THE DEFINITION OF ACOUSTIC IMPEDANCE AND ADMITTANCE
Multiple acoustic techniques and devices have been used to measure the average motion of the tympanic membrane (TM) in response to sound, where the basic task is to figure out what is the relationship between sound pressure as a function of time, p(t), and the rate of alternating volume displacement, u(t), (usually called the volume velocity) of the TM (Fig. 1). If the stimulus is a sinusoid of frequency f, that is, p(t) = P cos(2πft + θP), with an amplitude P and a phase argument θP (which specifies the value of the cosine function when time equals 0), and the sound intensity is within the range where themiddle ear is linear (sound levels less than 120 dB SPL, (Guinan & Peake 1967; Goode et al. 1994; Dalhoff et al. 2007), then the volume velocity at the TM is also a sinusoid at the stimulus frequency u(t) = U(f) cos(2πft + θU(f)), with a frequency-dependent magnitude U(f) and a phase θU(f)*. From these relationships we can define a frequency-dependent acoustic impedance Z(f), where Z is a complex number with a magnitude equal to the ratio of the magnitudes of the pressure and volume velocity at each stimulus frequency, |Z(f)| = P/U(f), and a phase angle ∠Z(f) that equals the difference between the phases of p(t) and u(t), that is, ∠Z(f) = θP − θU(f). (Note that while the sound pressure and volume velocity vary with time, the magnitude and phase angle of the impedance do not. All symbols are summarized and defined in Appendix B.)
Alternatively, we can use measurements of u(t) and p(t) to define an acoustic admittance Y(f), where the admittance magnitude |Y(f)|, is the ratio of U(f)/P, and its phase angle is the phase of the volume velocity relative to the phase of the sound pressure, that is, ∠Y(f) = θU(f) − θP. In terms of complex numbers †Z(f) = 1/Y(f). The simple relationship between Z(f) and Y(f) makes them effectively interchangeable; a fact that leads to the use of the term immittance to describe Z(f) and Y(f) collectively. The two quantities have different but related units. The Système International unit of acoustic impedance is the acoustic ohm: one acoustic ohm = 1 Pa-s-m−3. The unit of acoustic admittance is the acoustic siemen: one acoustic siemen = 1 m3-s−1-Pa−1. The pascal (Pa) is the Système International unit of pressure, where 1 Pa = 1 kg-m−1-s−2.
The techniques used to measure immittance in the ear canal are tabulated in Table 1, and include: (a) the combination of sound pressure measurements with the volume velocity output from a well-characterized sound source (determination of the equivalent source characteristics—as described in Appendix A—is required because it is difficult to measure directly volume velocity), (b) acoustic bridges that estimate Z(f) or Y(f) from direct comparisons of the sound pressures produced in unknown acoustic immittance and known acoustic immittances, (c) wave tubes that quantify the variation in sound pressure magnitude and phase angle along the length of a uniform tube coupled to a structure of unknown Z(f) or Y(f), and (d) wave tube methods that use sound pressure measurements along the length of the ear canal. Methods (1), and (2) are direct measurements of acoustic immittance determined from the measurement of sound pressure produced by a volume velocity source. Methods (3) and (4) actually measure acoustic reflectance within tubes and ear canals from which the immittance at the termination of the tube can be calculated.
WAVES IN TUBES
A long narrow rigid-walled tube of constant cross-section S acts like a one-dimensional acoustic device. The sound pressure varies along the long axis of the tube (the x axis) in the form of forward (p+) and backward (p−) traveling waves:
ω = 2πf, and k = ω/c.Figure 2 schematizes the presence of a forward and a backward wave traveling in a rigid tube of uniform cross-section. The tube is terminated on the right-hand side with an unknown acoustic immittance. The forward wave is initiated by a sound source on the left-hand side. The backward wave is initiated by reflection of the foward wave at the tube’s terminal end.
ACOUSTIC REFLECTANCE AND ABSORBANCE
When discussing acoustic reflectance, we need to distinguish between pressure reflectance and power or energy reflectance. Lets start with the more basic quantity. Acoustic pressure reflectance R(f, x) compares the complex sound pressure (magnitude and phase) in a sinusoidal wave that propagates toward a terminal boundary (the incident or forward wave) within the tube with the sound pressure of the reflected (backward traveling) wave that originates at the tubes terminal boundary. More specifically, R(f, x) is the ratio of the complex amplitude of the reflected and incident waves with a magnitude |R(f, x)| equal to the ratio of the magnitudes of the reflected and incident pressures wave and an angle ∠R(f, x) (which varies with distance from the source of the reflection) equal to the difference in phase angle between the reflected and incident waves at the measurement location. In terms of the description of the forward and backward waves of Eq. (1), the magnitude of the pressure reflection coefficient is:
and the phase of the pressure reflection coefficient is:
Reflectance is most easily understood and determined in a well-controlled system such as a wave tube (e.g., Fig. 2). A wave tube is a long cylindrical tube of constant (uniform) cross-sectional area with rigid walls. A sound source is placed at one end of the long tube, and an object with unknown immittance is used to terminate the other end of the tube. The tube construction includes some mechanism enabling measurements of sound pressure at varied x locations along the length of the tube, where the sound pressure at any x equals the sum of the forward going “incident” pressure wave and the backward “reflected” wave at the measurement location [Eq. (1)]. The relative magnitude of the backward and forward going waves depends on the magnitude of the acoustic immittance that terminates the tube. The relative phase angle of the two waves at position x depends on the phase angle of the waves at the termination and the distance between the measurement location and the termination. If the measurement tube walls are rigid, the tube is of uniform or slowly (variations in cross-section that are much smaller than a wavelength) and regularly (the derivative of the area with x is continuous and of a single sign) varying cross-section, and effects of the viscosity of air are small (a reasonable assumption for many circumstances), then |R(f,x)| is constant throughout the tube, and the phase angle of the reflectance ∠R(f,x) varies regularly with position along the tube [Eq. (2)]. Local maxima in sound pressure occur at positions in the tube where the phase angles of the incident and reflected waves are equal. Local minima occur at positions where the two waves are 180° out-of-phase. The ratio of the magnitude of the pressures at the local maxima and minima can be used to define the magnitude of the pressure reflectance |R(f,x)| within the tube produced by the termination, and the location of the minima together with the speed of sound and the frequency of the tonal signal defines the phase angle of the pressure reflectance ∠R(f,0) at the termination (Kinsler et al. 1982; Stinson et al. 1982).
The pressure reflectance at the termination (at position x = 0 in Fig. 2) can be quantified in terms of the acoustic impedance of the termination ZT(f) = Z(f, 0) and the characteristic impedance of the air-filled measurement tube Z0, where Z0 depends on the density of air ρ0, the speed of sound in air c, and the cross-sectional area of the tube S (Z0= ρ0c/S), such that:
You can also determine the pressure reflectance in the tube at any distance x away from the termination by comparing the impedance measured at that distance with Z0 based on the cross-sectional area at location x.:
At the terminal boundary, when the magnitude of ZT(f) is much larger than Z0, R(f, 0) is approximately 1, that is, the reflected pressure magnitude approximates the incident pressure magnitude, and the phase of the incident and reflected waves at the terminal boundary are nearly equal. When the magnitude of ZT(f) is much smaller than Z0, R(f, 0) is approximately −1, and the reflected pressure magnitude also approximates the incident pressure magnitude, but the phases of the incident and reflected waves are opposite (i.e., they differ by 180°). When the magnitude of ZT(f) equals Z0, the magnitude of the reflected wave is zero and R(f, 0) = 0. As noted in Eq. (2a), for a rigid-walled air-filled tube of uniform or slowly and regularly varying cross-section, the magnitude of the pressure reflectance along the tube is constant and equals the magnitude of the pressure reflectance at the termination where x = 0, that is, |R(f, x)|= |R(f, 0)|. The angle of the pressure reflectance measured along the tube equals the angle of the reflectance at the termination plus a factor related to the round trip travel time of the pressure wave between the measurement point and the termination [Eq. (2b)], that is, ∠R(f, x) = ∠R(f,0) +4π x/λ, where λ is the wavelength of sound at the stimulus frequency f and is equal to c/f. We can also express the right-hand term on the right side of the equation (4π x /λ) in terms of wave number (2kx) [e.g., Eq. (2b)].
The power reflectance ℜ describes the ratio of the power or energy in the reflected wave to the power or energy in the incident wave. Power reflectance (sometimes called energy reflectance) equals the square of the magnitude of the pressure reflectance: ℜ(f) =|R(f)|2. Note that the power reflectance is a real number and contains no phase information. Again, within an air-filled rigid tube of uniform or regularly varying cross-section, the terminating acoustic impedance and the tube’s cross-sectional area at the termination determine the power reflectance. This reflectance is constant along the tubes length, as long as losses are negligible.
While reflectance quantifies the amount of sound pressure or power reflected from the termination of the wave tube, absorbance defines the sound power that is absorbed by the termination. The absorbance is A(f) = 1 − ℜ(f). In some cases, it is useful to quantify the absorbance as a decibel value‡ where the absorbance level in decibels is 10 log10(A). This quantity has also been called transmittance (Allen et al. 2005).
MEASUREMENTS IN EAR CANALS
Modern techniques of estimating impedance and reflectance use measurements of sound pressure at a point x in the ear canal along with a completely characterized sound source to compute the impedance at the point of measured pressure Z(f, x). If power reflectance at the TM is the desired final measurement, Eq. (4) is used along with an estimate of the cross-sectional area of the ear canal at the measurement point (necessary to define Z0 at that point) to calculate the pressure reflectance R(f, x) at the measurement point and the power reflectance at the TM is assumed to equal the square of the magnitude of the pressure reflectance ℜ(f) =|R(f,x)|2. The accuracy of the measured ℜ(f), therefore, depends on the accuracy of the estimate of reflectance at the measurement point and the approximations that (1) the viscosity of air is negligible and (2) that the ear canal is a rigid-walled tube of constant or slowly and regularly varying cross-section. Figure 3 illustrates some schematic representations of the human ear canal that demonstrate that the ear canal is only approximated by a uniform tube, and that the assumption of a slowly and regularly varying cross-section is not accurate everywhere within the canal: near the eardrum termination, the cross-section of the canal doubles within a significant fraction (>10%) of the wavelength of a moderate- to high-frequency sound; and half-way down the length of the canal, there is a local minimum in the cross-sectional area (Farmer-Fedor & Rabbitt 2002).
One major advantage of using the reflectance is that as long as the approximations of slow, regular variations in cross-section and low-viscosity hold, the power reflectance at the point of measurement in the ear canal is a good approximation of the power reflectance at the TM. This is not the case with measurements of immittance in the ear canal, where the immittance magnitude and angle varies regularly with position in the canal. This variation occurs because the immittance measured at any point in the ear canal is influenced not only by the immittance at the TM (the middle ear input immittance) but also by the immittance of the air space between the TM and the measurement point (Moller 1960; Zwislocki 1962; Rabinowitz 1981). At frequencies where the length of the remaining canal lEC is less than a 10th of the sound wavelength, the air space in the canal acts like an acoustic compliance in parallel with the immittance of the TM such that
where the average cross-sectional area of the ear canal is S, lEC × S is the volume of the residual ear-canal space, and 1.4 × 105 Pa is the compressibility of air at one atmosphere of static pressure and 20°C. The parallel arrangement of the compliance and the TM immittance makes it convenient to use admittances in the calculation of Eq. (5). At frequencies where the length of the remaining canal is more than 0.1 λ, more complicated and less certain descriptions of the effect of the immittance of the intervening canal (Rabinowitz 1981; Lynch et al. 1994; Huang et al. 2000) can be used to estimate ZTM from the impedance measured at a location in the ear canal Z(f, lEC). Some of these techniques rely on measurements of the phase of the pressure reflectance to help determine the length of the canal between the measurement location and the TM (Keefe 2007; Voss et al. 2012).
One of the benefits of modern signal generation and averaging techniques is the ability to quickly measure the acoustic response of tubes and ears over a wide frequency range. These techniques rely on the use of wideband stimuli that are the sum of many sinusoidal waves of varied frequencies. Averaging the sound pressures produced by such wideband stimuli allows a fairly rapid computation of the immittance or reflectance over a wide frequency range (e.g., 0.2 to 6 kHz) with a relatively fine frequency resolution (e.g., a point every 100 Hz). The combination of fine resolution and wide bandwidth produces a more detailed look at the immittance and reflectance. Later articles in this volume will demonstrate that (1) finer frequency resolution can help detect variations in reflectance due to ossicular interruption and superior canal dehiscence (Nakajima et al. this issue, pp. 48S–53S) and (2) that the reflectance in the higher frequency range (1 to 4 kHz) can be a useful indicator of the presence of fluid-related conductive hearing loss in infants (Prieve et al. this issue, pp. 54S–59S).
COMPLICATIONS IN THE MEASUREMENT OF IMMITTANCE/REFLECTANCE
Variations in Canal Cross-Section
The wideband immittance and reflectance at the TM can be readily and accurately estimated from measurements made in a rigid ear canal of uniform cross-section. However, the ear-canal cross-section is not uniform along its length; the canal geometry (Fig. 3) includes local constrictions and expansions as well as bends and curves, and significant tapering of the canal near the TM (Johansen 1975; DiMaio & Tonndorf 1978; Stinson & Lawton 1989). Local constrictions and expansions in area are particularly critical for the generation of nonuniformities in the sound field (Farmer-Fedor & Rabbitt 2002). The significance of these deviations from uniformity increases with stimulus frequency.
One reflectance measurement technique that is less affected by the variations in canal cross-section along its length is based on finely spaced measurements of sound pressure within the ear canal (Stinson et al. 1982; Stinson 1990; Farmer-Fedor & Rabbitt 2002). Such reflectance estimates for the human TM can be readily calculated at frequencies as high as 20 kHz, and the fine spatial deviations in sound pressure observed in such studies are the best evidence for the significance of variations in canal cross-section.
Nonuniformities in the Sound Field Within a Given Cross-Section
The impedance and reflectance techniques that depend on the measurement of the sound pressure at a single point in the ear canal also assume that the pressure measured at that point describes the sound pressure throughout the canal cross-section (Fig. 2). This “uniform plane wave” assumption is common in describing sound flow in tubes, but it can break down in the presence of spatial variations in tubal cross-sections. Sudden or irregular variations in ear-canal area actually generate nonuniformities in sound pressure across a canal cross-section (Farmer-Fedor & Rabbitt 2002). Such nonuniform pressure waves generally only propagate short distances in the ear canal before dying out (this is true in the adult human ear canal at sound frequencies less than about 20 kHz), but they can be large enough to affect measurements at a single point. It is the effect of such nonuniformities on single-point sound pressure measurements within the ear canal that are the likely root of uncertainties in the reflectance and immittance at frequencies above 5 to 6 kHz in adult human ear canals (Farmer-Fedor & Rabbitt 2002).
Variation in the Rigidity of the Ear-Canal Walls
The walls of the adult human ear canal are not perfectly rigid. This is especially true of the cartilage and other soft tissues that compose the walls of the outer half of the ear canal. However, for the most part, the deviations from rigidity are small compared with the compliance of the air in the canal and can be ignored. The same cannot be said for newborn and infant ear canals in which the deviations from wall rigidity are even larger. Indeed, the increased compliance of the newborn and infant ear canal is known to play a significant role in tympanometric measurements where the compliance of the ear-canal walls can add to the measured compliance in both the static and pressurized conditions. Tympanometry depends on the response to both acoustic and static pressures, and the utility of such measurements in the infant and newborn population has been questioned (Holte et al. 1991; Prieve et al. this issue, pp. 54S–59S; Hunter et al. this issue, pp. 36S–42S). While the tissues of the canal wall are less stiff in infants and newborns, as sound frequency increases, it is the density of these tissues that determines the degree of their rigidity, and the effect of density increases as sound frequency increases. Experimental results suggest that at 1000 Hz and higher, the walls of the infant and newborn ear canals are significantly more rigid than the air in the canal. Acoustic measurements made at these higher frequencies are less affected by the ear canal and better reflect the acoustics and mechanics of the TM and middle ear (Prieve et al. this issue, pp. 54S–59S; Hunter et al. this issue, pp. 36S–42S).
The Effect of Viscosity
The viscosity of air is small and generally negligible; however, there are circumstances in which it can play a small role in measurements of immittance and reflectance. An area where its effect is minor is in the estimation of reflectance magnitude at the TM from measurements made in the ear canal. While viscous interactions between the air in the ear canal and the stationary walls of the canal will cause sound power to be absorbed as sound travels between the reflector and the measurement location, over dimensions relevant to measurements in humans, the losses involved are small (Huang et al. 2000; Voss et al. 2008; Keefe et al. 2010).
An area where the effect of viscosity is also small but more significant is in the calibration procedure used to describe the acoustic source in the first immittance measurement technique listed in Table 1. That method (Appendix A) uses measurements of sound pressure at the entrance of well-defined acoustic loads to characterize the output pressure and impedance of the sound source. The loads used are tubes of varied length. The fitting procedure used in the calculation works better if viscous looses (which are most significant at the resonant and antiresonant frequencies of the tube impedances) and “heat” losses (which are most significant at frequencies below 100 Hz) are included in the description of the calibration loads (Lynch et al. 1994).
Power Reflectance ℜ(f) Versus Pressure Reflectance R(f,x)
Descriptions of the power reflectance at the TM based on ear-canal measurements, by themselves, have advantages and disadvantages. The primary advantage of power reflectance (and the closely related absorbance A(f)) is that both are relatively independent of the measurement position within the ear canal, and they provide a simple frequency-dependent measure of the power transfer at the TM. A significant disadvantage is that power reflectance and absorbance depend only on the magnitude of the pressure reflectance and ignore significant information in the phase angle of the pressure reflectance and terminating immittance. Eqs. (2), (3), and (4) document that the terminating impedance of the ear canal ZT(f) and the associated pressure reflectance R(f,x) can be calculated from each other if one knows the diameter and the length of the intervening ear canal. However, the loss of the pressure reflectance phase information in the transformation to power reflectance makes it impossible to derive ZT(f) from ℜ(f) or A(f). This loss of information can be significant. For example, attempts to determine whether a low-power reflectance reading at low frequencies results from a leak of sound pressure out of the ear canal are greatly aided by inspection of the immittance phase (Voss et al. this issue, pp. 60S–64S; Huang et al. 2000; Voss et al. 2008; Keefe et al. 2010).
This article details the close theoretical relationship between acoustic immittance (impedance or admittance) and pressure and power reflectance. The relationship is strongest in cases of pressure reflectance at the termination of a uniform rigid ear-canal tube; the pressure reflectance in the canal depends only on the immittance at the TM, the canal cross-sectional area, and the distance between the TM and the measurements location within the canal. The power reflectance is simpler and only depends on the immittance at the TM and the cross-sectional area. This simplicity comes at a cost in that the power reflectance, by itself, does not completely describe the mechanics and acoustics at the TM.
A major advantage of power reflectance, or pressure reflectance magnitude, is that these measures are relatively independent of position within a canal with slowly and regularly varying cross-section. However, the pressure reflectance phase angle and the magnitude and angle of the immittance measured in the canal vary significantly at different locations within the ear canal. Indeed, measurements of immittance within the ear canal include effects of the canal space between the measurement location and the TM. The effect of this space on the measured immittance can be removed at frequencies where the space is small compared with a wavelength; such compensation is easiest and most accurate when the ear canal acts like a rigid uniform tube. Nonuniformities in the cross-section of the ear canal and other complications in ear-canal geometry can generate nonuniform sound pressure variations in the ear canal that complicate the measurement of reflectance and impedance with sound frequencies above 4 to 5 kHz when the single-point techniques is used. Reflectance and impedance estimated from measurements at multiple locations in the ear canal are less affected by such nonuniformities.
The ear-canal sound pressure, by itself, is not a sufficient descriptor of the sound drive to the ear canal and middle ear. A more complete description includes a measurement of sound volume velocity, where knowledge of both volume velocity and sound pressure is needed to define the acoustic immittance and calculate reflectance. To estimate the immittance and reflectance of the middle ear, from a measurement at a single location in the ear canal, the sound delivery system needs to be completely defined. A common characterization is to describe the source by its Norton or Thévenin equivalent circuit (Rabinowitz 1981; Allen 1986; Keefe et al. 1992; Lynch et al. 1994). The measurement of the sound pressure magnitude and angle (PL) at the entrance to the acoustic load (ZL) is illustrated in Figure A1 along with a Thévenin equivalent description of the sound source.
The source is described by an ideal sound pressure generator (PS) in series with a source impedance (ZS), which is in series with ZL. The impedances are complex and of the form Z = a + jb, where j is the imaginary number
. The volume velocity output from the source is the ratio of the pressure difference and the source impedance: (PS − PL) / ZS. The relation between PS and PL is also described by Eq. (A1):
Measurements of PL in two different loads (ZA and ZB) allow the calculation of the two equivalent circuit descriptors for the source (PS and ZS) (e.g., Rabinowitz 1981). Specifically, ZS can be directly estimated from the pressures measured at the entrance of the two loads (A and B):
and PS can then be determined from Eq. (A1) (Lynch et al. 1994).
The most common method to determine the equivalent source pressure and source impedance is by measurements in closed tubes of different lengths. The impedance at the entrance of a lossless tube with a closed ending is given in Eq. (A3).
Again, ρ0 is the density of the air, c is the speed of sound, S is the cross-section area, f the frequency, l is the length of the tube, and j is the imaginary number (Stevens 1998). The tubes are of a hard and smooth material consistent with the lossless approximation, and the cross-sectional area is chosen to be similar to the ear-canal cross-section area. We show the lossless equation for simplicity, the lossy versions are well described in the literature (e.g., Egolf 1977). The inclusion of losses becomes important at very low frequencies (∠100 Hz), where heat losses can be significant, and at the frequencies near the peaks and valleys in the cotangent function, where viscous losses dominate the impedance.
Examples of the calculated lossless impedance at the entrance of three tubes of different lengths (15, 25, and 40 mm) and a diameter of 8 mm are illustrated in Figure A2A; note the sudden and sharp phase transitions that occur at the frequencies with extremely sharp variations in magnitude. The calculated lossy impedances in the same three tubes are illustrated in Figure A2B; note the more rounded phase transitions and magnitude maxima and minima.
The parameters PS and ZS are often determined by measurements in more than two tubes, resulting in an overdetermined system. One way to estimate the source components in an overdetermined equation system is by using a least-squares approach. If the number of tubes for the determination of the source generator and source impedance is n, there are n measurements of PL. If Eq. (A1) is written in matrix form, the equation for the n measurements is
where A is the column array containing n rows of the product of the measured pressure in each load and the load impedance, B is the two-column array of n rows of the load impedance and −1 times the measured pressure, and C is a two-element column array with the elements PS and ZS. The estimated least-square values for the source,
where B−1 is the inverse of B and BT is the transpose of the matrix B.
APPENDIX A: DETERMINATION OF THE SOURCE CHARACTERISTICS APPENDIX B: LIST OF SYMBOLS
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* The frequency dependence of the volume velocity magnitude and phase angle results from the interaction of the controlled stimulus sound pressure and the mechanics and acoustics of the system that is opposing the sound pressure.
Complex numbers can be written as numbers that contain a real and an imaginary component or a magnitude and phase (polar coordinates). For example, if A= a+jb, where we use the electrical engineering convention that j =
, then the magnitude of the complex number is calculated via the Pythagorean theorem
, and the phase angle is the inverse tangent of the imaginary part of the complex number over the real part
. The mathematics of complex numbers are reviewed by Keefe and Feeney (2009). In this article, bold–italic symbols are used to represent complex values that have both a magnitude and an angle. Matrix symbols are bold.
‡ Absorbance is a measure of sound power and, therefore, we multiply the log10 of the absorbance by 10, not 20, to compute absorbance level.