Home  Basic Multifrequency Tympanometry
In tympanometry the mobility of the tympanic membrane is measured while the membrane is exposed to a (sinusoidal) tone of frequency f.
In the ear, the tympanic membrane is mechanically coupled with the middle ear ossicles to the oval window the interface between middle and inner ear. It is this entire system (membrane, middle ear, oval window) that is forced into oscillation. The oscillation is detected by a microphone. (A more detailed description is given in "Tympanometry in just seconds".)
A linear theory used to evaluate the signal from the microphone is presented here. The response of a linear system when driven by a periodic oscillation can be expressed in terms of the resistance with which the system responds to the excitation (called "impedance") or in terms of the ease with which it is set into motion (called "admittance"). Both expressions of the response are presented here next to each other in a table.
An excellent summary of the linear theory, and a review of its practical
applications and the reliability of multifrequency tympanometry in diagnosing
middle ear diseases is:
Robert H. Margolis, Lisa L. Hunter,
Acoustic Immittance Measurements, Chapter 17 of Audiology:
Diagnosis, by Ross J. Roeser, Michael Valente, Holly HosfordDunn (eds),
Thieme 2000.
Acoustic impedance 
Acoustic admittance 
Let V be a fast (adiabatic, i.e. heat nondissipating) change of a volume V of air and P the corresponding pressure change.  . 
Definition 3 (compressibility )
The adiabatic compressibility of air is defined as 
. 
Theorem 3
The compressibility can be expressed in terms of the density of the air and the speed of sound c in air: Proof can be found in textbooks of physics.

. 
Let volume V be approximated by a cylinder with base A (and a height h).  . 
Definition 4 (cross section A of air volume)
Then volume change V can be expressed as change z of the cylinder height 
. 
The corresponding pressure change P can be written in terms of the force F on A  . 
Definition 5 (Hooke's constant D for air, acoustic stiffness
K_{a})
Combining (11)  (14) the force F resulting from the volume change V can be written similarly as Hooke's law with the abbreviation

. 
Assumption 4 (friction R)
Let the volume V of air dissipate energy similarly as the mass m on a spring in (2):

. 
Assumption (rigid body of oscillating masses)
The periodic oscillation of the air in the ear canal wiggles at the tympanic membrane, the middle ear ossicles etc. This has been ignored in the system dealt with until now. Let us assume that all those masses comprise a rigid entity m_{eff} that oscillates as a whole and in phase with the air in the ear canal. In other words, the masses of which m_{eff} is composed do not oscillate separately and out of phase with the air. Definition 7 (oscillating mass m)
m = V + m_{eff} Thus, the force to overcome the inertia of m is 
. 
Theorem 4 (equation of motion)
As in the case of the mechanical oscillator, the resulting movement of the air particles in volume V can be calculated from the force balance F = F_{m} + F_{R} + F_{H}, where

. 
For a periodic pressure being applied by a loudspeaker to the ear canal air and mass m_{eff} (assumption 3), the system's response is analogous to (8) (note that again F = A p):  . 
Definition 6 (volume velocity U)
Volume velocity U is defined as the volume that flows through the air canal cross section per unit time:

. 
It is customary to replace v_{f} in (19)
with iU/A. Deviding both sides of (19) by A^{2}
we get the following expression chaaracterizing the system response
Definitions 7 (acoustic resistance R_{a}, acoustic inertance M) (1) To simplify the form of the equations, we will introduce the acoustic resistance . (2) Likewise, Kinsler and Frey (1962, p. 190 (Eq. 8.14)) introduced the definition of acoustic inertance .
Using (16), the last term on the right hand side can be simplified: 
. 
Theorem 5 (system response)
The final expression for the system response is (20). In analogy with (10) the ratio (22) is called acoustic impedance Z_{a} 
Theorem 5a (alternative system response)
Alternatively, the system response can be characterized by the inverse of ratio (22) 
Definition 8 (acoustic impedance Z_{a})
The impedance Z_{a} given in (22) has a real and an imaginary part (see (20)).
With definitions 7.1 (acoustic resistance R_{a}) and 7.2 (acoustic inertance M) and definition 5 (acoustic stiffness K_{a}) the acoustic impedance can be written in analogy with definition 2, and the following names are given: resistance R_{a}

Definition 8' (acoustic admittance
Y_{a}, eqs. (23'))
compliant or stiffness susceptance

. 
Fig. 1: G_{a}R_{a} and B_{a}R_{a} as a funtion of X_{a}/R_{a}. At X_{a}/R_{a} = 1 G_{a} R_{a} and B_{a} R_{a} have the same size. At resonance G_{a}R_{a} = 1 and B_{a}R_{a} = 0. Fig. 2: Oscillation plotted in {G_{a}R_{a},
B_{a}R_{a}} plane lies on a circle with radius 1/2, because
G_{a}R_{a}^{2}+ B_{a}R_{a}^{2}
= (1/2)^{2} for all X_{a}/R_{a}. The angle
will be used to calculate R_{a} from tympanographically meaasured
G_{a} and B_{a}.

.  Fit of R_{a} to multifrequency tympanogram G_{a}(f)
and B_{a}(f)
Definition of (see Fig. 2) From (23), (24) follows
Proof:
Multifrequency tympanogramm gives G_{a}(f) and B_{a}(f) . Thus (26) is a function of the immission frequency f. (26) can be solved for as a function of f. With (25) R_{a} can be fitted to the tympanogram 
Definition 9 (resonance frequency _{r})
Let the frequency at which the reactance X_{a} and susceptance
B_{a} vanish be called resonance frequency _{r}
of the system
.
Solving for _{r}
At resonance _{r} conductance and resistance are simple reciprocals of each other:
Data:
(d1) At f = 226 Hz (d2) Plugging (d1) and (d2) into the definition of X_{ca} above (d3) A volume V = 1 cm^{3} of air has a compliant reactance 
At high positive or negative ear canal
pressures the tympanic membrane is almost fixed and the middle ear is nearly
motionless (m_{eff} approx. 0, Ra approx. 0) the admittance Y_{a}
= B_{a} = B_{ca} (the latter because X_{ma} <<
X_{ca}) with
A volume V = 1 cm^{3} of air has a compliant susceptance 
Use definitions 8 (23) of X_{ma} and X_{ca}:
In detail (see Fig. 3):
Fig. 3: Extrapolation of X_{ca}(f) and
X_{m}(f) yields V and m/A^{2}.
Another possibility, using (28), see
Fig. 4:
Ear canal cross section A together with oscillating mass
m can be fitted to the resonance frequency f_{r}.
Fig. 4: Plot of contours of constant resonance
frequency f_{r} as a funtion of the ear canal radius r and the
oscillating effective mass m_{eff}.
Example marked by arrows: for r = 0.37 cm and m_{eff}
= 0.002 g the resonance frequency is f_{r} = 1140 Hz.
As the contour plot Fig. 4 shows, a possible choice for f_{r} = 1140 Hz is:
= 0.00129
g/cm^{3}
V = 1.36 cm^{3}
m = V
+ m_{eff} = (0.0018 + 0.002) g = 0.0038 g.
Data used in Example:
(31) V_{0} = 1.36 cm^{3}, TW = 40 daPa = 400 Pa (with 1daPa = 10 Pa)
(31a) m_{eff} = 0.002 g, Ra = 1000 ohm, r
= 0.37 cm, = 0.00129
g/cm^{3}.
Fig. 5: Plot of the two components of the reactance
as functions of immission frequency f. Heavy curves represent mass
reactances, light curves compliant reactances.
Curve parameter is the ear canal pressure p.
Curves are plotted for p = 0 and p = 400 daPa. (for implementation of p
see (29) and (30)).

. 
Fig. 6: Plot of total reactance as a function of immission frequency f for fixed ear canal pressures p = 0 and p = 400 daPa. At resonance X_{a} = 0. 
. 
Fig. 7: Plot of total reactance as a function of ear canal pressure p. Curve parameter is the immission frequency f. Curves are plotted for f =113 Hz and the following 6 octaves above 113 Hz. 
. 
Fig. 8: G_{a}R_{a} as a function
of both ear canal pressure p (daPa) and immission frequency f (log f is
used, with f in Hz).
3Dplot: p is plotted along the xaxis (range: 400 daPa
<p < 400 daPa), log f is plotted along the yaxis (range: 2 <
log f < 3.7).
2Dplot: G_{a}R_{a}(p) is plotted with
f as parameter, i.e for f fixed at 226 Hz and the 5 following octaves above
226 Hz.
Fig. 9: Detail of Fig. 8 near resonance at zero
ear canal pressure, p = 0.
Fig. 10: B_{a}R_{a} as a
function of both ear canal pressure p (daPa) and immission frequency f
(log f is used, with f in Hz).
Left: 3Dplot, p is plotted along the xaxis (range:
400 daPa < p < 400 daPa), log f is plotted along the yaxis (range:
2 < log f < 3.7).
Right: 2Dplot B_{a}R_{a}(p) with f as
parameter.
Fig. 11: Detail of Fig. 10 near equivalence frequency at zero ear canal pressure p = 0.
Fig. 12: Y_{a}R_{a} as a
function of both ear canal pressure p (daPa) and immission frequency f
(log f is used, with f in Hz).
Left: 3Dplot, p is plotted along the xaxis (range:
400 daPa < p < 400 daPa), log f is plotted along the yaxis (range:
2 < log f < 3.7).
Right: 2Dplot Y_{a}R_{a}(p) with f as
parameter.
Fig. 13: Detail of Fig. 12 near resonance
at zero ear canal pressure p = 0.
Fig. 14: Graphical explanation of shapes of curves
in Figs. 8  11:
Lower plots: X_{a}/R_{a} as functions
of f for fixed p = 0 and p = 400 daPa (see Fig. 6).
Upper plots: G_{a}R_{a} and B_{a}R_{a}
as functions of X_{a}/R_{a} (see Fig. 1).
To obtain a value G_{a}R_{a} for a given
immission frequency f
(1) choose f and read X_{a}/R_{a} from
lower plot (follow line 1 in direction of arrow),
(2) then read G_{a}R_{a} for X_{a}/R_{a}
(follow line 2 in direction of arrow).
m_{eff1} := 0.1 g; R_{1} := 1000 ohm;
m_{eff2} := 0.01 g; R_{2} := 300 ohm;
r_{1} := 0.4 cm; r_{2} := 0.37 cm;
V_{1} := 0.9 cm^{3}; V_{2} := 0.2 cm^{3};
_{1} :=
0.001 g/cm3; _{2}
:= 0.00129 g/cm3;
TW = 40 daPa.
Fig. 16: Conductance G_{a} and susceptance
B_{a }plotted as functions of immission frequency f. Because of
(36')
G_{a} = G_{a1}+ G_{a2}, and B_{a} = B_{a1}
+ B_{a2}.
Fig. 17: Resistance R of the composite system as
a function of immission frequency f. By (36') 1/R
= 1/R_{1} + 1/R_{2}.
Fig. 18: Oscillation of composite system in {G_{a}, B_{a}} plane. The point {G_{a},(f) B_{a}(f)} runs on the curve in the direction indicated by the arrows, when f runs from 100 Hz to 4111 Hz. The circle has been drawn to emphasize noncircular form of curve.
Fig. 19: Oscillation of composite system plotted
in {G_{a}R_{a}, B_{a}R_{a}} plane lies
on circle with radius 1/2. The reason for this is the linearity of the
composite system: The oscillation of each subsystem lies on this circle
(see Fig. 2), thus the linear composition of
these oscillations lies on that circle, too. The curve drawn by hand indicates
how the point {GaRa, BaRa} runs on the circle when f runs from 110 Hz (arrow
near {0.6, 0.4}) to 4060 Hz (arrow ending near {0.1, 0.1}).
Fig. 22 results when these B_{a} are plotted vs. G_{a}.
The data are then analysed with a linear model. This means that the deviation of the curve in Fig. 22 from a circle will be interpreted as resulting from a frequency dependent resistance R_{a}(f) according to (27). This may or may not be justified. It is simply a method of condensing the measured data into a set of equations (the ones developed in this paper) and corresponding parameters (necessary to evaluate the equations).
After calculating R_{a}(f) with (27) (Fig.
23), B_{a}R_{a} is plotted vs. G_{a}R_{a},
resulting in the circle presented in Fig. 24. The curve in Fig. 24 drawn
by hand indicates how the point {G_{a}R_{a}, B_{a}R_{a}}
runs first clockwise and finally counterclockwise on the circle when f
runs between 230 Hz (arrow at beginning of clockwise part) and 1930 Hz
(arrow at end of counterclockwise part). The circle crosses the abscissa
(G_{a}R_{a}axis) at f = 1350 Hz (by definition
the resonance f_{r }of the system).
Fig. 20: Conductance G_{a} as a function
of the immission frequency f. The ear canal pressure is p =  250 daPa.
Data from Margolis and Hunter.
Fig. 21: Susceptance B_{a} as a function
of the immission frequency f. The ear canal pressure is  250 daPa. Resonance
frequency f_{r} is defined here as the frequency at which B_{a}
= 0 (f_{r }= 1350 Hz, dashed line). Data from Margolis and Hunter.
Fig. 22: B_{a}(f) plotted vs. G_{a}(f).
B_{a}(f) and G_{a}(f) as presented in Fig. 20, 21. The
curve starts at f_{i} = 226 Hz and ends at f_{i}
= 2000 Hz.
Fig. 23: Resistance extracted from oscillation
presented in Fig. 22 with method given by eq. (27).
Immission frequencies f_{i} used by multifrequency tympanometer
are marked as dots in lower part of graph. They start at f_{i}
= 226 Hz and end at f_{i} = 2000 Hz. Dashed line marks resonance
frequency f_{r} = 1350 Hz. Sampled frequncies f_{i} miss
resonance f_{r}.
Fig. 24: B_{a}(f)R_{a}(f) plotted vs. G_{a}(f)R_{a}(f). Data from Margolis and Hunter.