Medical Physics Tutorial 3 - The Physics of Hearing


Acoustic Quantities
The Ear

Anatomy of the Ear

Frequency Response of the Ear

Intensity Response

The Decibel Scale

Hearing Loss


Sound is a longitudinal wave.  It travels as a series of compressions and rarefactions.  We know that in air the speed = 336 m s-1.  If you are not sure about the nature of longitudinal waves, revise waves in Waves Tutorial 2.  Sound also travels in other materials.  In steel the speed is 6000 m s-1.  In water sound travels at 1500 m s-1.  There is formula that can be used to relate the speed of sound to the density and the bulk modulus of a solid.  You are not expected to know it for the exam.


Acoustic Quantities

Some materials conduct sound better than others.  We can use an analogy to electrical resistance.  We use the term acoustic impedance, which is the product of the density and the speed of sound in the material.  A material with low acoustic impedance conducts sound well.  One with a high acoustic impedance conducts sound badly.  The equation is given below:


Z - acoustic impedance (kg m-2 s-1); r - density (kg m-3);  v - speed  (m s-1). 


At the boundary of two materials with widely differing acoustic impedances, most of the sound is reflected and little is transmitted.  This has important implications in ultrasound scanning.


The intensity of a sound is the power per unit area.  The further you are away from any source, the lower the intensity.  Like all progressive waves the intensity decreases with an inverse square law relationship.  Double the distance and the intensity drops to a quarter. 


Intensity is given the Physics code I and is measured in watts per square metre (W m-2).


Materials with a high acoustic impedance attenuate sound waves, which means the intensity is reduced.


The Ear

The ear is an amazing piece of kit.  About the size of a matchbox, it contains a high fidelity sound detection system that can pick up a huge range in intensity.  It can be tuned to be sensitive to very quiet sound, or adapt to exposure of very loud sounds.  It can discriminate particular sounds above a whole cacophony of other noise.  No electronic instrument has the capabilities of the ear.  Added to that, it is also important for maintaining balance. 


We have two ears (really?) for a purpose.  Not only do two ears allow us to hear the sounds, but also give important spatial information about the direction of the sounds and the layout of the sounds.  This had important evolutionary consequences.  We could tell which direction a predator was coming from. If we listen to a well set-up stereo system, we can tell the positions of all the instruments in the orchestra.




Question 1

Explain how each of the listeners A, B, C hear the sound from the sound source the way they do.



The ear can discriminate a time difference of about 10-5 s.


Question 2

What path difference does a time difference of 1.0 × 10-5 s represent if the speed of sound in air is 340 m s-1?



In many animals the pinna of the ear (the flapper) can be moved about to funnel the sound into the ear to get a more accurate location.  In the human, the muscles are too weak to move the ear, but electrical activity can be detected if a sound is made.  This is a test that doctors use on babies to test their hearing.



Anatomy of the Ear

The structure of the ear is shown below:


Graphic by Dan Pickard, Wikimedia Commons


The ossicles are three small bones that amplify the sound by increasing the movement through a system of levers.  Their names are:


However in loud environments there are muscles that act to restrict the movement and reduce the amplification effect.  This helps to avoid damage.  The muscles take time to react and very sudden intense sounds can do major damage to the ear. 


The cochlea is a fluid filled canal which is lined with hair cells.  The sound impulses are sent through the fluid to be vibrate the hair cells.  The picture below shows the hair cells.



There are different theories as to how the sounds of different frequencies are picked up in the ear.  These are summed up in the picture below:




Frequency Response of the Ear

The human ear can detect sounds of frequency about 20 Hz.  Sounds below this frequency are called infrasound and are felt rather than heard.  In a young person, the upper limit is about 20000 Hz, although the upper limit comes down with age.  A middle aged person will hear frequencies up to about 15000 Hz, where an elderly person has an upper limit of about 10000 Hz.


The human ear has a peak sensitivity of 3000 Hz, which causes a sense of unease.  The human scream is at this frequency, and alarms are designed to sound at 3000 Hz.  The reason for this is that the cochlea has a tube length of about 2.5 cm, closed at one end and open at the other. 


Question 3

Show that the standing wave formed in such a system has a wavelength of about 10 cm.


Question 4

What frequency does the ear resonate at?



The frequency response of the ear is shown below:



Notice that:

The graph here shows the sensitivity of the ear to a fixed reference level:



The ear can discriminate the difference between frequencies.  Above 10 000 Hz this ability is poor.  In the range 60 - 1000 Hz, a change of 3 Hz can be detected.  Certain ratios of frequency are particularly pleasing to the ear, and these form the basis of music.


Question 5 

What is:

(a) the intensity of the minimum threshold of hearing and what range does this occur over?

(b) the intensity of the discomfort and pain.  Are they frequency dependent?

(c) the audible frequency range at an intensity of 10-6 W m-2?





Intensity Response (Loudness)

The loudness of a sound depends not just on the intensity, but also of the energy transfer characteristics of the ear.  Loudness is measured in units called phon.  The graph below shows how the ear responds to sounds of equal loudness, but of different frequencies.



The key point is that the perception of changes in loudness does not correspond directly to the change in intensity.  There is a strong frequency element in this.  Small changes in intensity can be detected at 3000 Hz, but not at 100 Hz.


Measuring the frequency response to the ear is very subjective; you cannot dismantle somebody's head and connect the ear to a CRO.



The Decibel Scale

The range of intensities we have seen is huge and we use logarithmic scales to compress the graph into something more manageable.  In the decibel scale we use the minimum threshold of hearing, 1 × 10-12 W m-2 as a reference point, and we give the reference point the physics code I0.   We can write an expression for the change in intensity DI.



1 Bel is the intensity change from 10-12 to 10-10 W m-2 .  The Bel (B) is rather a big unit and we use the decibel (dB) instead.


Worked Example

The maximum limit that is acceptable in a noisy environment is 10-3 W m-2 .  What is this in decibels?



DI = 10 × log (10-3 W m-1/10-12 W m-2) = 10 × log (109)

DI = 90 dB



Question 6

The sound intensity levels in a school classroom vary from 30 dB to 80 dB.  What is the change in sound intensity as a ratio?



A doubling of intensity gives a 3 dB increase.   The ear can just about detect this.


The dBA scale is used to take into account the frequency dependence.  Remember that the ear is most sensitive at  frequencies between 100 to 10 000 Hz .  The table shows the levels of certain noises:


Level (dBA)




Threshold of hearing



Blood pulsing



ticking watch



Quiet conversation



Quiet street



Hoover in a room



Road drill at 7 m

Prolonged exposure can lead to hearing damage


Noisy factory



Loud discothèque

Threshold of discomfort


Aircraft at 25 m

Threshold of pain


Rifle close to ear

Ear drum ruptured



Question 7

You are walking in the countryside when you are suddenly "buzzed" by a jet aeroplane flying at 25 metres.  What is the change in intensity in decibels?




Hearing Loss

Loss of hearing can be a result of:

Disease can stop the ossicles from moving.  This can be corrected by surgery.  Alternatively a hearing aid can be used that transmits the vibrations through the bones of the skull.  Loss of ear sensitivity can be compensated for by a hearing aid.  If disease destroys the nerve fibres to the cochlea, then nothing can be done.  The cochlea is surrounded by bone and very difficult to get to.


Tinnitus is the term given for a continuous ringing or hissing in the ear.  This can be a temporary effect after a loud rock concert, or can be a long term and distressing condition where the patient cannot sleep. 


Serious hearing damage can result from prolonged exposure to loud noises.  Noise is not easy to define, as one person's pleasant sounds may be a nightmare to others.  Some noises can be ignored at first but if repeated often enough can lead to stress.  A man in Middlesbrough was murdered a few years ago by a neighbour infuriated by his DIY activities.


The Health and Safety at Work Act now makes it mandatory that ear defenders are used by employees using noisy machinery.  However that can be undone if the employee wears a personal stereo on the way to work that can be heard by the whole carriage.  Excessive noise can break off the hears of the hair cells.  The graph show the hearing of a normal 40 year old and one that has been exposed to a loud environment.



Aging also leads to hearing loss.  The graph here shows the hearing loss of a 65 year old compared to a 40 year old.  Neither has been exposed to prolonged periods of excessive noise.

If the 65 year old had been exposed to excessive noise the response line would be much lower.  Notice that the higher frequencies are much more affected.


Question 8

Sketch a graph showing the hearing response of a 40 year old office worker compared to a 65 year old who has worked with noisy machinery all his life.