simultaneously make them
available in these pages, hoping for additional valuable feedback from
the Wikibooks community.

Postscript: I have not been able to add chapters and sections as fast as
I would have liked, but it\'s going to happen. I haven\'t given up on
this project!\--Koantum 03:05, 27 September
2007 (UTC)


# This Quantum World/Atoms

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

None of these images depicts an atom *as it is*. This is because it is
impossible to even visualize an atom *as it is*. Whereas the best you
can do with the images in the first row is to erase them from your
memory---they represent a way of viewing the atom that is too simplified
for the way we want to start thinking about it---the eight fuzzy images
in the next row deserve scrutiny. Each represents an aspect of a
stationary state of atomic hydrogen. You see neither the nucleus (a
proton) nor the electron. What you see is a fuzzy position. To be
precise, what you see are cloud-like blurs, which are symmetrical about
the vertical and horizontal axes, and which represent the atom\'s
internal relative position---the position of the electron relative to
the proton *or* the position of the proton relative to the electron.

-   What is the *state* of an atom?
-   What is a *stationary* state?
-   What exactly is a *fuzzy* position?
-   How does such a blur represent the atom\'s internal relative
    position?
-   Why can we not describe the atom\'s internal relative position *as
    it is*?

## Quantum states

In quantum mechanics, **states** are
probability algorithms. We use them to calculate the probabilities of
the possible outcomes of
measurements on the
basis of actual measurement outcomes. A quantum state takes as its input

-   one or several measurement outcomes,
-   a measurement M,
-   the time of M,

and it yields as its output the probabilities of the possible outcomes
of M.

A quantum state is called **stationary** if the probabilities it assigns
are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density
function
$\rho(\boldsymbol{r})$. Imagine a small region $R$ like the little box
inside the first blur. And suppose that this is a region of the
(mathematical) space of positions relative to the proton. If you
integrate $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. If it
were inside, the probability of finding it outside would be zero, and if
it were outside, the probability of finding it inside would be zero.
*After* the measurement, on the other hand, the electron is either
inside or outside $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    is neither true nor false; it lacks a (definite) truth
    value.
-   A measurement generally changes the state of the system on which it
    is performed.

As mentioned before, probabilities are assigned not only *to*
measurement outcomes but also *on the basis of* measurement outcomes.
Each density function $\rho_{nlm}$ serves to assign probabilities to the
possible outcomes of a measurement of the electron\'s position relative
to the proton. And in each case the assignment is based on the outcomes
of a simultaneous measurement of three observables: the atom\'s energy
(specified by the value of the principal quantum number $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ is neither 0 nor 1.

What about observables that are generally described as continuous, like
a position?

The description of an observable as \"continuous\" is potentially
misleading. For one thing, we cannot separate an observable and its
possible values from a measurement and its possible outcomes, and a
measurement with an uncountable set of possible outcomes is not even in
principle possible. For another, there is not a single observable called
\"position\". Different partitions of space define different position
measurements with different sets of possible outcomes.

-   Corollary: The possible outcomes of a position measurement (or the
    possible values of a position observable) are defined by a partition
    of space. They make up a finite or countable set of *regions* of
    space. An exact position is therefore neither a possible measurement
    outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron\'s fuzzy
position relative to the proton? Strictly speaking, they graphically
represent probability densities in the mathematical space of exact
relative positions, rather than fuzzy positions. It is these probability
densities that represent fuzzy positions by allowing us to calculate the
probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom\'s internal
relative position *as it is*. To describe a fuzzy observable is to
assign probabilities to the possible outcomes of a measurement. But a
description that rests on the assumption that a measurement is made,
does not describe an observable *as it is* (by itself, *regardless of
measurements*).

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