3.1.

Phase Sampling Based on $2+1$ Frame Local Correlation

The $2+1$ frame LC phase sampling method¹⁷ is based on hybrid spatiotemporal phase sampling methods^15,16 that quantify changes in the state of deformation of an object throughout the duration of a transient event. In the LC method, two sets of frames are acquired. The first set consists of two temporally phase shifted reference frames gathered just before object deformation and the second set consists of a series of rapidly acquired frames gathered during object deformation. The method quantifies the double-exposure phase change at each instance by correlating the deformed and reference states. The correlation of individual frames relies on the assumption that the object is sufficiently stable during the acquisition of the two phase shifted reference frames, which we will demonstrate is the case in our measurements.

Consider two individually recorded camera frames at a reference,$I_{\mathrm{ref}}$, and at a deformed, $I_{\mathrm{def}}$, state with intensities

By assuming a constant beam ratio, $\varphi$, the phase change of interest, $\varphi$, in space, $(m,n)$, corresponding to the deformation of the object at a specific time instance,$\varphi$, can be computed by the combination of Eqs. (4) and (6) as

The computational process to recover $\varphi (m,n,t)$, schematically illustrated in Fig. 1, utilizes data sets from two phase shifted reference images, $I_{\mathrm{ref}}$ and $I_{\mathrm{ref}+\pi /2}$, recorded at times $t_{\mathrm{ref}}$ and $t_{\mathrm{ref}+\pi /2}$ before object excitation and from deformed images, $I_{\mathrm{def}}$, at time instances, $t_{\mathrm{def}}$, during and after object excitation. A temporally estimated DC is subtracted from the intensity values of $I_{\mathrm{ref}}$, $I_{\mathrm{ref}+\pi /2}$, and $I_{\mathrm{def}}$. The DC-compensated frames $I_{\mathrm{ref}}$ and $I_{\mathrm{def}}$ are then used for the evaluation of the correlation coefficient, $\rho (I_{\mathrm{ref}},I_{\mathrm{def}})$, at every measurement point on the frame. The numerical calculation of $\rho (I_{\mathrm{ref}},I_{\mathrm{def}})$ is implemented based on a computationally efficient Pearson product-moment correlation coefficient method^17,25 that uses sets of intensity values from spatial kernels (e.g., $3\times 3\text{pixels}$) around each measurement point of frames $I_{\mathrm{ref}}$ and $I_{\mathrm{def}}$. An analogous procedure is applied to the evaluation of the Pearson correlation coefficient for $\rho (I_{\mathrm{ref}+\pi /2},I_{\mathrm{def}})$. By computing $\rho (I_{\mathrm{ref}},I_{\mathrm{def}})$ and $\rho (I_{\mathrm{ref}+\pi /2},I_{\mathrm{def}})$ for each postdeformation measured time point using Eq. (7), $\varphi (m,n,t)$ is computed for modulus $2\pi$, which requires further processing to obtain a continuous phase distribution by application of spatiotemporal phase unwrapping algorithms.^2,26

Fig. 1

Flow chart illustrating the pointwise implementation of the $2+1$ frame local correlation (LC) phase sampling algorithm to recover the phase change, $\varphi (m,n,t)$, corresponding to transient deformations of an object. The correlation coefficient, $\rho$, at a measurement point between any two consecutive frames is based on the evaluation of intensities defined by a kernel within the proximity of the measurement point. The zero of the time axis corresponds to the initiation of the acoustic excitation and signifies the first of the set of deformed frames.

The pointwise evaluation of the correlation coefficients assumes that $\varphi (m,n,t)$ varies slower in space $(m,n)$ than all of the other parameters in Eqs. (1) and (2) and is sufficiently constant within a spatial kernel¹⁶ (i.e., small spatial neighborhood of measurement points). This assumption is adequate for the acoustically induced response of the human TM under typical loading conditions since the resulting fringe density within the hologram is small ($<4\text{fringes}/\mathrm{mm}$) compared to the spatial resolution ($>25\mathrm{lp}/\mathrm{mm}$) of a typical HSC.²⁵

The use of a spatial kernel to represent each measurement point in the LC method is equivalent to applying a spatial band-pass filter. In addition, the intensity values within the kernel can be individually weighted to obtain the frequency response of specific spatial filters, such as mean or Gaussian.¹⁶ Variation of the size of the spatial kernel also allows for control of the spatial frequency range of filtration.

The LC algorithm we describe is independently executed at every measurement point, which allows for the simultaneous evaluation of $\varphi$ at all measurement points across each deformed frame, $I_{\mathrm{def}}$, in parallel. By using this computational parallelism, we have developed multithreaded and GPU accelerated software that improves computational speed and efficiency.

3.2.

High-Speed $2+N$ Frame Acquisition

To measure the transient deformations of the TM, a high-speed $2+N$ frame acquisition approach based on the $2+1$ frame LC phase sampling method has been developed and implemented. In this approach, two reference frames, $I_{\mathrm{ref}}$ and $I_{\mathrm{ref}+\pi /2}$, and $N$ consecutive frames, ${(I_{\mathrm{def}})}_{i,i\in \mathrm{1,2}\dots N}$, are recorded before and throughout the evolution of an event.

Figure 2 shows the timing diagrams of the events that occur during the high-speed $2+N$ acquisition, including camera exposure, acoustic excitation, phase shifting, and the TM’s response. According to this diagram, the two reference frames, $I_{\mathrm{ref}}$ and $I_{\mathrm{ref}+\pi /2}$, are recorded with a temporal separation that is longer than the piezoelectric transducer (PZT) settling time after the introduction of the initial $\pi /2$ phase shift.²⁷

Fig. 2

Timing diagrams of the events occurring during high-speed $2+N$ frame acquisition with the LC phase sampling method. $N$ deformed frames are acquired at the maximum frame rate of the camera and referenced relative to frames captured during two periods prior to acoustic excitation. Acquisition takes into account the settling time of the PZT phase shifter, while minimizing the overall acquisition time. PZT is returned to its original position at the end of the acquisition. Due to the distance between the sound source and the sample, there is an acoustic transmission delay. Representative values of major parameters are indicated.

For efficient acquisition of ${(I_{\mathrm{def}})}_i$ frames, the PZT is kept at its final position after phase shifting, which leads to the ${(I_{\mathrm{def}})}_i$ frames containing a constant $\pi /2$ phase offset expressed as ${(I_{\mathrm{def}+\pi /2})}_i$. Therefore, the optical phase, $\varphi$, within any ${(I_{\mathrm{def}+\pi /2})}_i$ frame, and corresponding to the deformation of the TM at a specific instance, can be determined from Eq. (7) with

All deformed frames, ${(I_{\mathrm{def}+\pi /2})}_{i,i\in \mathrm{1,2}\dots N}$, before and throughout the evolution of the transient response of the TM, are relative to the same reference frames, $I_{\mathrm{ref}}$ and $I_{\mathrm{ref}+\pi /2}$, before acoustic excitation.

The camera continuously records frames during the full duration of the measurement, including during the settling time of the PZT, with the maximum sampling rate of acquisition constrained by the frame rate of the camera. During a typical acquisition, multiple frames (i.e., $\sim 20$ frames at 42 k fps) at each reference position are captured and temporally averaged in order to compensate for potential external disturbances. In addition, and by confirmation with laser Doppler vibrometer (LDV) measurements, 1.5 ms are allocated for the settling time of the PZT and the frames recorded over this time are discarded.

3.3.

Hardware and Software Implementation of the Acquisition

The hardware implementation of the high-speed acquisition involves the synchronization of the procedures for acoustic excitation of the sample, temporal phase shifting, and frame acquisition. This is achieved by the development and implementation of a control architecture that consists of four major hardware modules, which include electronic I/O control (I/O), sound presentation and measurement (SP), phase shifter (PS), and HSC, as shown in Fig. 3.^17,25,26 Hardware modules are managed by a unified control software with a user interface for setting and customizing acquisition and synchronization parameters.²⁸

Fig. 3

Schematic showing the major modules of the high-speed acquisition hardware and software. The components within the I/O control module communicate with the sound presentation and measurement, phase shifter, and HS camera (HSC) modules that are synchronized by the digital output component serving as the master clock.

To perform high-speed transient measurements of a TM sample, the user is required to specify the following parameters:

During a measurement, the control software utilizes the digital output component of the I/O module to automatically trigger the execution of all procedures through correlated timing signals with a temporal accuracy $>1\mu \mathrm{s}$. The applied sound pressure level (SPL) and frequency content of the acoustic excitation is automatically quantified with a calibrated microphone within the SP module interfaced to an analog input port of the I/O module.^5,29

4.1.

Experimental Setup

The components of the different modules of the HHS are shown in Fig. 4. The sound presentation module consists of a calibrated microphone (Etymotic Research ER-7C, Elk Grove Village, Illinois) and a speaker (SB Acoustics SB29RDC-C000-4, Brookfield, Wisconsin).²⁹ The laser delivery module includes a continuous wave laser (Oxxius SLIM-532, 50 mW, Lannion, France), variable ratio beam splitter, and a PZT PS. The HSC module is a Photron SA5 1000k (Tokyo, Japan) (42 K fps at ${384}^2\text{pixels}$). For validation of the transient measurement capabilities of the HHS, an LDV is incorporated within the HHS experimental setup.

Fig. 4

Experimental setup and artificial sample: (a) schematic of the HHS setup that includes sound presentation, laser delivery, and high-speed camera modules; (b) latex membrane with six retroreflective markers; (c) and (d) time waveform and power spectrum of an average of 20 acoustic clicks indicating 10 Pa or 115 dB peak sound pressure level (SPL) based on a calibrated microphone. The wedge in the HSC module redirects 95% of the object beam and 5% of the reference beam power to the sensor of the HSC. A laser Doppler vibrometer (LDV) is incorporated for characterization of the HHS measuring capabilities.

A schematic of the 10-mm-diameter circular latex membrane used for transient measurements is shown in Fig. 4(b). Retroreflective markers were applied at several predefined points on the surface of the sample in order to improve the signal quality for LDV measurements. Five markers were distributed equidistantly to allow for even coverage of the latex surface (points 1 to 5), as shown in Fig. 4(b). In order to monitor any rigid body motion, a sixth marker was placed at the top on the locking ring that clamped the latex membrane to its cylindrical support (point 6).

Acoustic excitation for all dynamic measurements was based on a $50\text{-}\mu \mathrm{s}$ acoustic click produced by the signal generator in the I/O module and introduced to the sample by the speaker in the SP module. The time waveform and power spectrum of an average of 20 clicks produced by the speaker are recorded by the calibrated microphone and are shown in Figs. 4(c) to 4(d).

4.2.

Validation of the $2+1$ Frame LC Phase Sampling Method

The noise floor and accuracy of the $2+1$ frame LC phase sampling method were quantified independent of the high-speed acquisition method through a set of experiments performed in static conditions.

4.2.1.

Noise floor

In order to quantify the noise floor of the $2+1$ frame LC phase sampling method, we measured the displacement of a solid object (i.e., 12.7-mm-thick aluminum plate) under static conditions and no external excitation. We assumed that the changes of the sample during the measurements are insignificant and any detected deformations were associated with the noise floor of the phase sampling method. During the noise floor measurements, all optical and sampling conditions were kept identical to ones used with the high-speed acquisition. The spatial distribution of the noise floor, as shown in Fig. 5, indicates a standard deviation (STD) of 8.7 nm or $\lambda /31$.

Fig. 5

Representative results of the spatial distribution of the noise floor of the $2+1$ frame LC phase sampling method under static conditions and no external excitation of the object: (a) map of the spatial distribution of the noise signal; (b) horizontal and (c) vertical cross-sections of (a); and (d) histogram of (a). The $\pm 1\sigma$ (dashed line) and the $\pm 2\sigma$ (dash dot line) of the noise measurement map cross-sections are highlighted.

4.2.2.

Displacement accuracy

To assess the accuracy of the $2+1$ frame LC phase sampling method, we compared the displacement measurements of a statically deformed sample against a four-frame temporal phase sampling method.^2,3 In order to maintain identical conditions for both phase sampling methods, we used two phase shifted references and one of the deformed frames of the four-frame data set and applied it to the $2+1$ frame LC phase sampling method. Because of the static loading conditions and the longer ($>100\mathrm{ms}$) measurement time of the four-frame acquisition method, the camera’s sampling rate was reduced to 60 Hz (the minimum for Photron SA5).

The sample was a 12-mm-diameter aluminum membrane under static loading with a constant force applied normal to the back surface. Aside from the sampling rate, the optical setup and illumination conditions for the accuracy measurements were identical to the high-speed acquisition settings.

The comparison of the modulation and phase maps of the four-frame double exposure and $2+1$ frame LC phase sampling methods is shown in Fig. 6. Based on the phase maps, the corresponding object displacements are calculated and their differences are shown in Figs. 6(e) and 6(f). The STD of the displacement differences, which can be used to assess the accuracy of the LC method, is within 11 nm or $\lambda /25$.

Fig. 6

Characterization of the accuracy of $2+1$ LC relative to the four-frame phase sampling method: (a) modulation and (b) phase of the $2+1$ LC phase sampling method; (c) modulation and (d) phase of the four-frame phase sampling method; (e) displacement differences between the two methods; and (f) histogram of (e). The difference in displacement has a mean of 0 with a standard deviation of 11 nm or $\lambda /25$.

4.3.

Validation of the High-Speed Acquisition

4.3.1.

Phase stepping at high speed

In order to optimize the timing of the high-speed acquisition method, we characterized the dynamic response of the PS (measured at the center of the mirror) by measuring its velocity time waveform with an LDV, as shown in Fig. 7. Analysis of the data indicates a 0.5 ms time constant, with $<1.5\mathrm{ms}$ settling time with 6% residual fluctuations, and a noise floor of $<4\%$ of the maximum response.

Fig. 7

Representative results of the velocity time-waveform of the response of the PZT phase shifter (PS) indicating 0.5 ms time constant and $<1.5\mathrm{ms}$ settling time with $<6\%$ residual vibrations relative to the maximum response. The PZT control input (dash dot line), PZT velocity response (solid line), and fitted decay curve (dashed line) are shown.

4.3.2.

Temporal accuracy

We characterized the temporal accuracy of the high-speed acquisition method relative to an LDV by correlating coincident LDV and HHS measurements of controlled transient acoustic stimulation of the latex sample. The LDV sampled the motion at the six locations described in Sec. 4.1. In order to characterize the accuracy in position and velocity, and due to the HHS and LDV measuring domains, time waveforms were either differentiated or integrated.

The HHS was set at a sampling rate of 42 kHz (corresponding to $23.8\mu \mathrm{s}$ interframe time) with $6.62\mu \mathrm{s}$ exposure time (corresponding to a shutter speed of 151 kHz), in order to be consistent with the sampling parameters specified in Table 1. With these settings, the HSC has a spatial resolution of $384\times 384\text{pixels}$ corresponding to a spatial resolution of $>25\mathrm{lp}/\mathrm{mm}$ at the object plane. The LDV was sampled at 84 kHz through a 16-bit analog input at the I/O module. Representative HHS and LDV displacement and velocity results, depicted in Fig. 8, show the time waveforms at two of the discrete locations of the sample, as identified in Sec. 4.1, which correspond to points with the minimum (point 5) and maximum responses (point 1). The acoustical excitation was a $50\text{-}\mu \mathrm{s}$ click with a peak SPL of 104 dB.

Fig. 8

Representative results of the temporal accuracy of the high-speed acquisition relative to LDV based on the minimum (point 5) and maximum (point 1) responses of the latex membrane shown in the top and bottom rows of graphs, respectively. Acoustic excitation was a $50\text{-}\mu \mathrm{s}$ click with a 104 dB maximum SPL. The time axis is relative to the beginning of the transient acoustic stimuli. Correlation between the time waveforms of each method is on average $>95\%$ for both displacement and velocity.

Correlation between the time waveforms is, on average, $>95\%$ for both displacements and velocities at all points (1 to 5) on the surface of the sample, as specified in Sec. 4.1. The average differences between the time waveforms are $<5\%$ relative to the p-p response of the sample. Differences between temporal locations of the local maxima and minima in the time waveforms of the HHS and LDV are within $10\mu \mathrm{s}$.

5.1.

Full-Field-of-View Transient Displacement of the Human TM

The HHS provides high temporal ($>42\mathrm{kHz}$) and spatial ($>100\mathrm{k}$ data points) resolutions that enable the simultaneous measurements of the spatial distribution and the temporal and frequency domain of the displacement of transient response of the human TM. Representative measurements of the spatial, temporal, and frequency measurement capabilities of the HHS, based on the acoustically induced transient response of a human TM, are shown in Fig. 9. Displacement time waveforms of the response of the umbo, as shown in Fig. 9(a), indicate $>90\%$ correlation between the HHS and LDV measurements above the noise floor ($<11\mathrm{nm}$) of the HHS. The average correlation of the time waveforms of the HHS and LDV at points (1 and 2) on the TM’s surface was $>94\%$. According to Figs. 8 and 9(a), it is observed that the magnitude of the correlation coefficient is related to the magnitude of the response. Lower correlation levels (i.e., $\sim 90\%$) are associated with displacement measurements closer to the noise floor ($<11\mathrm{nm}$) of the HHS, as shown in the waveform corresponding to the minimum response of the human TM in Fig. 9(b).

Fig. 9

Representative measurement of the spatial, temporal, and frequency measurement capabilities of the HHS based on the acoustically induced transient response of a human TM sample: (a) displacement versus time waveform of the umbo measured with HHS and LDV indicating 90% correlation of the time waveforms; (b) power spectrum of the displacement transfer function at the umbo measured with HSS (solid line) and LDV (dotted line); and (c) full-field HHS displacement maps at five instances 80 to $390\mu \mathrm{s}$ after the arrival of the acoustic pulse indicating a $0.83\mu \mathrm{m}$ p-p maximum along the periphery of a human TM. The excitation was a $50\text{-}\mu \mathrm{s}$ click with a peak SPL of 115 dB. Dashed lines in (b) refer to automatically determined modal frequencies from the HHS measurement. With the exception of the lowest modal frequency (0.98 kHz, which was not identifiable in the LDV), there was a $<5\%$ difference between HHS and LDV determined modal frequencies. Outlines of the outer boundary of the membrane and the manubrium (the handle of the malleus that is attached to the TM) in (c) are indicated with solid lines.

Based on the frequency domain of the HHS and the calibrated microphone measurements of the displacement and acoustical pressure time waveforms, respectively, the transfer function (TF) of each point across the surface of the TM can be calculated,⁵ and the frequencies of maximum motion associated with the modal frequencies can be identified by automatic quantification of the local maxima of the TF, as shown in Fig. 9(b). Due to the short duration of the transient event (i.e., $<5\mathrm{ms}$), frequency components $<200\mathrm{Hz}$ (i.e., $>5\mathrm{ms}$ period) are disregarded and are not considered in the modal frequencies’ identification. The detected modal frequencies at the umbo, shown in Fig. 9(b), based on the HHS and LDV differ by $<5\%$. The difference between magnitude of the TF measured with HHS and LDV in the frequency range of 1 to 8 kHz is within 5 dB. Based on measurements on a latex membrane, described in Sec. 4.3.2, the noise floor of the TF was estimated as $\sim 20\mathrm{dB}$ below the average signal level for frequencies $<8\mathrm{kHz}$. The difference between the HHS and LDV at $\sim 6\mathrm{kHz}$ could be associated with the acoustical properties of the experimental setup as well as the response of the speaker used. In particular, the speaker could have band gaps in its response, which could result in local higher noise floor (lower signal-to-noise ratio) for the HHS, while the LDV data have been averaged over multiple runs (i.e., $>10$) and exhibit fewer effects from that phenomenon. Future work will be focused on specifying an acoustic source with a flatter click response.

Based on the spatiotemporal evolution of the HHS displacement measurements, as shown in Fig. 9(c), it can be seen that the transient response of the human TM undergoes two distinct stages—global initiation of the surface motion and local surface wave propagation. The first 30 to $100\mu \mathrm{s}$ of the initial stages of the transient displacement of the human TM, shown in Fig. 10, indicate motion that is approximately in-phase across the full surface of the visible TM. It can be seen that there is $<30\mu \mathrm{s}$ delay between the arrival of the acoustic pressure front at the surface of the TM and the beginning of its transient response. This delay can be contributed to the temporal resolution of the HSS system ($24\mu \mathrm{s}$ interframe time) and the reaction time of the speaker.

Fig. 10

Representative results of the 30 to $100\mu \mathrm{s}$ of the initial stages of the transient response of the TM indicating mostly in-phase displacement across the full surface. The acoustic excitation was a $50\text{-}\mu \mathrm{s}$ click with a 115 dB maximum SPL. Outlines of the outer boundary of the membrane and the manubrium are indicated with solid lines.

The further spatiotemporal evolution of the transient response of the TM indicates circumferentially traveling surface waves propagating symmetrically relative to the manubrium and radially from the inferior to the superior parts of the TM, as shown in Fig. 11. The local phase velocity of the surface waves can be estimated by automatically identifying the shift of the spatial location of the local minima and maxima of the displacement maps between successive frames. Preliminary surface wave speed estimations indicate $24\mathrm{m}/\mathrm{s}$, which is in agreement with previous research.^5,21

Fig. 11

Representative results at 320 to $390\mu \mathrm{s}$ of the transient response of the surface of the TM indicating circumferentially traveling surface waves propagating at $24\mathrm{m}/\mathrm{s}$ symmetrically relative to the manubrium and radially from the inferior to the superior parts of the TM as indicated by solid arrows. The acoustic excitation is a $50\text{-}\mu \mathrm{s}$ click with a peak SPL of 115 dB. Outlines of the outer boundary of the membrane and the manubrium are indicated with solid lines.

5.2.

Acoustic Delay and Dominant Modal Frequency Maps

The HHS provides the time waveforms of the transient response of all points across the surface of the human TM sample, allowing for estimation of the spatial dependence of motion parameters, such as acoustic delays and dominant modal frequencies, as shown in Fig. 12.

Fig. 12

Spatial dependence of the (a) acoustic delay of each point on the TM relative to the peak time of the center of the umbo (marked with $\oplus$); histogram of (b); spatial distribution map of the dominant frequency at each point on the surface of the TM; and (d) histogram of (c). Outline of the manubrium is indicated with a solid line.

The acoustical delay map, as shown in Fig. 12(a), is calculated by automatically identifying the peak time of the first local extrema of the time waveform of every point across the surface of the TM and referencing it to the peak time of the umbo, indicated with $\oplus$ in Fig. 12(a). This quantifies the spatial distribution of the delay of the time-domain response of each point relative to the umbo. The range of the measured acoustical delays across the surface of the human TM, as shown in Fig. 12(b), is within previously reported data.^5,21

The acoustic delay map and histogram indicate that the peak motion response of $>50\%$ of the surface is within $\pm 50\mu \mathrm{s}$ of the peak motion of the umbo, which supports the observations of predominantly in-phase motion during the initial response of the surface of the TM, as shown in Fig. 10. The acoustic delay data, shown in Fig. 12(a), also indicate that the surface of the TM at the interior and posterior boundary moves with a $-25\mu \mathrm{s}$ acoustical delay relative to the umbo. This suggests that the acousto-mechanical response of the TM reaches its first extrema in the region between the TM boundary and the umbo, which agrees with theories of acousto-mechanical energy transfer from the TM periphery to the umbo.⁴ However, since our measure of delay depends on the time to the first temporal extrema, it includes the time associated with responses to different natural frequencies.

Based on automatic identification of the local maxima of the TF, shown in Fig. 9(b), at every point across the TM, we can determine the dominant modal frequency and its spatial distribution, as shown in Fig. 12(c). The dominant frequency map indicates noticeable differences (three- to fourfold) between the regions of the TM near the manubrium and the central region midway between the manubrium and the TM boundary. Assuming that the spatial distribution of the dominant frequency is representative of the local variations of stiffness and thickness of the TM, our observations can be related to previous studies indicating more than a threefold increase in the local thickness of the TM near the manubrium relative to the central region of the TM.³⁰

In order to relate the data in Figs. 12(a) and 12(c), we assume that the initial displacement (i.e., $<100\mathrm{us}$) of the click response of the TM, as shown in Fig. 10, exhibits a single tone decayed response based on a dominant frequency spatially varying across the TM as shown in Fig. 12(c). Based on this assumption and the approximately in-phase start of the motion of all points suggested by Fig. 10, the acoustical delay (i.e., the temporal location of the first local extrema) between any two points should be within a ¼ of the difference of the periods of oscillation of the points. Figure 12 indicates dominant frequencies of $\sim 1\mathrm{kHz}$ ($\sim 250\mu \mathrm{s}$ for ¼ cycle) at the umbo and an average of 3.5 kHz ($\sim 70\mu \mathrm{s}$ for ¼ cycle) across the central region midway between the manubrium and the TM boundary. This suggests $\sim 180\mu \mathrm{s}$ of maximum acoustic delay that can be associated with the difference in the natural frequency of the response between the umbo and the region midway between the umbo and the rim. Such a delay is in agreement with the range of the measured delay to the first extrema ($-25$ to $75\mu \mathrm{s}$) indicated in Fig. 12(b).