How can we hear the shape of the drum using Fournier analysis? by OGBONNAYA AKPARA
Ogbonnaya Akpara
Math Summer Research
Mentor: Dr. Terrence Blackman
August 23 rd , 2013
How can we hear the shape of the drum using Fournier analysis?
The basic question is this: Is it possible to distinguish between some shapes by using the
frequency spectrum alone. What do I mean? Can we “hear” the difference between a rectangular drum
and a circular drum or “what can you tell about a drum from its frequency spectrum”? Some features
are evident, others are not. Although the frequency domain representation of a signal may not tell us
everything about its source, it can indeed provide us with some information. Using techniques of
Fournier analysis and frequency domain representation, we will find out the information about the
shapes of drums that we cannot see. These techniques help us to determine the chemical composition
of star millions of light years away. Breaking up a sound or other signals into frequency components can
help us uncover fundamental information about origin and structure that is not otherwise evident.
Have you noticed that music and math are related? Stories were told that the violin prodigy
exceled in calculus while the musical composer was good at prime numbers. That’s how music and math
are connected to each other. The talent of music and math are related in even abstract ways. The
societal, psychological, and perhaps even biological factors involves in the coincidence of math and
music. Music and math are connected by wave phenomena and the understanding of sound.
Understanding sound as an example of wave phenomena provides a good forum for the
communication of ideas from music, physics and mathematics. Also, understanding the nature of sound,
such as Fournier analysis can be generalized to mathematical understanding of sound that has helped
foster the development of new technologies that extend the possibilities for musical exploration.
Music is an important role in many cultures. In some cultures, music is an active art in which
everyone encouraged will partake. In other cultures, music is a form of worship or entertainment to be
practiced by few individuals but appreciated by many. Music plays a key role in Greek society. It was an
element of religious ceremonies, sporting event and feast and it was part of Greek theatre. In making
their music, Greeks used techniques that are still commonly used today, employing strings, reed and
resonant chambers to create and control tones and melodies. The Greeks were the first people to apply
math to music. They did a pretty good job understanding what math would bring to music. The Greeks
noticed a relationship between harmonic intervals and rational numbers. They had the understanding of
the most basic musical concept of all (sound). Sound is caused by compression and rarefaction of air
molecules. Aristotle suspected that sound was some sort of “disturbance” that is propagated through
the air.
How do we hear? The ability to hear is such an integral part of life that many of us take for
granted. The ear is a complex organ comprising of three parts. The outer ear, inner ear and the middle
ear. It is important that each part is functioning correctly for us to hear properly. The outer ear is made
up of the skin and cartilage on the outside, and the ear canal that leads into your head. Also, it is the
part that collects sounds and directs them down the ear. The middle ear which begins at the ear drum
about 2.5 cm inside your head and includes the little bones that carry the sound vibrations to the area
where hearing really begins. The middle ear contains 3 small bones (n the hammer, anvil and stirrup),
these tiny bones are connected to both the eardrum and inner ear and they are shaped like a cone of a
loud speaker. The inner ear where those vibrations are changed into the signal that is carried to your
brain, which you experience as sound. This part of your ear controls your balance. In the inner ear, it is
mainly responsible for sound detection and balance. The inner ear is found in all vertebrates, with
substantial variations in form and function. The inner ear is innervated by the eight cranial nerve in all
vertebrates.
Sound waves travel down the ear canal and hit the eardrum in the middle of the ear. This
causes the eardrum to vibrate. The eardrum passes the vibrations through the middle ear bones or
ossicles into the inner ear. The inner ear is shaped like a snail and it is called the cochlea. Inside the
cochlea, there are thousands of tiny hair cells. Hair cells change vibrations into electrical signals that are
sent to the brain through the hearing nerve. The brain tells you that you are hearing a sound and what
that sound is. A string of length L can be idealized by the interval [0,L] for all real numbers x between
and including 0 and L (0<x<L). We represent a possible configuration of the vibrating string as a function
f (x, t) defined for x in [0,L].
Math Summer Research
Mentor: Dr. Terrence Blackman
August 23 rd , 2013
How can we hear the shape of the drum using Fournier analysis?
The basic question is this: Is it possible to distinguish between some shapes by using the
frequency spectrum alone. What do I mean? Can we “hear” the difference between a rectangular drum
and a circular drum or “what can you tell about a drum from its frequency spectrum”? Some features
are evident, others are not. Although the frequency domain representation of a signal may not tell us
everything about its source, it can indeed provide us with some information. Using techniques of
Fournier analysis and frequency domain representation, we will find out the information about the
shapes of drums that we cannot see. These techniques help us to determine the chemical composition
of star millions of light years away. Breaking up a sound or other signals into frequency components can
help us uncover fundamental information about origin and structure that is not otherwise evident.
Have you noticed that music and math are related? Stories were told that the violin prodigy
exceled in calculus while the musical composer was good at prime numbers. That’s how music and math
are connected to each other. The talent of music and math are related in even abstract ways. The
societal, psychological, and perhaps even biological factors involves in the coincidence of math and
music. Music and math are connected by wave phenomena and the understanding of sound.
Understanding sound as an example of wave phenomena provides a good forum for the
communication of ideas from music, physics and mathematics. Also, understanding the nature of sound,
such as Fournier analysis can be generalized to mathematical understanding of sound that has helped
foster the development of new technologies that extend the possibilities for musical exploration.
Music is an important role in many cultures. In some cultures, music is an active art in which
everyone encouraged will partake. In other cultures, music is a form of worship or entertainment to be
practiced by few individuals but appreciated by many. Music plays a key role in Greek society. It was an
element of religious ceremonies, sporting event and feast and it was part of Greek theatre. In making
their music, Greeks used techniques that are still commonly used today, employing strings, reed and
resonant chambers to create and control tones and melodies. The Greeks were the first people to apply
math to music. They did a pretty good job understanding what math would bring to music. The Greeks
noticed a relationship between harmonic intervals and rational numbers. They had the understanding of
the most basic musical concept of all (sound). Sound is caused by compression and rarefaction of air
molecules. Aristotle suspected that sound was some sort of “disturbance” that is propagated through
the air.
How do we hear? The ability to hear is such an integral part of life that many of us take for
granted. The ear is a complex organ comprising of three parts. The outer ear, inner ear and the middle
ear. It is important that each part is functioning correctly for us to hear properly. The outer ear is made
up of the skin and cartilage on the outside, and the ear canal that leads into your head. Also, it is the
part that collects sounds and directs them down the ear. The middle ear which begins at the ear drum
about 2.5 cm inside your head and includes the little bones that carry the sound vibrations to the area
where hearing really begins. The middle ear contains 3 small bones (n the hammer, anvil and stirrup),
these tiny bones are connected to both the eardrum and inner ear and they are shaped like a cone of a
loud speaker. The inner ear where those vibrations are changed into the signal that is carried to your
brain, which you experience as sound. This part of your ear controls your balance. In the inner ear, it is
mainly responsible for sound detection and balance. The inner ear is found in all vertebrates, with
substantial variations in form and function. The inner ear is innervated by the eight cranial nerve in all
vertebrates.
Sound waves travel down the ear canal and hit the eardrum in the middle of the ear. This
causes the eardrum to vibrate. The eardrum passes the vibrations through the middle ear bones or
ossicles into the inner ear. The inner ear is shaped like a snail and it is called the cochlea. Inside the
cochlea, there are thousands of tiny hair cells. Hair cells change vibrations into electrical signals that are
sent to the brain through the hearing nerve. The brain tells you that you are hearing a sound and what
that sound is. A string of length L can be idealized by the interval [0,L] for all real numbers x between
and including 0 and L (0<x<L). We represent a possible configuration of the vibrating string as a function
f (x, t) defined for x in [0,L].
Comments
Post a Comment