Impedance

Impedance (Z) is the interaction existing between a force (F) which works on a mechanical system and the resulting speed of displacement of that system (v) : Z = F/v

For any given force (F), a system having an impedance (Z) will have a low speed: v = F/Z

The specific acoustic impedance for water (Z eau = volume x ??acceleration of sound = r x c = 1.5 x 10 6 N.s.m -3) is 4000 times greater then that of air (4.102 N.s.m -3). Any given acoustic pressure (measured in Newton.m 2 or Pascal) causes molecular displacements 2000 times smaller in water than in air!

In the inner ear, the sensory cells must be stimulated mechanically to give a response. At the threshold of hearing at 1000 Hz (0dB SPL, at 2.1-5Pa), the displacement of the molecules of air are between 10 -11 and 10 -12 m (between 1/10 th and 1/100 th of the diameter of an hydrogen atom!) (Or, the displacement of air molecules necessary to stimulate a mechanical response, ie the threshold of response, is between 10 -11 and 10 -12 m (between 1/10 th and 1/100 th of the diameter of an hydrogen atom!)

If the displacement of liquid molecules and the sensory structures of the inner ear were 2000 times smaller, (10 -15 m), they would never be able to detect sound at the threshold (there would be noise due to thermal currents, indeed the limits of ?quantum mechanics…)

 

We have seen that the speed (v) of a molecule (which corresponds directly to its amplitude for any given frequency) is defined by the pressure and impedance: v = P/Z. Since impedance Z is given “by construction” for air and for water (the product of mass x ??acceleration of sound), to increase “v” it is necessary to increase “P”!

To understand this better, let us examine the formula that represents the impedance of a mechanical system: Z = ?[R 2 + (Mw – K/w) 2] (where w = 2pf)

One can see that when the frequency (f) is low (and so w is small), the influence of mass (M) is reduced and that of rigidity (K) is important: at low frequencies the displacement of the tympano-ossicular chain and the “efficiency” of the middle ear are limited by the rigidity of the tympanic membrane, ligaments, the volume of air in the middle ear and so on. When the frequency (f) is higher (and so when w is larger), mass (M) has an important influence, whereas rigidity (K) is less important. Therefore at middle and high frequencies the “efficiency” of the middle ear is limited by the mass of its components (and by their friction, R).