The Ear and the Auditory System
The ear is a remarkably complex transducer, converting energy carried by sound waves ultimately into electrical energy which is transmitted by nerve cells into the brain for final processing. There are several energy changes that take place, following the sequence shown below.
The sensitivity of our auditory system spans a broad range
- in frequency: 20 Hz < f < 20 kHz, giving the ratio of the highest to lowest audible frequencies of ~1,000. Contrast this to the frequency sensitivity of the eye to light waves: for vision, the ratio of the highest frequency to lowest frequency light waves we can see is only a factor of two!
- in amplitude: the ear’s sensitivity spans a range of 1012 (corresponding to one million million) in loudness, between the softest and loudest audible sounds.
Today, we’ll talk about some of the basic physics involved in hearing.
The visible part of the ear is called the pinna. The pinna serve as a collector of sound waves, acting in much the same manner as a satellite dish for radio waves. Sound waves make air pressure oscillations in the auditory canal, which is a short pipe, roughly 2 cm in length, open at the pinna and closed at the other end by the eardrum. The auditory canal causes standing sound wave resonances in the middle range of audible frequencies, serving to boost hearing sensitivity in this range.
The ear drum is the first transducer in the ear. It is nothing more than a thin membrane which is stretched with an adjustable tension (the adjustment is used in the acoustic reflex to protect the inner ear from very loud sounds). This eardrum membrane acts just like a drumhead. For the latter, oscillations in a drumhead are produced by mechanical motion ultimately producing sound waves. The mechanical energy is from drumsticks hitting a drumhead. For an eardrum, this process is reversed: air pressure oscillations associated with sound waves cause vibrations of the eardrum resulting in motion of small bones called ossicles that lie on the membrane.
The auditory system responds to pressure changes (related to loudness) ranging from very small values of 10-5 N/m2 up to very large values of 10 N/m2. The resulting force on the eardrum associated with sound waves can be found by knowing the area of the eardrum; it’s roughly circular, with a radius of ~2 mm, meaning its area is 10-5m2. This means that even for the loudest sounds, the force on the eardrum is small
The oscillations of the eardrum in response to sound waves exert a force on the small bones, known as ossicles, in the middle ear. To a good approximation, the three bones in the ossicles act as a lever, increasing the force on the eardrum to be a ~50% larger force on the oval window on the cochlea.
The oval window has approximately twenty times smaller area than the eardrum. The oscillating force on the oval window, exerted by the stirrup shaped bone of the ossicles, is larger than F1 by a factor of 1.5 and causes pressure changes inside of the cochlea
The fact that pressure oscillations in the cochlea are 30 times larger than the air pressure oscillations associated with sound means that this portion of the ear has worked as an amplifier! The pressure oscillations of the air at the eardrum have been transmitted into pressure oscillations inside the cochlea and increased by a factor of 30.
The cochlea is a coiled-up tube shaped like a snail. Uncoiled, it’s about 4 cm long. It is filled with liquid and divided down the middle by the basilar membrane. Pressure oscillations in the oval window on the cochlea result in a rippling motion of the basilar membrane that depends on frequency. Small hair cells (called cilia), attached to the organ of Corti, which rests on the basilar membrane, are bent as the basilar membrane ripples in response to sound. The hair cell motions produce electrical activity in nerve cells which are ultimately transported through the auditory nerve to the brain for final processing resulting in sound perception.
It’s the manner in which the basilar membrane moves that results in the first discrimination between sounds of different pitch.
- High frequency sounds produce the greatest motion of the basilar membrane near the oval window.
- Low frequency sounds produce the greatest motion of the basilar membrane farthest from the oval window.
This results in different nerve cells, distributed along the organ of Corti, producing electrical pulses depending on the frequency of the sound waves.
Nerves along the organ of Corti respond to a narrow band of frequencies, called critical bands. Each frequency band has a width delta_f, called the critical bandwidth. Sounds that differ only a little bit in frequency are within the same critical band and cause the same nerve cells to fire. At audible frequencies above ~500 Hz the ratio of the critical bandwidth to the central frequency of the critical band is roughly constant and equal to 12%. This means that the same neurons fire in a range of frequencies equal to 0.12f0. Below ~500 Hz, the critical bandwidth is constant and equals ~90 Hz.
Perception of sound is then ultimately related to which nerve cells within the basilar membrane respond to sound waves. It is the signals from these nerve cells that our brains decode to perceive sound.
|Sound quality||Nerve Signal|
|pitch||which nerves fire?|
|loudness||how large is the signal from a given set of nerve cells?|
|timbre||which sets of nerve cells fire together?|
|duration||how lond do the nerve cells fire?|