CHAPTER M 4 (Model)

Cochlear fine-tuning: a surface acoustic wave resonator

4.2 Basic description of the model

4.3 Parallels between SAW devices and the cochlea

4.4 Prima facie validity

4.5 What is the wave propagation mode?

This chapter introduces the idea that the cochlea analyses sound in a way

resembling electronic surface acoustic wave (SAW) devices. The outer hair cells

(OHCs) of our cochlea typically occur in three distinct rows1, and it is suggested the

rows act like the interdigital electrodes of a single-port SAW resonator. Frequency

analysis occurs through sympathetic resonance of a graded bank of these resonators.

This active process relies on the motor properties of OHCs: as well as being sensors,

OHCs undergo cycle-by-cycle length changes in response to stimulation. The SAW

analogy works if the motility of one row of cells can be communicated to a

neighbouring row, allowing positive feedback between one cell’s body (an effector)

and another’s stereocilia (a sensor). If the wavelength of the disturbance were

comparable to the row spacing then inter-row reverberation could occur. A candidate

wave possessing the required properties has been identified – a ‘squirting wave’

known in ultrasonics – and this wave, operating in the subtectorial space between the

tectorial membrane and the plateau of Corti, is described in Chapter R5.

1 As we will see in Chapter R6, occasional extra rows are not a problem.

4.1 Introduction

M 4 [2]

Standing-wave resonance between the rows could thus provide amplification

and high Q, characteristics underlying the ‘cochlear amplifier’ – the device whose

action is evident to auditory science but the identity of which has not yet been

established. Also emerging naturally from such a local interaction on the partition are

spontaneous, evoked, and distortion-product emissions (Chapter R7).

The SAW mechanism, as outlined in this chapter, may provide a second filter

for a primary traveling wave stimulus, and this cannot be ruled out without further

experiments. The advantage of taking this approach is conformity with the standard

model. The drive of this thesis, however, is towards a simple resonance theory of

hearing along the lines of Helmholtz, in which case we require that the input to the

SAW is the fast pressure wave stimulus. The advantage here is simplification of

cochlear mechanics. A later chapter (Chapter D8) strengthens the supposition about

pressure sensitivity.

4.1 Introduction

Earlier chapters have focused on the cochlea’s behaviour as an active

transducer, a necessary consideration flowing from Kemp’s seminal discovery2 of

sound emerging from the ear. An essential element of the auditory organ is therefore

a ‘cochlear amplifier’ whose action improves gain and tuning. If the gain is excessive

at some frequency, the cochlea will spontaneously oscillate, a possible source of

‘spontaneous otoacoustic emissions’ – soft, pure tones – that may be detected with a

microphone placed in the ear canal. Most human ears continuously emit such tones at

frequencies of 1–4 kHz and with bandwidths ranging down to 1 Hz or less. A

detailed review3 of these developments was given by Robinette and Glattke in 2002.

What sort of biological structure could produce such pure tones? The outer

hair cells (OHCs) must be intimately involved, for these sensing cells are active,

possessing a property known as ‘electromotility’. When an audio-frequency voltage

2 Kemp, D. T. (1979). Evidence of mechanical nonlinearity and frequency selective wave amplification in the cochlea. Arch. Otorhinolaryngol. 224: 37-45. 3 Robinette, M. S. and T. J. Glattke (2002). Otoacoustic Emissions: Clinical Applications. (Thieme: New York).

M 4 [3]

is applied to an isolated cell, it synchronously changes length4. Nevertheless, how the

cochlear amplifier harnesses electromotility is unknown. This thesis offers a possible

solution.

The standard cochlear model assumes that stimulation of hair cells occurs by

a hydrodynamic traveling wave moving along the partition. The difficulty has been

to understand how the response of such an intrinsically broad-tuned system can be

sharpened to give the fine frequency resolution it displays5. This chapter puts forward

a proposal whereby the unique structure of the sensing surface of the cochlea can

achieve narrow-band frequency analysis. It conjectures that the cochlear amplifier is

based upon the cooperative activity of neighbouring OHCs, which work together like

the interdigital electrodes of the SAW resonator6,7.

4.2 Basic description of the model

The SAW model builds on a remarkable fact: in all higher animals, including

humans, OHCs lie in three (or more) well-defined rows (Fig. 4.1), hitherto

unexplained.

SAW devices are signal-processing modules in which finger-like electrodes

are interleaved on the surface of a piezoelectric substrate to create slowly

propagating electromechanical ripples of wavelength equal to the periodicity of the

interdigital electrodes8,9. The minimum number of electrode fingers is three

(Fig. 4.2), an arrangement giving the widest bandwidth response10.

4 Brownell, W. E., et al. (1985). Evoked mechanical responses of isolated cochlear outer hair cells. Science 227: 194-196. 5 Patuzzi, R. B. (1996). Cochlear micromechanics and macromechanics. In: The Cochlea, edited by P. Dallos et al. (Springer: New York), 186−257. 6 Campbell, C. K. (1998). Surface Acoustic Wave Devices for Mobile and Wireless Communications. (Academic Press: San Diego). 7 Ballantine, D. S., et al. (1997). Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications. (Academic: San Diego). 8 Bell, D. T. and R. C. Li (1976). Surface-acoustic-wave resonators. Proc. IEEE 64: 711-721. 9 Smith, W. R. (1981). Circuit-model analysis and design of interdigital transducers for surface acoustic wave devices. In: Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic: New York), 15, 99-189. 10 Campbell (1998), p. 501.

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Fig. 4.1. Three rows of outer hair cells (bottom) and one row of inner hair cells (top) in the cochlea of a rabbit. Spacing between OHC rows is about 15 µm. [From Counter et al. (1993)11 with permission of S. A. Counter (Karolinska Institute) and Elsevier Science]

Fig. 4.2. The simplest interdigital electrode has three fingers.

The SAW resonator (Fig. 4.3) has a topology in which the electromechanical

waves on the surface are arranged in a feedback loop between two sets of electrodes

– a ‘two-port’ system, normally operated as a delay line, in which one set of

electrodes launches the ripples and a similar set some distance away detects them.

11 Counter, S. A. (1993). Electromagnetic stimulation of the auditory system: effects and side-effects. Scand. Audiol. Suppl. 37: 1-32.

+–

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Fig. 4.3. Schematic of surface acoustic wave delay line (after Maines and Paige, 1976). Electromechanical ripples are generated and detected by two sets of interdigital electrodes on a piezoelectric substrate to give a delay line. Driving and sensing functions can be combined into a single set of electrodes with resonance between the fingers.

SAW modules are used whenever a number of cycles of signal need to be

stored and operated on. Feeding the output of the second set of electrodes back to the

input set creates a highly tuned resonance (Q ≈ 103–104) typically in the megahertz

range. Resonance can also occur when the two sets of electrodes are merged into a

‘single-port’ resonator12; in this degenerate case, ripples now reverberate back and

forth between the fingers instead of between the two electrode sets. The present

hypothesis is that audiofrequency resonances arise in the cochlea in a similar way

(Fig. 4.4).

–

+

–Fig. 4.4. The rows of outer hair cells are assumed to act like the 3 fingers of an interdigital transducer.

12 Bell and Li (1976); Campbell (1998), Ch. 11.3.

surface acoustic wave

electrical input

absorber

highly polished piezoelectric crystal

detector output

interdigital electrode

roughened and/or waxed back surface

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The model is most easily conveyed by reference to Fig. 4.5, which shows a

cross-section of the cochlear partition. The key components are the three rows of

OHCs overlain by the soft gel13 of the tectorial membrane (TM) in which the tips of

the hair-like stereocilia are embedded. Forced oscillation of an OHC could send out

waves that, if sufficiently slow, will have wavelengths comparable to the OHC row

spacing.

Fig. 4.5. Because outer hair cells are both sensors and effectors, ripples could reverberate between them to produce a standing wave. Double-headed arrows indicate contraction/ elongation of cells (electromotility) in response to a common stimulus.

As described later on (§M 4.5), a literature search into wave propagation at

liquid–solid interfaces revealed a prime candidate for such a wave: a slow, highly

dispersive wave known in ultrasonics as a symmetric Lloyd–Redwood wave or

‘squirting’ wave14. These waves can arise when a thin fluid layer is sandwiched

between two deformable plates, as occurs in the cochlea (Fig. 4.5). Because squirting

waves rely on the interaction between the mass of the fluid and the elastic restoring

force of the plates, they are characteristically slow – measured in the cochlear case in

millimetres per second.

Squirting waves provide a ready basis for positive mechanical feedback and

amplification in the partition (Chapter R5). The proximity of motors – OHC bodies –

to sensors – OHC stereocilia – invites feedback, and if the phase delay of the wave

13 Steel, K. P. (1983). The tectorial membrane of mammals. Hear. Res. 9: 327–359. 14 Bell, A. and N. H. Fletcher (2004). The cochlear amplifier as a standing wave: "squirting" waves between rows of outer hair cells? J. Acoust. Soc. Am. 116: 1016-1024.

(reticular lamina)

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reaches 360º, oscillation between the rows of OHCs will occur. Modeling

(Chapter R6) shows that after applying an input signal to the rows of OHCs, a

standing wave appears between the outermost rows (OHC1 and OHC3). The

standing wave can occur either in a whole wavelength mode (if the polarity of the

input to all rows is the same) or half-wavelength mode (if the polarity of OHC1 is the

inverse of that of OHC3). At the same time, a progressive, but attenuating, wave will

move towards the inner hair cells (IHCs), the ear’s detectors. We now have a SAW

resonator amplifying an input signal and passing it to a detector.

This scheme meets Gold’s prescription for a ‘regenerative receiver’ in the

cochlea where, to avoid compromising signal-to-noise ratio, positive feedback is

used to amplify a signal before its detection. Later (Gold, 1987), he drew an analogy

to the functioning of an ‘underwater piano’: only by introducing a sensor onto each

string and supplying positive feedback could such a viscously loaded instrument be

made to operate, and this is what the SAW model achieves. The SAW resonator can

be identified with the cochlear amplifier and would explain spontaneous emissions

and other active aspects of cochlear mechanics.

The arrangement also fits the description by Hudspeth15 of a distributed

amplifier. Reinterpreting Hudspeth, the separate OHCs could be likened to a

pendulum clock (a distributed mechanical amplifier in which a pendulum is driven

by a coiled spring through an escapement); thus, their oscillation frequencies would

be determined by one physical property (the length of the SAW cavity, equivalent to

the length of a pendulum), while the timed release of energy (from the OHC motor –

the escapement) overcomes damping and sustains activity.

The key assumptions of the SAW hypothesis are set out below.

1. The OHCs and their surroundings have properties conducive to the

propagation of a slow wave – probably a squirting wave – communicating the

motion of one row to the next.

2. The speed of the waves varies from base to apex in a systematic way,

thereby supplying the cochlea’s tonotopic tuning. Halfway along the cochlea, where

the spacing between OHC1 and OHC3 is about 30 µm and a characteristic frequency

(CF) of 1 kHz is typical, the wave will need to have a propagation speed of

15 Hudspeth, A. J. (1997). Mechanical amplification of stimuli by hair cells. Curr. Opin. Neurobiol. 7: 480-486.

M 4 [8]

30 µm/ms (= 30 mm/s) to establish whole-wavelength resonance. Near the base, with

a CF of say 10 kHz and spacing is 20 µm, the necessary speed is 200 mm/s; in

contrast, near the apex where CF may be 0.1 kHz and the spacing 50 µm, the

corresponding speed would fall to 5 mm/s. Chapter R5 shows that the dispersive

properties of a squirting wave allow the human cochlea to be tuned over its full

frequency range (20–20 000 Hz) based largely on the spacing of OHC rows.

3. Rows 1 and 3 respond with the same phase to a stimulus, while row 2

probably responds in anti-phase. Modeling in Chapter R6 shows two likely modes: a

full wavelength mode, like the first overtone of an organ pipe open at both ends, in

which the response of OHC2 is in antiphase to OHC1 and OHC3; and a half-

wavelength mode, like the fundamental mode of the open-ended organ pipe, in which

the responses of OHC1 are in antiphase to OHC3 (and OHC2 is a displacement

node)16. Evidence for bi-phasic activity in OHCs is presented in §R 9.1.

The result of the interaction is a standing wave, a mechanical cochlear

amplifier. That is, IHCs respond to wave energy delivered to them by a squirting

wave generated by the combined activity of the outer hair cells.

4.3 Parallels between SAW devices and the cochlea

There are strong analogies between a SAW resonator and the anatomy of the

cochlea. Comparison is aided by reference to Fig. 4.5 and anatomical texts17.

1. The three rows of OHCs are the interdigital electrodes. It is significant that

the required minimum number of fingers for an electronic SAW device is three, and

in all vertebrate animals there are three or more OHC rows18. Additional rows,

sometimes present, would supply extra gain.

2. Wave energy propagating on the surface of a SAW resonator can be

absorbed or reflected by impedance discontinuities, and, when required, this is

16 A dynamic display of organ pipe modes can be found at http://www.cord.edu/dept/physics/p128/lecture99_35.html 17 Such as Slepecky, N. B. (1996). Structure of the mammalian cochlea. In: The Cochlea, edited by P. Dallos et al. (Springer: New York), 44−129. 18 Bredberg, G. (1968). Cellular pattern and nerve supply of the human organ of Corti. Acta Otolaryngol. Suppl. 236: 1-135.

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normally achieved by etching grooves or placing strips of material on the surface of

the device19. The TM possesses Hensen’s stripe20, a rounded feature located above the

IHC stereocilia (see Fig. 4.5) which may act as a mechanical impedance

discontinuity: this might redirect wave energy emerging from the OHC cavity and

send it to this detector.

3. Energy escaping the OHC cavity towards the outer edge of the TM is a

potential problem and needs to be reflected so as to re-enter the cavity with

appropriate phase delay. At the outer edge of the TM another aggregation of material

is found, a rounded thickening known as the marginal band21, which may act this way

(Fig. 4.5).

4. To absorb and disperse unwanted bulk propagation modes (multiple fast

compressional waves that propagate throughout the substrate of the device), the back

of a SAW resonator is either roughened or waxed22. In the cochlea the top of the TM

is criss-crossed with a covering net23.

5. Towards the inner edge of the TM we find a sharp discontinuity – the

vestibular lip (Fig. 4.5) – and here reflections could occur, possibly acting to return

wave energy back into the amplifying cavity and allowing real-time convolution and

autocorrelation of the signal to take place23,24,25. The key idea here is that the IHC

would then sit as a central detector between the signal source (the OHC cavity) and

its reflected image. As explained by Fergason and Newhouse (1973), when two

surface waves pass through each other from opposite directions and interact

nonlinearly (to produce sum and difference frequencies), a centrally placed detector

integrates the sum frequency to form the convolution and the difference frequency

supplies the correlation26.

6. The speed of electromechanical ripples in a solid-state SAW resonator is

about 5 orders of magnitude lower than the speed of the electrical signal in its input

19 Smith (1981), p. 102. 20 Hardesty, I. (1908). On the nature of the tectorial membrane and its probable role in the anatomy of hearing. Am. J. Anat. 8: 109–179 (+ plates). 21 Edge, R. M., et al. (1998). Morphology of the unfixed cochlea. Hear. Res. 124: 1-16. 22 Smith (1981), p. 130. 23 Campbell 1998, Ch. 17.3. 24 Quate, C. F. and R. B. Thompson (1970). Convolution and correlation in real time with nonlinear acoustics. Appl. Phys. Lett. 16: 494-496. 25 Kino, G. S. (1976). Acoustoelectric interactions in acoustic-surface-wave devices. Proc. IEEE 64: 724-748. (p. 728 and Fig. 6) 26 Furgason, E. S. and V. L. Newhouse (1973). Convolution and correlation using nonlinear interactions of Lamb waves. IEEE Transactions on Sonics and Ultrasonics SU-20: 360-364.

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leads, a reduction that makes it possible to compactly store many cycles for signal

analysis27. Thus, it is possible to store in an inch of crystal the information that would

otherwise fill a mile-long cable28. In a similar way, the speed of the hypothetical

squirting wave is 4–5 orders of magnitude lower than the speed of a sonic pressure

wave in the surrounding cochlear fluids, some 1500 m/s. In the width of the tectorial

membrane, then, may be stored up to a dozen cycles of acoustic signal. One

experiment involving a microphone in a human ear canal29 observed a 1280-Hz tone

continually waxing and waning every 5.8 ms– that is, an envelope repeating every

7.4 waves.

4.4 Prima facie validity

The structural parallels are suggestive, but by themselves they are not

conclusive. A number of lines of evidence can be found in the literature, and a more

detailed presentation of some of them is set out in Chapter D9. However, in order to

justify the modeling that forms the core of this thesis (Chapter R6), it is worth noting

two main predictions of the SAW model and the major evidence supporting them.

A. Inverted response of different rows. A distinctive feature of SAW devices

is the alternating polarity of the interdigital electrodes. Translated to the cochlea, this

means that the response polarity of one or more rows of OHCs should be inverted

compared to the others. One key finding here is that the cochlear microphonic, an

electrical potential recordable from outside the cochlea, exhibits ‘dual’ polarities30, as

if it were composed of two potentials of opposite polarity (which could in turn derive

from two populations of OHCs of opposite polarity). Moreover, isolated OHCs can

either lengthen or shorten in a tuned manner in response to oscillating pressure31, and

27 Maines, J. D. and E. G. S. Paige (1976). Surface-acoustic-wave devices for signal processing applications. Proc. IEEE 64: 639-652. 28 Claiborne, L. T., et al. (1976). Scanning the issue. Proc. IEEE 64: 579-580. 29 Wit, H. P. and R. J. Ritsma (1980). Evoked acoustical responses from the human ear: some experimental results. Hear. Res. 2: 253-261. The wave continued for more than 280 ms. 30 Pierson, M. and A. R. Møller (1980). Some dualistic properties of the cochlear microphonic. Hear. Res. 2: 135-149. 31 Brundin, L., Flock, Å., and Canlon, B. (1989). Sound-induced motility of isolated cochlear outer hair cells is frequency-specific. Nature 342: 814–816.

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direct antiphasic activity of OHC rows has been observed, even (surprisingly) in an

explant32.

B. Radial wave motion. If the SAW model is valid, we would expect closely

spaced phase changes across the partition. Indeed, relative phase changes of up to

180º between locations only 10 µm apart radially have been observed on the basilar

membrane of a guinea pig33. Nilsen and Russell drilled a hole in the cochlea and

measured, in a profile across the basilar membrane, the motion of the membrane in

response to sound stimulation. By using a beam from a self-mixing laser diode,

focused to a 5-µm spot, they avoided having to place reflective beads on the

membrane, a routine practice that conceivably might interfere with normal cochlear

activity. The unique Nilsen and Russell results are highly suggestive, even though

several other workers have only seen smaller phase gradients and others have not

seen any (see §D 9.3/b). Rapid phase variations strongly support the SAW model,

and are otherwise difficult to explain.

4.5 What is the wave propagation mode?

SAW devices can, in principle, operate using a wide range of wave

propagation modes: Rayleigh waves are the most common, but Lamb, Love,

Bleustein–Gulyaev–Shimizu, Stonely, Sezawa, and other wave modes are

employed34,35. Deciding which mode operates in the cochlea is difficult due to

complex structure and little-explored physical properties of the layers bounding the

OHC stereocilia – the tectorial membrane (TM) at the top (in which the tips of the

longest stereocilia are embedded) and the plateau of Corti (reticular lamina, or RL)

underneath. The TM is a fibre-reinforced viscoelastic gel covered with surface layers

(Hardesty’s membrane and covering net) and immersed in electrolytes and an

electrical field; the RL, for its part, is a very thin (∼1 µm) membrane composed of

32 Scherer, M. P. and A. W. Gummer (2004). Vibration pattern of the organ of Corti up to 50 kHz: evidence for resonant electromechanical force. Proc. Nat. Acad. Sci. 101: 17652-17657. 33 Nilsen, K. E. and I. J. Russell (1999). Timing of cochlear feedback: spatial and temporal representation of a tone across the basilar membrane. Nat. Neurosci. 2: 642-648. 34 Ballantine, et al. Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications. Chapters 2 and 3. 35 Campbell (1998), Chapter 2.

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the interlocking flattened heads of phalangeal processes that in turn originate from

Deiters cells (see §R7).

Wave modes can be categorised in two main ways: longitudinal, in which the

particles move back and forth in the direction of wave propagation; and transverse, in

which they move at right angles to this direction. In geophysics, the first are known

as P-waves whereas the second are S-waves. P-waves depend on compressional

moduli, and hence are relatively fast compared to S-waves which rely on shear

moduli and can be much slower in gel-like materials. In seeking a remarkably slow

wave in the cochlea, shear waves appear the better candidates.

Simple capillary (surface tension) waves, with their characteristically slow

speed, are also worth consideration since surface tension effects have been observed

in the cochlea36. The speed, c, of a surface tension wave37 of wavelength λ on the

surface between a liquid and a gas is:

c = (2πT/λρ)½ (5.1)

where T is the surface tension and ρ is the density of the underlying liquid. Surface

tension may have a role in the cochlea, but it is difficult to see how this parameter

can be made to vary systematically from base to apex by a factor of 104 or more in

order to tune the partition over a frequency range of at least 2 decades. Note from

Eq. 5.1 that capillary waves are dispersive, with speed varying as ω 0.5.

A shear wave of some type is a strong possibility because the shear moduli of

gels are small, much less than their compressional (Youngs) moduli. In its most basic

form, the speed, v, of a bulk shear wave is given by

v = (G/ρ)½ (5.2)

where G is the shear modulus and ρ is the density. These waves are non-dispersive

(speed independent of frequency); a Rayleigh wave, a shear wave on the surface of a

solid that is thick compared with the wavelength involved, travels at about 90% of

this speed. Acoustic measurements38,39 of soft ‘ringing gels’ (showing the peculiar

36 Olson, E. S. and D. C. Mountain (1990). In vivo measurement of basilar membrane stiffness. In: Mechanics and Biophysics of Hearing, edited by P. Dallos et al. (Springer: Berlin), 296-303. The attraction between the basilar membrane and a probe only occurred pre-mortem; it was also noted by Olson in subsequent work. 37 Lighthill, J. (1978). Waves in Fluids. (Cambridge University Press: Cambridge). 38 Sinn, C. (2000). Acoustic spectroscopy of aerogel precursors. Progr. Colloid Polym. Sci 115: 325-328.

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property of ringing like a bell when struck), give shear moduli as low as 14 kPa and a

corresponding shear-wave velocity of 4.1 m/s, which is not low enough for audio-

frequency SAW operation in the cochlea – some millimetres per second (point 2 on

page M 4 [8]).

Measurements of the shear modulus (elasticity) of the actual TM have

returned a remarkably wide range, as Table 4.1 shows. The table includes values of

the less precise ‘point stiffness’, which earlier experimenters used, a measure that

depends on the sharpness of the probe (elasticity of 1 kPa ≈ point stiffness of

0.01 N/m). Values vary from as low as 0.0004 N/m to as high as 10 N/m, but even

the lowest value (G ≈ 0.2 kPa) is, after using equation 5.2, still too high for our

purposes (giving a c of 800 mm/s).

For comparison, measurements on the elasticity of the gel covering the

otoliths in an amphibian are also given in Table 4.1. These values are very low, and

the unsupported gel would probably collapse under its own weight in air, but even

this material gives a shear wave speed no lower than 50 mm/s. Experiments with a

vibrating magnetic bead inside a cell40 give a shear modulus of the cell contents as

20–735 Pa, and one cannot hope to go lower than that. Taken together, the evidence

suggests that a shear wave does not seem to be the answer.

A more likely avenue is to look at bending or flexural waves in plates, which

in general have lower velocities than waves based on shear. These waves also tend to

be highly dispersive. The basic surface wave is a Lamb wave, and they come in

symmetric and antisymmetric forms41,42. SAW devices based on Lamb waves can be

lowered further in frequency by immersing them in liquid or loading them with a

layer of high density and lower shear modulus, which retards the wave speed and

tends to trap wave energy at the surface. As a result, other interface modes such as

Stoneley, Sezawa, and Scholte waves begin to appear43. In this connection, we note

that the TM is not a homogeneous layer, but has its underside covered with a very

thin layer about 1 µm thick44 called Hardesty’s membrane.

39 Sinn, C. (2004). When jelly gets the blues: audible sound generation with gels and its origins. Journal of Non-Crystalline Solids 347: 11-17. 40 Bausch, A. R., et al. (1999). Measurement of local viscoelasticity and forces in living cells by magnetic tweezers. Biophys. J. 76: 573-579. 41 Wenzel, S. W. and R. M. White (1988). A multisensor employing an ultrasonic Lamb-wave oscillator. IEEE Transactions on Electron Devices 35: 735-743. 42 White, R. M. (1970). Surface elastic waves. Proc. IEEE 58: 1238-1277. 43 Ballantine et al. (1997), Ch. 3, pp. 89 ff. 44 Hardesty (1908).

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Table 4.1. Measurements of tectorial membrane stiffness

Experimenter Point stiffness (N/m) Elasticity (kPa)

Békésy (1960) 45 0.1–10

Zwislocki & Cefaratti (1989) 46 0.13 0.6

Hemmert et al. (2000) 47 0.1–0.6

Abnet & Freeman (2000) 48 0.07–1 7–100

Freeman et al. (2003) 49 0.06–0.34

Shoelson et al. (2003) 50 0.0004–0.003 0.7–3.9

Shoelson et al. (2004) 51 3–20

Measurements of otolith gel

Kondrachuk (1991) 52 1

Kondrachuk (2002) 53 0.01

The way in which liquid and gel layers combine to form low propagation-

speed layers prompts further investigation. Ballantine et al. (1997) discuss lossless

Scholte waves (their p. 124) and they mention that its phase velocity approaches zero

for very thin layers (which they show in their Fig. 3.46). A literature search on

Scholte waves returned a most astonishing reference – Hassan and Nagy (1997) – the

abstract of which describes the experimental finding of “a mode slower than the

slowest-order bending mode of the plates and [which] asymptotically approaches the

45 Békésy, G. v. (1960). Experiments in Hearing. (McGraw-Hill: New York). 46 Zwislocki, J. J. and L. K. Cefaratti (1989). Tectorial membrane II: stiffness measurements in vivo. Hear. Res. 42: 211-228. 47 Hemmert, W., et al. (2000). Mechanical impedance of the tectorial membrane. Midwinter Meeting, Florida, Association for Research on Otolaryngology http://www.aro.org/archives/2000/4838.html. 48 Abnet, C. C. and D. M. Freeman (2000). Deformations of the isolated mouse tectorial membrane produced by oscillatory forces. Hear. Res. 144: 29-46. 49 Freeman, D. M., et al. (2003). Dynamic material properties of the tectorial membrane: a summary. Hear. Res. 180: 1-10. 50 Shoelson, B., et al. (2003). Theoretical and experimental considerations for the study of anisotropic elastic moduli of the mammalian tectorial membrane. Mid-Winter Meeting, Florida, Association for Research in Otolaryngology 51 Shoelson, B., et al. (2004). Evidence and implications of inhomogeneity in tectorial membrane elasticity. Biophys. J. 87: 2768-2777. 52 Kondrachuk, A. V., et al. (1991). Mathematical simulation of the gravity receptor. The Physiologist 34: S212-S213. 53 Kondrachuk, A. V. (2002). Models of otolithic membrane–hair cell bundle interaction. Hear. Res. 166: 96-112.

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Stonely–Scholte mode” of a solid–liquid interface54. The mode, which involves the

motion of fluid in the gap between two symmetrically flexing plates, was first

predicted in 1965 by Lloyd and Redwood, and they called it a ‘squirting wave’ after

the way in which fluid squeezed between two closely spaced plates behaves. This

wave, which I call a symmetric Lloyd–Redwood (SLR) wave, applies well to the

anatomy of the cochlea’s sensing surface. Moreover, the predicted wave speed –

based on the dimensions and properties of the subtectorial space – gives a good

match to the speeds required to operate the SAW resonator. This prime candidate is

described in detail in the next chapter.

54 Hassan, W. and P. B. Nagy (1997). On the low-frequency oscillation of a fluid layer between two elastic plates. J. Acoust. Soc. Am. 102: 3343-3348. (p. 3343).