Lecture 6Fourier AnalysisSympathetic VibrationsThe Human Ear

Instructor: David Kirkby (dkirkby@uci.edu)

                                                                                 

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MiscellaneousProblem set #1 handed back today. The average was 77%.

Problem set #2 due today.

Problem set #3 distributed today (also available from the course web site).

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The last question on Problem Set #1 asked you to estimate the speed of sound from two measurements:

•The time taken for 20 claps•The distance to the wall

Both of these measurements should have a spread, and these spreads shouldtranslate into a spreadin your answers.

Your answer can betoo good! Expected

spread

Too good!

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Sympathetic VibrationsIf two resonant objects are in contact, then driving one object can indirectly drive the other. The vibrations in the second object are called sympathetic vibrations.

FirstObject

SecondObject

Energy appliedDirectly to first object…

…can indirectly drivea second object.

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Sympathetic vibrations are important in musical instruments where the primary resonator (e.g., a vibrating string) does not itself move enough air to be audible.

primary resonator

secondary resonator

coupling

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Sympathetic Vibrations with Tuning Forks

A tuning fork vibrates (too a good approximation) at a single frequency. A tube open at both ends whose fundamental resonant frequencymatches the tuning fork willstore energy from the vibrationsof the tuning fork as astanding wave.

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Sympathetic Vibrations in a PianoTry this experiment the next time you are sitting at a piano:

Press gently on the C one octave below middle C (to free the string without striking it) then strike middle C sharply and release it.

After you release middle C you will continue to hear its tone. Why?

(This will only work if the piano is in tune. You can do a similar experiment with a guitar.)

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Understanding Complex VibrationsWe introduced the Principle of Superposition (PoS) as a tool to help analyze a complex vibration in terms of the superposition of many simpler vibrations due to Simple Harmonic Motion (SHM).

With our new understanding of resonances, this job has become a lot easier:

Instead of considering a continuum of possible SHM frequencies that might contribute, we only need to consider a well-defined set of resonant frequencies!

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Two Amazing Facts: Fourier AnalysisThere are two amazing facts that make the analysis of complex vibrations even simpler still:

(1) Any possible vibration (eg, of a string) can be described as a superposition of simple harmonic motions.

(2) The set of SHMs that make up a complex vibration, as well as their proportions, are unique.

The breakdown of a complex vibration intoits harmonic components is known asFourier Analysis.

Jean Baptiste Joseph Fourier (1768-1830)

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Amazing Fact #1Any possible vibration of a string can be described as a superposition of simple harmonic motions.

This means that you can sketch any curve you want between two points and it will be equivalent to a sum of sine (SHM) curves.

Try this demo to see examples of how non-SHM looking curves can be built out of SHM curves:

http://www.jhu.edu/~signals/fourier2/

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There are a few catches though…

In general, you need add an infinite number of SHM curves to match an arbitrary curve. But if you start with the lowest frequencies and then add higher frequencies, you quickly get the overall shape right and the higher frequencies are just refinements.

The second catch is that you need to specify more than how much of which frequencies to add. You also need to specify their relative phases. This is not very important for musical sound, so we will ignore this complication.

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Amazing Fact #2The set of SHMs that make up a complex vibration, as well as their proportions, are unique.

Because of this uniqueness, these two graphs give the same information about this sound:

Frequencyspectrum

Fun

dam

en

tal

1st

Overt

on

e

2n

d O

vert

on

e

3rd

Overt

on

e

4th

Overt

on

e

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(Actually, the frequency spectrum is missing the phase information that I said earlier we would ignore.)

A frequency spectrum often gives more insight into the resonances that are responsible for producing a musical sound and so we will use them often when studying instruments later in the course.

In the harmonica example, we learned that the resonances involved are approximately harmonic and that the 2nd harmonic is louder than the fundamental (=1st harmonic).

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Review of Unit I:Physical Foundations of Sound

In this unit, we studied the physical foundations of sound.

Some of the fundamental concepts we covered are:•Force•Acceleration•Pressure•Newton’s second law

The key equations we encountered are:•v = f•fn = n f0

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We also studied several recurring patterns in physical phenomena that are relevant for the production, transmission, and detection of sound:

•Simple harmonic motion•Dissipation and damping•Waves (reflection, refraction, diffraction)•Resonance

Finally, we learned about some powerful techniques for solving a broad array of problems:

•Principle of Superposition•Limiting cases•Fourier analysis

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Introduction to Unit II:The Perception of Sound

The goal of this unit is to make connections between two ways of describing (musical) sound:

•Physical: an objective description in terms of fundamental physical parameters (force, pressure,…)

•Psychological: a subjective description of how we perceive sound (pitch, tone, location,…)

The scientific study of the connections between human perception (not just sound) and physics is called pyschophysics.

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? Questions ?Some of the key questions we will address in this unit are:

Do we always hear what is really there?

Can one sound mask another?

Why are some sounds pleasing the to ear and others not?

How can we distinguish between different sources of sound? (i.e., how do we undo the Principle of Superposition?)

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As the first part of our study, we will learn:

•how the human ear works as a physical system, and

• explore the limits of human hearing.

Since sound and light are both waves, we will also compare the capabilities of our senses of hearing and vision.

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The Human EarWe will follow the crest of a sound wave on its journey through the human ear.

This journey has three main segments:

•The outer ear,

•The middle ear,

•The inner ear.

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The Outer EarSound enters the outer ear through the pinnae (wings) and into the ear canal.

Reflections from the many folds on the pinna help to focus short-wavelength (low-frequency) sound into the ear canal.

More sound is collected from infront than behind, helping tolocalize thesource ofa sound.

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The ear flaps (pinnae) are the main feature that distinguishes the human hearing system from other mammals.

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Ear CanalThe ear canal is about 1cm across and 2-3 cm long (about half a Q-tip!) and passes through a hole in the temporal bone of the skull.

The ear canal behaves roughly like an organ pipe andresonates at frequencies in the range 2000-5000 Hz, thusboosting our hearing in this range.

Temporalbone

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EardrumThe eardrum (or typanum) is the interface between the outer and middle ears and makes an airtight seal.

The eardrum is a thin disc of fibrous tissue that is held in place and stretched tight by a muscle.

This muscle is quicklytightened (by reflex) whena loud sound is heard, inorder to protect the moresensitive inner ear fromexcessive vibrationsand damage.

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The eardrum bulges in and out in response to the force of sound waves hitting it.

Its job is to convert the sound pressure waves into a mechanical motion.

The eardrum and subsequent processing are remarkably sensitive to small pressure variations, and can detect motion of as little as 10-11 meters (smaller than the size of a Hydrogen atom!)

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The Middle EarThe in-and-out motion of the eardrum is picked up by a set of three small bones called the ossicles.

Malleus(hammer)

Incus(anvil)

Stapes(stirrup)

Dime(10c)

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HammerThe first bone (hammer) is connected directly to the eardrum:

The anvil and stirrup bones transmit the motion of the hammer to the inner ear, through a small oval window.

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Lever ActionThe three osscicle bones are arranged to provide a lever action. This magnifies the motion that is transmitted to the inner ear by about 50%.

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Pressure GainIn addtion to this 50% gain in motion, there is a gain of about 20x in the pressure applied to the inner ear, due to the small area of the oval window (into the inner ear) compared with the area of the eardrum (into the outer ear).

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Eustachian TubeA pressure difference between the outer and middle ears of just 1 part in 100,000 corresponds to a painful level of sound.

But the normal atmospheric pressure variations due to changes of the weatherare much larger than this.

The Eustachian tubeconnect the middle ear tothe throat and thereforeequalize the pressure onboth sides of theeardrum.

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Inner EarThe inner ear consists of the cochlea and the semicircular canals, carefully nested into passages in the temporal bone. The semicircular canals are used for balance but not for hearing.

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CochleaThe cochlea is a shaped like a snail shell and is filled with an incompressible fluid.

It has two flexible windows into the middle ear: the oval window and the round window. The stirrup is in contact with the oval window, but is quickly pulled back (by reflex) when you hear a loud sound, in order to protect your inner ear.

Ovalwindow

Roundwindow

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CochleaUnrolling the cochlea would reveal a tapered tube about 3.5 cm long. A cross-section through this tube reveals three chambers:

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The stirrup bone transmits the ear drum’s motion (magnified by 50%) to the oval window, which in turn puts pressure on the liquid in the upper two chambers.

Since the walls of thecochlea are rigid, and thefluid is incompressible, theonly way to relieve thispressure is by bulging theother (round) window.

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The pressure on the oval window has to be relieved at the round window, and takes the shortest path to achieve this.

The shortest path is through the flexible basilar membrane that separates the upper and lower chambers.

But the point at which the vibrations can cross the basilar membrane depends on the sound’s wavelength.

Therefore the basilar membrane acts as a wavelength (or equivalently, frequency) analyzer of the incoming sound.

How does this work?

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Another Toy AnalogyImagine a toy where marbles are insertedin the oval hole and then appear laterat the round hole.

How would we design this toy so thatlarge marbles (large wavelength, lowfrequency) take longer to makethe journey than small marbles(short wavelength, highfrequency)?

oval

round

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Large marbles (large wavelength sounds) pass through theslot (basilar membrane) further down the toy (cochlea) thansmall marbles (short wavelength sounds).

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Basilar MembraneBy choosing the shortest path, the sound causes the basilar membrane to vibrate at a position that measures its wavelength (and therefore also frequency).

400 Hz

4000 Hz

How would the basilar membrane respond to thesuperposition of 400Hz and 4000Hz sounds?

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Organ of CortiVibrations of the basilar membrane are converted into electrical nerve signals in the Organ of Corti.

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There are about 20,000 hair cells in the Organ of Corti that each send an electrical signal on individual nerve fibers to the brain via the auditory nerve.

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Into the BrainThe auditory nerves from both ears bring signals into a set of specialized processing centers and then into the brain near its base.

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Signal CharacteristicsA nerve signal from a particular hair cell tells the brain where along the basilar membrane the sound vibrations passed through.

This in turn roughly encodes the wavelength (and therefore frequency) of one component of the sound.

In general, you are listening to many frequencies simultaneously and so many regions of your basilar membrane are vibrating at once.

The signal from each hair switches on and off at a rate that encodes the frequency of vibrations. Frequency andwavelength (f = v) are usually redundant but not always…

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End of the JourneyWe have now completed our journey, starting from sound waves entering the outer ear, all the way through to the electrical signals entering the brain.

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Review of Lecture 6We finished our study of the Physical Foundations of Sound with:

•Sympathetic vibrations•Fourier analysis

We started our study of the Perception of Hearing with an anatomical tour of the human ear, following sound on its journey through:

•The outer ear (ear flaps, ear canal, ear drum)•The middle ear (hammer, anvil, stirrup bones)•The inner ear (cochlea, basilar membrane, organ of Corti, auditory nerve)

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Review QuestionsWhat reflex safety mechanisms does your ear use to protect you from potentially damaging loud sounds?

How does your middle ear magnify the sound pressure incident on the ear drum about 30 times before passing it on to the cochlea?

Knowing how the basilar membrane responds to wavelength, when would you expect one sound to mask another? How might the brain separate two sounds being heard simultaneously?