Sound Research

Derek Bartelt

Patrick Behringer

Caleb Cooper

Jordan Leisener

Mark Moser

Outline

Thesis- Since they were discovered, acoustic vibrations have certain mathematical properties.

I.          Intro

II.         History

A.     Developers            

B.     Materials

C.     Practical uses

III.       The mathematics behind sound

A.     Waves

B.     Frequency

C.     Math behind music

IV.       Acoustic Sounds

A.     Instruments

B.     Modeling Sound Waves

C.     Major advances in sound technology

V.        Conclusion

            “Boom, Boom, Boom!!!” You can feel the house shake as the lightning strikes next to your house.  The thunder is so loud that you can feel your bed shake as you’re lying down to go to sleep.  Sound is an amazing and powerful thing that intrigues many.  It can shatter glass or make beautiful noise to our ears.  Although not highly thought of as mathematical, there are multiple properties that relate sound to math.  Let us now explore the mathematical properties of resonance, or vibration.

            Bells have a rich history. Although there is no record as to when the first bell was created, it is widely accepted that bells came into existence no later than the second century. Exodus mentions bells as a part of Hebrew worship and they are shown  as decorations on the robes of priests. After this, the Chinese were one of the first to create an instrument that depended on vibration to make sound.  They used a portable set of bells, which were tuned to intervals of the major scale. These bells are depicted in many pictures from early Chinese dynasties.  It is supposed that bells were first used as an alarm, which was to signify the approach of an important person or an approaching enemy.  Greek warriors were rumored to have small bells concealed within their shields, which they were required to ring at certain intervals while on guard. Romans used large portable bells to summon their servants. Later on, bells were used in towers and attached to clock like mechanisms to strike the hours. This regular tolling became very important to people in early days. It was the only way they could keep track of time. Most of the more modern bells like we use today were first produced in France because that had become the art center of the world. The use of bells and bell making, however, diminished around the French revolution. Later in the nineteenth century, English bell makers rediscovered the tuning that had been used by the French.  The English also improved the bell and how to strike it creating the modern bells and chimes that we have today.

             Throughout history, bells have been associated with religion. They also have been used in connection with all major religions except Islam. In ancient times, bells were worn by priests. At that time they had no resemblance to the present shape of bells. These early bells were merely flat metallic disks, a number of which were suspended by cords on the priests’ robes. When walking, the bells would strike against each other, causing a vibration that notified those within hearing distance that a high official was approaching. When the Christian church became accepted in Rome, priests would hand bells outside of their churches.  In medieval times, bells were steeped in superstition. They were baptized and hung in doorways to ward of evil spells and spirits.  During the Middle Ages, sets of bells were used for musical instruction and to accompany the chants in the churches.  In the 1700's, bell ringers discovered that they could ring tunes, so they began to ring carols and hymns.  Today hand bells are still rung in churches and many churches all over the world have bell towers.

            Bells are not the only musical instrument that uses vibration to create sound.  The lyre also harnesses vibrations, except the lyre uses strings, which are plucked to create sound.   The strings are stretched between the crossbar and the sound box and are plucked with the fingers or with a plectrum. In ancient times Sumer, Babylonia, Israel, and Egypt had various sorts of lyres. Ancient Greece had two lyres—the kithara, which was the larger instrument used by the professional musician, and the lyre, the smaller instrument of the amateur. Each had from 3 to 12 strings, made of hemp. The tuning and playing techniques of modern lyres in E Africa are thought to be similar to those of ancient Greece and Egypt. After the 10th century, the lyres of Northern European countries were bowed instead of being plucked. The bowed lyre that persisted longest was the Welsh creth, known as early as the 11th century and still in use in the early 19th century. At some time in its history a fingerboard was added, making it an early member of the violin family.

            How do bells and stringed instruments work? The simple answer to this is vibration. Vibration is an oscillatory motion– a movement in one direction, and then back again in the opposite direction. A struck bell or string vibrates in multiple ways (modes), but only a few of them cause the surrounding air to vibrate strongly enough to be audible.  The simplest node gives rise to the lowest frequency vibration (the lowest pitch called the “fundamental”). The fundamental comes primarily from the lip of the bell. The whole vibration is a repeating process. When you ring the bell, the clapper pushes the bell outwards at the top of the bell. Because there is a fixed amount of metal in the bell, the sides must move inward.  This in turn squeezed the bottom of the bell, pushing it outward.  The net result is that the circular shape of the bell is turned into an ellipse.  The stresses caused by this distortion causes the bell to return to its circular shape, but at the instant the circle is attained the metal is moving.  This inertia leads to another ellipse, this time with the top and bottom inside the circle and the sides outside. Where these ellipses intersect are nodes – motionless lines that extend down the full length of the bell.  A pair of nodal lines that are directly opposite each other defines a geometric plane including the axis of the bell.  Hence the fundamental is said to consist of two “nodal planes” separated by an angle of 90 degrees.  The overtone that essentially makes a hand bell sound different from a tuning fork is the twelfth. The twelfth arises from an oscillation that has three nodal planes, separated by an angle of 60 degrees. The twelfth is emitted most strongly from the curved part of the bell -- about a third of the distance from the lip to the top. Both the fundamental and the twelfth are radial vibrations -- the motion of the metal is more-or-less directly towards or away from the axis of the bell. The air outside the bell is caused to vibrate radially as well, so the note will be loudest radially outward. It's also louder if a point between nodes (an "antinode") is pointed towards the listener. Since the clapper drives an antinode at both the strike point and the point 180 degrees around the bell, there isn't much difference in which point the clapper actually strikes. 

            A wave is a vibration in material.  A standing wave is a wave that is fixed at both ends and continually reflects off both of its boundaries to form nodes (where amplitude = 0 for both waves going in both directions) and antinodes (where amplitude is maximum).

The length of the material is represented by the letter (L) and (u) is the deflection from the equilibrium of the wave. Our goal is to represent (u) in terms of position (x) and time (t) for all x between 0 and L and all t > or = 0.  We can use partial derivatives to relate the three variables (u, x, t) Based on Newton’s 2^nd law, F= ma, (where F and a are vectors in the same direction) we can derive the equation  u²/L² = c² times u²/x² , where the left side represents acceleration of the wave and, c² is a constant which depends on mass and tension of the string, and the right derivative represents the curvature. One of the main characteristics of standing waves is that they can be represented by both sine and cosine functions in one equation. Therefore u (x ,t) = (c₁ cos (x) + c₂ sin(x))(k₁ cos(x) + k₂ sin (x))  where x= (2L)/n  and n is the harmonics, the number of times the wave crosses the x-axis.

            Frequency, or pitch, is the number of cycles of a sound wave in one second.  It is specified in hertz.  The frequency of a sound increases as the number of cycles per second increase.  High-pitched sounds, such as a police whistle, have a high frequency with thousands of cycles per second.  Low-pitched sounds, such as far away thunder or a tuba, have a low frequency with only a few cycles per second.  The deepest sound that a young, healthy individual will be able to hear has a frequency of 20 hertz, while the lightest has a frequency of approximately 20,000 hertz.  Most natural sounds do not have just one frequency.  They can have many frequencies at different amplitudes.  This quality of sound is called a timbre.  This is what makes a trombone and a clarinet sound different even when they are playing exactly the same note. 

            Music is both beautiful and profoundly mathematical.  In fact, the entire musical scale of whole and half steps come from simple requirements of ratios.  The idea of equal temperament is and approximation that comes from the twelfth root of two.  This determines the length of piano strings, the spacing of holes on a flute, and the location of frets on a guitar.

            There are many instruments that use vibration to create sound.  The bell uses a hammer in its center to strike the metal sides.  The guitar or lyre uses the vibration of a long string, which repeatedly moves back and forth to create sound.  The violin also uses strings, which are stroked with a bow to create sound.  Drums use the repeated banging of a stick to the hollowed out box (basically).  The human voice can also be thought of as an instrument.  As air passes through the vocal box, it vibrates chords that produce sound. 

            Sound waves look very much like sine or cosine lines on a graph.  There is a repeated pattern that goes from x = negative infinity to when x = positive infinity.  It also has a crest that is constant for infinity and a trough that is constant for infinity.  The number of times that the line goes through this cycle can be related to the frequency of a sound wave. At the midpoint between the crest and the trough of each wave, the length of the wave stays the same.  A sound wave also almost represents an exponential equation.  On a guitar, if you play a note or a chord it starts off at its loudest point.  Then, the amplitude slowly decreases until it is no longer audible. 

            Throughout this century there have been many advances in mapping sound waves.  Before the 20^th century, there was a lack of sophisticated equipment for processing and analyzing audio signals; so little progress was made in the study of sound until this century.  Much of our current knowledge about sound and acoustics began with the creation of electronic, acoustical and optical testing equipment, such as mikes, amplifiers and the cathode-ray oscilloscope.  Computers have aided in the very complicated process of analyzing sound waves by mathematical methods.  Scientific study in this area is now a multidisciplinary effort, relying on international exchanges of information and a plethora of sophisticated devices and computer equipment. 

            Resonance and sound waves have very many mathematical properties.  Basically, every instrument relies on resonance to create its musical tone.  Without vibrations there wouldn’t even be any sound.  Even though we have discovered a lot of new information on sound, there is still much to learn about it. We have barely tapped into the possibilities that sound has to hold.

Works Cited

Addell, James.  “The Physics of Bells.”  Directors’ Seminars; 1996.

            <http://www.ling.upenn.edu/~kingsbur/acoustics.html>

Harrison, David.  “Acoustic Resonance.”  Bell & Sons; 1994.

            <http://www.upscale.utoronto.ca/IYearLab/stwaves.pdf>

Marcolm, Jake.  “Sound Frequencies.”  Allacritude, LLC; 2002.

<http://www.encyclopedia.com/sound/frequency>

Morris, Timothy.  “A History of Bells.”  Pagewise, Inc.; 2002.

<http://oror.essortment.com/bellshistorycu_plj.htm>

Pillinger, Ian.  “The Nature of a Sound Wave.”  The Physics Classroom; 1999.

            <http://www.glenbrook.k12.il.us/gbssci/phys/Class/sound/u11l1a.html>

“The Mathematics of Sound.”  Yale New-Haven Teachers Institute; 2003.

            <http://www.yale.edu/ynhti/curriculum/units/2000/5/00.05.10.x.html#c>