Comparison to previous studies
The time-averaged holograms in show similarities and differences with TAHs that have been previously described. The best-described set of TAH data is that of Khanna and Tonndorf (1972)
in cat and Tonndorf and Khanna (1972)
in human temporal bone. In those groundbreaking studies Khanna and Tonndorf described simple patterns
of TM surface displacement that occurred at frequencies of less than 2 kHz, where the displacement patterns were consistent with in-phase displacement of the entire membrane surface with one to three spatial maxima located on the membrane surface. At frequencies between 2 and 8 kHz, the patterns of displacement became more complicated; the number of local maxima increased, the shape of each maxima increased in complexity and the different maxima where separated by high light intensity (bright) regions of greatly reduced motion (nodes).
These two distinct patterns are observable in our data. We see simple patterns of displacement in chinchillas at frequencies less than 0.8 kHz, and in cat and human at frequencies less than 2 kHz. The complex patterns with multiple maxima separated by nodal lines of negligible motion and high light intensity occur in chinchillas at frequencies between 1 and 3 kHz and at frequencies between 2 and 8 kHz in cat and human. In addition, our data demonstrate that at higher frequencies, all of the specimens show a transition from complex to ordered patterns where a large number of local displacement maxima are orderly arranged on the membrane surface. As we will see, these ordered patterns of displacement are consistent with modal displacement patterns in a simple acousto-mechanical system.
While we describe three step-like differences in displacement pattern (simple, complex, ordered), these different classifications describe prominent positions along a progression of patterns. For example, once we recognized the existence of the ordered pattern at higher frequencies, we could trace its origin back in frequency into what we originally considered the complex domain. Indeed, one of the major differences between complex and ordered is not in the position or shape of the node-separated modal maxima (the ‘pearls’ and ‘strings’) but in their number and size. The larger the number of the pearls and strings and the smaller the dimensions of the pearls, the more apparent is their ordered arrangement.
Consistent with a progression of displacement pattern, the ordered
patterns show additional order in how they vary with increasing frequency. As stimulus frequency increases the patterns show regular changes consistent with an increase in the number (or density) of the local maxima (the pearls) as well as an increase in the orderliness of their arrangement. The increase in density is also associated with a decrease in the spatial area of each local maxima. The regular variation in the size, number and location of these local maxima as frequency increase, makes it unlikely that their location and size are tied to inhomogeneities within the structure of the membrane such as variations in the membrane thickness or in the density of radial or circumferential fibers within the central layer of the TM (Lim 1968
; Funnell & Laszlo 1982
; Decraemer & Funnell 2008
Methodological Issues and Potential Artifacts
The effects of painting the TM surface
Another point of commonality between this study and most holographic studies of TM motion (the exception is Løkberg et al. 1979
) was the need to coat the TM surface with an opaque paint or metal powder in order to decrease the transparency of the measurement surface. We have presented laser-Doppler vibrometer data () that demonstrate that the effect of our TiO2
painting procedures on sound-induced stapes velocity is relatively small at frequencies below 8000 Hz, but it can produce narrow-band reductions in stapes velocity magnitude of as much as 20 dB at higher frequencies. An unanswered question is: How the coating affects the patterns of TM displacement? We have made some preliminary measurements in human temporal bones of TAHs with and without painting. The thicker epidermal layer of the human tympanic membrane is more opaque than in animals (especially in older humans) and like Løkberg et al. (1979)
, we were able to obtain a few ‘muddy’ TAHs with more poorly defined fringes without painting the membrane (). With low-frequency stimulation the painted and non-painted TMs show similar-sized areas of gray in similar locations, though no fringes are visible within the gray area of the unpainted TM. With high frequencies the typical ‘pearl and string’ pattern is visible in the unpainted TMs and the location of the pearls and strings is comparable to that observed after painting. The pearls on the unpainted membrane are less well defined and appear larger. We conclude from these measurements that painting the TM does not evoke the basic patterns of displacement that we observe; it does however make them visible.
Figure 6 A comparison of the time-averaged holograms measured with a cadaveric human temporal bone at two frequencies before and after applying a thin layer of TiO2 paint on the TM surface. The TM is oriented in an identical manner in each figure. The line drawings (more ...) The need for high sound levels
Because (1) the THA is a measure of the sound-induced displacement of the TM and our blue laser requires a relatively large displacement value of 0.1 micrometers to evoke the first holographic fringe, and (2) the displacement of the TM produced by a constant SPL decreases as frequency increases, the TAHs we illustrate were gathered with the high sound levels of 80 to 130 dB SPL. At the highest levels of this range, we expect measurable nonlinear responses, visible as harmonic distortion: in the sound source, in the microphone, in the sound-field itself, as well as in the middle ear’s response to sound. Measures of distortion made with the probe microphone noted visible harmonic distortion, but such distortions were at least 20 dB smaller than the stimulus-frequency at 130 dB, and smaller at lower stimulus levels. Also measures of ossicular motion show responses consistent with low-distortion (at least 20 dB down) at stimulus levels that approach 130 dB SPL (Guinan and Peake 1967
; Goode et al. 1994
). It has also been noted that most of the measurements we illustrate were gathered at sound levels less than 116 dB SPL, and that all of the patterns of motion that we describe are visible in ears at sound levels less than 100 dB SPL. Therefore, the basic results of this report are not greatly affected by nonlinearities in the sound stimulus or in the TM’s response to high-level sound.
Spatial variations in the high-frequency sound field
Another potential acoustic artifact in our results that we have discussed is the presence of variations in the sound field in space, specifically variations between the sound pressure at different points on the membrane surface and at the probe microphone placed within 1 mm of the TM’s rim. Such variations are expected to be more significant at high frequencies where the dimensions of the TM (4.5 mm radius for chinchilla and human and 3–3.5 mm radius for cat) are a significant fraction of the sound wave length: The wave length of a 20 kHz tone in air at standard temperature and pressure is 17 mm, and wavelength varies inversely with frequency. While we have not investigated spatial variations in the sound field acting on our experimental membranes, the methods summarize some control measurements made on an artificial membrane of similar dimensions and displacement magnitudes. These controls suggest that spatial variations in pressure are smaller than 3 dB in magnitude at frequencies of 20 kHz and less, but do also describe more significant variations at higher sound frequencies. Descriptions of the motion of the human, chinchilla and cat at frequencies above 20 kHz will require better understanding of such variations.
Nodes and natural frequencies
The holographic patterns we see on the various TMs measured in response to low-frequency stimuli appear as areas of gray punctuated with black contours. This pattern suggests that at these low frequencies, the entire membrane is set into motion with distinct areas moving with larger magnitude. As frequency increases, the number of local displacement maxima increases and we start to see these areas separated by brighter regions with an intensity similar to that of the non-moving bone that surrounds the TMs. These delimiting areas of reduced motion are reminiscent of nodes in response patterns that result from the combination of ‘natural’ responses of the membrane to the stimulus, where each natural response of the membrane is best elicited by a stimulus at a ‘natural frequency’.
The natural modes of motion of a lossless circular membrane of uniform mechanical properties and radius a that is bound at its rim are described by Equations 8
, which define the signed amplitude of displacement at any point on the surface of the membrane d(r,)
in terms of two integer “mode numbers”, m
, a wave number kmn
(determined by the mechanical properties of the membrane), and the radial distance r
and angular position from the center of the membrane (Kinsler et al. 1982
; Fletcher 1992
is the mth
-ordered Bessel function of the first kind. Equation 9
constrains the values of kmn
to satisfy the boundary condition of zero motion at the membrane’s rim (r=a
), thereby restricting the possible values of kmn
to natural frequencies (fmn
), where kmn
= 2 / mn
describes the spacing of the different modal maxima and minima, and mn
is the wave length associated with the spacing of the maxima and minima. If these modal patterns result from the interference of traveling transverse waves on the membrane surface, then mn
, and cmn
is equivalent to the propagation velocity of surface waves moving along the membrane for a specific natural frequency. The Bessel function term (Jm
) in Equation 8
defines a series of n
circular regions of zero displacement (nodal circles), while the cosine term defines a series of m
diameters of zero displacement (nodal diameters) that separate the TM into similarly-sized wedges (Kinsler et al.1982
; Fletcher 1992
An example of a ‘natural’ modal pattern is illustrated in . In this figure we see 5 regularly spaced nodal diameters (m=5) and
3 nodal circles (n=3
), where d
, ) =0
. The modal pattern in , and the computed TAH associated with it (), approximate the ‘ordered’ holographic patterns seen at higher frequency in all four specimens of . The computed modal pattern and the TAH both show a collection of regions of large displacement magnitude separated by nodal regions zero displacement. The regions of large motion magnitude, the ‘pearls’, are circularly arranged on ‘strings’. The largest magnitudes of motions, the deepest blues (outward motions) and reds (inward motions) in 7A and the highest concentrations of concentric dark fringes in 7B, occur near the center of the membrane while the local displacement maxima and minima are of smaller magnitude closer to the TM rim. One difference between the computed modal pattern () with its associated nodes and the TAH () is the rectification that occurs in the TAH computation, where the TAH image intensity is the square of the Bessel function of the displacement (Equation 7
). Because of this rectification, the TAH gives no information about the relative phases of the different maxima, whereas the computed modal patterns displayed in predict that positive outward moving modal maxima are surrounded by negative inward moving modal maxima, with nodal regions of zero displacement separating the modal peaks and valleys.
Figure 7 The modal pattern of displacement of a circular membrane computed from equations 8 and 9, with m=5 and n=3. (A.) The signed magnitude of the displacement is coded by color as a function of position, with zero displacement coded as a pale blue-green. The (more ...)
Estimating node numbers
While the TM is certainly more complicated in structure than a flat circular lossless membrane, the similarity of and the lower rows of suggest that we can use counts of nodal circles and diameters to describe the modal patterns visible in the TAHs of . (An analysis of such node counts assumes that the measured TAH primarily reflect a single natural modal pattern. The strength of this assumption is a point of later discussion.) We counted the nodal circles and diameters on the four specimens of for a range of stimulus frequencies from 5 to 25 kHz. shows the number of ‘m’ nodal diameters we counted in each specimen at the varied stimulus frequencies; illustrates the number of ‘n’ nodal circles in the same specimens. While the ‘n’ value was rather easy to count in each specimen at frequencies greater than 1–4 kHz, it was more difficult to count ‘m’: We could not always see a distribution of m nodal diameters and modal maxima at all locations on the membrane surface, and when we did, it was sometimes difficult to separate one local displacement maxima from another. (This ‘blurring’ of the nodal diameters is responsible for the fewer data points in .) Because of these difficulties, our estimates of ‘m’ are often based on extrapolations of counts made over known segments (about one-quarter) of the TM surface.
Estimates of the ‘m’ and ‘n’ node numbers from the measured TAHs in our four specimens as a function of frequency.
indicates a strong frequency dependence in the number of ‘m’ nodal diameters in our 4 specimens, where m in chinchilla is about 2 to 3 times that observed in human and cats at similar frequencies, with m varying from 27 to almost 50 in chinchilla, and between 5 and 25 in human and cat. The increase in m with frequency is consistent with the observations that the size and separation of the ‘pearls’ on the strings decreased as frequency increased. indicates that the number of ‘n’ circular nodes we observed was independent of the species we used but highly dependent on sound frequency (correlation coefficient of 0.926); n varied from 2 to 6 as the sound frequency was varied from 5 to 25 kHz. The increase in n with frequency quantifies the observation that the number of the strings of pearls in the TAHs increased as stimulus frequency increased.
The analysis of m
nodes used above assumes that the TAHs we record are dominated by a single modal pattern with a natural frequency nearby the stimulus frequency. In actuality, the TAHs we record represent the summation of the weighted response of a group of modal patterns. One weighting factor is inversely proportional to the difference between the frequency of the stimulus and the natural frequency of each pattern, and generally the modes with natural frequencies nearest the stimulus frequency dominate. However, other weighting factors, such as the average displacement of the membrane at each modal frequency, also play a role and may interfere with our assumption (Fletcher 1992
Nodes, modes and standing waves
The presence of nodes, locations of decreased motion, on the surface of the TM is consistent with the presence of modes or standing waves that result either (1) from the cancellation of multiple transverse waves on the surface of the TM, or (2) from instantaneous excitation of modal surface-motion patterns on the TM. However, these two possibilities are not easily separable. For example waves on a string produced either by (1) an oscillating displacement source at the one end of the string or (2) instantaneously produced waves that result from plucking a string fixed at both ends, can both be described either in terms of the localized summation and cancellation of bi-directional traveling waves as they reflect at the boundaries and propagate back and forth along the string, or in terms of the summation of a family of temporally invariant standing waves.
Regardless of their derivation, we can use observations of the displacement patterns to quantify parameters relevant to either model of TM response. One measure of the relative magnitude of the reflections occurring at the boundaries of a wave-conducting media, or the losses within the media, is the standing wave ratio or SWR: the ratio of the time averaged magnitudes of the (mode associated) spatial displacement maximum and the nodal displacement minimum. In our TAH images we often see displacement magnitude maxima with TAH patterns that include at least 3 holographic fringes. For the blue laser used in our system, suggests that the third fringe corresponds to a magnitude of 320 nm. While imperfect knowledge of the illuminating intensity prevents us from precisely defining the magnitude minima within the nodal ‘white’ areas in the TAHs of , it is clear from and the holograms in the middle and lower rows of that the nodal ‘white’ areas have gray values that are least half the maximum intensity of the gray areas observed between fringes 1 and 2. This suggests that the nodal magnitude is smaller than 80 nm. Therefore, the SWR is greater than 4 (320/80).
In the surface-wave model of TM excitation (case 1), a standing wave ratio of 4 or greater implies that at least 36% of the energy in any surface wave traveling along the membrane is reflected at the wave boundaries (Kinsler et al. 1982
). In the ordered regime, the boundaries seem to include not only the rim of the TM, but also the attachment of the TM to the relatively immobile malleus. The demonstration of a significant SWR in our specimens is consistent with some models of TM function that suggest significant surface standing waves on the TM (e.g. Puria and Allen 1998
) but not consistent with models that suggest the TM acts as a lossless, matched transmission line in which there is little reflection as waves progress from the membrane rim to its center (e.g. Parent and Allen 2007
In the instantaneous-excitation model (case 2), a standing wave ratio of 4 implies the presence of energy losses within the membrane or its attachments. The magnitude of these losses is described by one minus the power-reflection coefficient. Since we estimate a power reflectance of at least 36%, we place an upper limit on power absorption (1 minus power reflection) of 64%. This estimate suggests that such losses are significant.
Waves on the TM surface?: Estimates of wave speed and wave delay
The TAH technique we use describes the magnitude of motion of each point on the TM surface averaged over the time it takes to collect four camera frames. With this instrument, pure radially traveling waves of constant amplitude along the membranes surface will produce changes in light intensity that are uniform along the membrane surface, where the magnitude of the measured intensity would depend on the magnitude of displacement and the Bessel function that relates the two (, Equation 7
). Realistically, such unidirectional traveling waves cannot occur in a membrane that is rigidly supported at its rim, where interactions of traveling waves with the rim will produce reflected traveling waves moving in the opposite direction. The interactions of such bidirectional traveling waves will produce standing waves with local spatial minima (nodes), consistent with the nodes observed in . While we have argued that the nodal minima we observe could also result from uniform stimulation of the entire membrane surface that produces natural modal patterns of excitation, we will at this point use our measurements to produce estimates of surface wave travel for comparison with other estimates in the literature.
The strongest arguments for traveling TM surface waves are the observations of delay in middle-ear transmission (as estimated from measurements of group-delay in middle-ear transfer functions: Olson 1998
; Overstreet & Ruggero 2002
; Ravicz, Cooper & Rosowski. 2008
) and the use of TM surface delays to fit functional middle-ear data (Puria & Allen 1998
; O’Connor & Puria 2008
). We can use our data to estimate the phase velocity of waves propagating radially along the TM surface by assuming that the distance between the easily-recognized circular nodes n
(counted in ) is half of a wavelength, i.e. c
= 2f n
. shows estimates of the speed of propagations computed from the circular (n
) nodal patterns we observe on the four specimens. These velocities vary from 25 to 65 m/s and in general, the estimated phase velocity increases as frequency increases. The phase velocities we estimate in cat vary less with frequency and tend to be lower than the phase velocities computed from the human and chinchilla TAHs at similar frequencies. The 30 to 40 m/s phase velocities we compute in cat fall in between phase velocities determined for cat TM by others (Puria & Allen 1998
; Fay et al. 2005
). (It should be noted that our estimate of wave velocity based on a count of the number of waves lengths along a radius of the TM, produces an average measure of wave velocity along that radius and does not account for variations in velocity along the radius. It also does not account for the compression of wavelength with radial distance that is predicted by the Bessel functions we use to model the radial modal patterns.)
Speed of propagation of radially traveling waves on the 4 TMs based on counts of ‘n’ nodal circles in .
The hypothesized propagation delay (Puria & Allen 1998
; Parent and Allen 2007
; O’Connor and Puria 2008
) for the travel of radial waves on the TM surface can be estimated by the ratio of the radial distance of the TM and the group velocity. lists estimates of the wave delay (based on the phase velocities of ) associated with radial travel along TMs that are 4 mm in radius for chinchilla and human, and 3 mm for cat. These delays vary from 0.06 to 0.16 ms in chinchilla, and from 0.06 to 0.09 ms in human and 0.075 to 0.1 cat, with the longer times at the lower frequencies. The delays we estimate in chinchilla are 1.5 to 6 times larger than the 0.025 to 0.040 ms middle-ear transmission delays observed in gerbil and cats, while the delays we estimate for cat are about twice the estimated middle-ear delay (Puria & Allen 1998
; Olson 1998
; Overstreet & Ruggero 2002
; Ravicz, Cooper & Rosowski 2008
), but the calculated delays in human TM are similar to the 0.04 to 0.09 ms delays predicted from models of TM sound transmission (O’Connor & Puria 2008
) and measurements of middle-ear function in human temporal bones (Nakajima et al. 2009
There is a significant difference in the methods we have used to quantify the delay associated with wave travel across the TM. The other reports in the literature in the literature (e.g. Olson 1998
; Overstreet & Ruggero 2002
) estimated the group delay based on measurements of changes in response phase with changes in frequency. Such delays can be associated with delays in energy transmission ((Brillouin 1960
). Our estimates of delay come from estimates of velocity based on observations of wavelength and phase delay using tonal (single-frequency) stimulation. The relationship between our measured phase delays and velocity of energy transmission is not clear cut, and the differences between previous delay estimates and those we calculate may reflect some of those complexities. Better estimates of group delays on the surface of the TM would come from observations of the TM surface displacements produced by transient stimuli.
Transverse waves versus modal displacements
While the existence of nodes in the patterns of TM surface displacement measured by the TAH technique is consistent with the presence of standing transverse waves on the TM surface, whether the nodes are produced by transverse surface waves is unclear. Repeating our earlier analogy of a string: A modal or standing-wave pattern can be produced on a string either by (a) the interaction of a forward-going and reflected transverse wave, when one end of the string is driven while the other is constrained, or (b) by initially plucking a string, which is bound at both ends, along its entire length thereby evoking a modal pattern of displacement. In either case, the steady-state motion along the length of the string can be described mathematically either in terms of multiple modal standing waves whose interactions produce traveling waves, or by bidirectional traveling waves whose interactions produce standing waves. Others have argued that the existence of middle-ear delay is suggestive of significant transverse wave travel along the surface of the TM (Puria & Allen 1998
; Fay et al . 2005
; Parent & Allen 2007
). An argument in favor of the patterns resulting from more instantaneous modal displacement of the surface is the disparity in the wave speeds estimated for transverse waves traveling along the surface of the TM (generally less than 70 m/s) and the significantly higher wave speed of sound in air at body temperature and standard pressure (nearly 350 m/s). The TAH technique we employ is unable to distinguish between these possibilities. However, future applications of stroboscopic holography techniques in which one can estimate the magnitude and phase of TM surface motion (e.g. Furlong et al. 2009
) will address this issue more directly.