A theory for stethoscope acoustics
Maximilian NUSSBAUMER(1) , Leyre TROYAS-MARTINEZ(2) , Anurag AGARWAL(3)(1) University of Cambridge, United Kingdom, mn406@cam.ac.uk(2) University of Cambridge, United Kingdom, lt397@alumni.cam.ac.uk(3) University of Cambridge, United Kingdom, anurag.agarwal@eng.cam.ac.uk

Abstract
Maximising the signal to noise ratio while considering ergonomics and aesthetics is the key design challenge formodern stethoscopes. In order to optimise the design, there is a need for a well-validated model for the transfer
function from a source within the chest to the output signal obtained. Most variants of the stethoscope are aircoupled sensors. Here we propose a new theory for the acoustics of this type of sensor, which takes into accountthe coupling between the sensor and the human chest. We have conducted rigorous experiments to characterise
the transfer function of the chest-stethoscope system and have investigated the effects of key design parameters.
Our data confirms traditional findings on the effects of bell geometry and diaphragm usage, but also highlights
the importance of the coupling between the sensor and the chest, and reveals features of the transfer functionthat are not captured by existing models. Our model employs a transmission matrix formulation and discretises
the system into lumped element components. It can be used to inform design choices for acoustic, electronicand dual-mode stethoscopes, opening up the possibility of an optimum design that maximises the signal to noise
ratio for a desired application.

Keywords: Stethoscope, Lumped-Element, Coupling

1

INTRODUCTION

It has long been known that biological sounds from the heart and lungs convey useful diagnostic information.
The initial stethoscope, invented just over 200 years ago by the French physician René Laennec, was a simple
cylinder in which the enclosed air coupled the chest to the ears of the physician. Over two hundred years later,
air-coupled sensors are still the predominant method of listening to and recording body sounds.
Despite their ubiquitous use, there is no simple or standardised way to compare stethoscopes or evaluate new
designs. This is largely due to the fact that an evaluation of the standalone acoustic properties of the stethoscope (as performed e.g. by Abella et al. [1]) is insufficient in assessing its performance when coupled to thehuman body. This is because the application of a stethoscope modifies the vibration of the chest surface.
Over the years many researchers have carried out experiments to compare sets of stethoscopes [1] [2]. Unfortunately is is difficult to compare results across studies because the properties of the coupling between thesensor and the chest are difficult to control and quantify. One approach has been to carry out experiments using
the sensors directly on human subjects [2] [3]. The issue here is that there is no way to accurately control
the input signal, or for other researchers to replicate the exact physical properties of the test subject. Anotherapproach is to use an experimental rig built to mimic the human chest (commonly referred to as a ‘phantom’[4] ). In this case it is possible to measure an input signal (although it is still fairly arbitrary as to how andwhere this should be defined). Kraman et al. [5] and Mansy et al. [4] have both carried out extensive researchon optimising the design of chest phantoms, however neither group’s testing system has been adopted by thewider research community.

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Nowak and Nowak [3] argue that any attempt to use a phantom in testing stethoscopes will fail to accurately
replicate the stethoscope’s use in a diagnostic situation. Their preferred approach is to collect data on a large
number of human subjects and use statistical methods to account for variations in input signal.
Beyond experimental comparisons between stethoscopes, it is of interest to develop a physical understanding of
how these systems work. Perhaps the most complete attempt at discussing the physics of air-coupled sensors
can be found in the chapter on “Sensing by Acoustic Biosignals” by Kaniusas [6]. Kaniusas makes extensive
use of early research by Ertel [7] and others to discuss the resonant properties of various sub-systems of thecoupled chest-sensor system. In doing so it is possible to make qualitative statements about how changes indesign parameters may affect the overall response. However, a combined model for the coupled system is notpresented.
The most complete attempts to model the coupled chest-sensor system come from Wodicka et al. [8] andSuzuki and Nakayama [9]. These models are for air-coupled sensors - essentially stethoscope chestpieces with amicrophone attached in place of the tubing used in acoustic stethoscopes. Wodicka et al. [8] present an acoustic
impedance analogy circuit for the combined system using a simple series model for the chest impedance, a twoport model for the air cavity and a series-impedance model for the microphone end load. Their model fulfils
its aim of matching the experimentally observed trend due to the variation of one design parameter. However,it does not capture all of the essential features of the coupled system and does not lead to a validated theory
for the acoustics of the chest-stethoscope system. Suzuki and Nakayama [9] present two (conflicting) models,
both of which are able to match different observed trends, but neither of which paint an accurate picture of thephysics of the system.
In this paper we present an approach to modelling the complete coupled system by carefully taking into accountthe coupling between the chest, the body of the sensor and the cavity. The model is validated by a systematic
set of experiments, leading to a new theory for the acoustics of the stethoscope.

2

METHODS - EXPERIMENTS

Figure 1. Schematic of ‘chest rig’ and detail of an air-coupled sensor.
A ‘chest phantom’ was designed and built in the style of those commonly used in the literature [5][4]. A
schematic for the setup is shown in Figure 1. A silicon rubber (EcoflexTM - density 1.07 kg/l and elastic modulus 69 kPa) was used to replicate the viscoelastic properties of chest tissue. The silicon rubber was cast ina cylindrical perspex tube with an inner diameter of 90 mm. Two rigid plates were embedded in the silicon4992

rubber. The first plate, at the base of the phantom, was connected to the shaker while the second plate wasembedded at an intermediate depth. A piezoelectric accelerometer (PCB M353B15) was attached to the second
plate to obtain a reference input signal for computing the air-coupled sensor response.
Stethoscope chest pieces were 3D-printed in Objet VeroWhitePlus RGD835 on an Objet24 3D printer to a variety of geometric specifications to allow the effect of various design parameters on the acoustic response to
be investigated. The sensor cavity was terminated by a 1/4 inch microphone (G.R.A.S. 46BL) in each case. A
range of masses were applied to the sensor to mimic differences in application pressure. The combined massof the sensor, microphone and load mass was measured in each case.
An NI cDAQ-9174 chassis with an NI 9234 input module and an NI 9263 output module were used for dataacquisition, with MATLAB used as the interface software. The input signal used was a series of chirps overthe desired frequency range (5 Hz to 2500 Hz). Signal to noise ratio was high, and averaging over many Fast
Fourier Transforms gave smooth (Cross-) Power Spectral Density estimates with minimal noise.

3

METHODS - MODEL

The highest frequency of interest in auscultation is debated but is typically below 3 kHz for most applications[7]. At these frequencies the wavelength of sound (in both the chest tissue and the air cavity) is much larger
than the spatial dimensions of the sensor (excluding tubing). Therefore, a lumped element formulation canbe used [8]. Two models are presented in this paper. The first is for an ‘air-coupled microphone’ which
is a microphone coupled to the chest via a small air cavity. The second model is for a traditional acoustic
stethoscope - again the chest is coupled to a small air cavity, but in this case the cavity is connected to tubing
and the ears of the clinician.
3.1 Air-Coupled Microphone Sensor
A model for the air-coupled microphone (see Figure 1) needs to encompass the constitutive properties of theunderlying material, the interaction between the rim of the sensor and the material, the motion of the material
surface within the cavity and the effect of the compliance of the air cavity.
Kaniusas [6] suggests that the chest surface within the bell acts as a ‘natural diaphragm’ vibrating in its firstmode, which is pre-stressed by the static load from the sensor. However, the assumption that surface tension
plays the dominant role in dictating the behaviour of the chest surface within the cavity is not necessarily valid.
It is, however, possible to reduce the relation between the rim motion and the interior free surface to a single
degree of freedom lumped element model, provided there is only one dominant mode in the frequency range ofinterest.
Figure 2 shows a simplified mechanical model for the chest-sensor system. In this model x1 represents aninternal displacement (defined at a chosen depth) while x2 represents the average displacement of the skinsurface within the bell. x2 is linked to x1 by a constitutive model for the chest medium. For the constitutive
model of the chest an inerter (see Smith [10]) is used in parallel with a standard Kelvin-Voigt model to accountfor kinetic energy of the chest - which will be largely associated with out-of-plane vibration of the medium(due to low compressibility).

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Figure 2. Air-coupled sensor schematic with displacements defined, and a lumped element model for chestsensor system. Dashpots in parallel with springs have been omitted from the schematic for ease of interpretation
but should be included in the analysis.
The displacement of the sensor (mass ms ) is represented by x3 . The skin surface moves with the sensor at therim, while the surface within the cavity is linked to this motion by the stiffness of the material. In the lumped
element model this is represented by the term km . The skin surface and the sensor are further related by thestiffness of the air-cavity (ka ). The small size of the cavity with respect to the wavelength of interest meansthat it can be assumed to be at uniform pressure. The mechanical stiffness of the cavity can thus be calculatedas:ka = ρc20 A2c / V(1)
where Ac is the contact area between the cavity and the ‘skin’, ρ is the density of air, c20 is the speed ofsound in the cavity and V is the volume of the cavity. Note that at the frequencies of interest it is a fairly
good assumption that the microphone can be taken as an infinite impedance, and a ‘perfect’ measurement of
the pressure within the chamber.
The discrete system in Figure 2 can be modelled in matrix form by assuming time-harmonic motion and excitation in the usual manner. This leads to:
Ax = f(2)
where A is a 3 × 3 matrix capturing the mass, stiffness and damping terms, x is a vector of the displacements
and f is a vector of the corresponding forces. For the freely vibrating system, the only term in f is the input
force f1 applied at x1 . The matrix equation can be solved to express x2 − x3 in terms of ẍ1 or f1 as required.
The pressure in the cavity is then given by:pcavity = ka (x2 − x3 ) / Ac(3)

where it should be noted that x2 is the displacement of the chest surface averaged over the cavity, such thatAc (x2 − x3 ) is a change in cavity volume.
3.2 Stethoscope
In order to extend the analysis to include acoustical elements, such as the tubing of the stethoscope, an acoustic
impedance electrical analogy circuit can be formulated (see e.g. Beranek and Mellow [11]).

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Figure 3. Mechanical impedance analogy for the chest sensor system, coupled to the acoustical impedance
analogy circuit for the cavity by a transformer with area ratio Ac . Note that v1 = ẋ1 .
The left side of Figure 3 shows a mechanical impedance analogy equivalent circuit of the lumped element
model in Figure 2. By using a transformer to represent the interface between the mechanical and acousticalsystems it is possible to then add further acoustical elements to the equivalent circuit as shown on the rightside of Figure 3.
The length of the tubing in an acoustic stethoscope means that it does not satisfy the condition for lumped
element analysis, and thus it must be included as a transmission line in the equivalent circuit. This is shownin Figure 4 which also has the mechanical terms brought to the acoustical side of the transformer by applying
the appropriate scaling factors. The entire circuit is now an acoustical impedance analogy circuit with pressure
as the ‘voltage’ and volume velocity as the ‘current’. The end load in Figure 4 is an open circuit. In the
impedance analogy an open circuit represents a rigid (infinite impedance) termination which is a reasonableapproximation given that the impedance of the microphone is very high compared to the other elements inthe system. To investigate instead the case where the end load is a human ear a standard ear model can be
employed - see e.g. Voss et al. [12].

Figure 4. Acoustical impedance analogy equivalent circuit for the complete chest-sensor system including tubing.
A transmission matrix formulation can be employed to link the pressure at the microphone to the pressure andvolume velocity terms at the input (see e.g. Beranek and Mellow [11]). For validation against the experimental
data obtained in this study the response of the microphone pressure with respect to x¨1 is computed using
MATLAB (R2018b).

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4

RESULTS

Figure 5 shows the experimental frequency response function (FRF) for the pressure in the cavity with respect tothe acceleration of the embedded base plate within the phantom rig. Several interesting features can be observedin the experimental FRF. There is a low frequency resonance at around 40 Hz, followed by a -40 dB/decade rolloff. There is then an anti-resonance at around 400 Hz, followed by a jump in magnitude and a subsequent
further region of -40 dB/decade roll off. The jump in response after the anti-resonance is characteristic of amass shedding effect. These features are of course specific to the rig-sensor combination used, by setting thecomponent values in the model to match this scenario the modelling approach can be validated.

Figure 5. Frequency response function of microphone pressure with respect to base acceleration for an aircoupled microphone with a contact area of 200 mm2 , a cavity volume of 1200 mm2 and a combined sensor massof 215 g. Spectrum averaged over 1240 FFTs with a Hann window, 50% overlap and a frequency resolution of
1.6 Hz. Magnitude is expressed in decibels relative to 1 Pa/ms−2 .
Figure 6 shows a simulation of the FRF for the same sensor geometry and load mass which was computed using
the model described in this paper. Model parameters were estimated for the phantom rig. The fit between theexperimental data and the model is good, however, there are discrepancies in the exact shape of the response.
These discrepancies arise from two main sources: the estimation of parameter values for the constitutive modelof the phantom rig was fairly crude, and furthermore, a lumped element model can never fully describe thebehaviour of a continuous system. At higher frequencies (beyond those shown here) higher order modes of thephysical system come into play and significantly complicate the response. As the range of frequencies usedin auscultation is fairly low this does not impact the validity of the model. In the frequency range shown
the model captures the low frequency resonance, the gradual drop-off and the anti-resonance observed in theexperimental response. Further to capturing features of the response, the model also accurately predicts theeffect of changing cavity volume and contact area, thus providing further validation.

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Figure 6. Model Simulation of the frequency response function for cavity pressure (pcavity ) with respect to baseacceleration (x¨1 ) for an air-coupled microphone with the same dimensions and loading as in Figure 5.
The model allows a range of parametric studies to be carried out far more efficiently than by experiments,
allowing a wide design space to be explored. It is found that increasing cavity volume lowers the amplitude
response, while increasing cavity area increases the response amplitude. Both changes also shift the resonancesof the system. Using the acoustical impedance analogy equivalent circuit it is also possible to study the effectof tubing. In general tubing significantly decreases the baseline response (especially at low frequencies) andadds several standing-wave peaks within the frequency range of interest for auscultation. The result is that thefrequency profile of lung sounds heard by a physician using an acoustic stethoscope can be very different tothat measured at the chest.

5

CONCLUSIONS

We have introduced a new approach to modelling the stethoscope motivated by a careful consideration of themechanical and acoustical coupling between the chest and the sensor. We have validated our model against asystematic set of experiments carried out on a ‘phantom-rig’. The model was able to capture the key featuresof the frequency response, but further work is needed to adapt the model to the human chest and re-validate itin this setting.
Plotting the response of cavity pressure to the acceleration at a defined depth within the phantom rig is usefulfor comparison with experimental results, however, as there is no single origin for biological chest sounds, thefrequency response to excitation at a single fixed point within the chest is not ultimately the quantity of interest.
It is more interesting to consider how the stethoscope responds to vibrations that have reached the chest surface.
The issue here is that the loading of the stethoscope affects the amplitude and properties of the vibrations at
the surface. As such, a general reference that is independent of loading is required. There are two options
here. One is to use the unloaded ‘free-surface’ vibration, and the other (suggested by Suzuki [9]) is to use the‘stop-surface’ pressure. Neither of these can be easily measured experimentally. However, when using a validated acoustical impedance analogy equivalent circuit it is straightforward to compute the sensor response as afunction of one of these references, thus opening up the possibility of a more objective method of classifying
sensor performance.
While previous work in the literature has already provided simple models for many of the trends observed
between design parameters and acoustic response, this paper goes a step further by introducing a methodology
for modelling the complete system, which is not only able to capture these trends independently, but can also
provide quantitative predictions and be used to compare designs with several design parameters varied. This
allows the model to be used for design optimisation which is an area for future work. In order to optimise thedesign of an acoustic stethoscope it is further necessary to consider the characteristics of the biological sounds
of interest, external noise sources and the selective hearing of the human user.

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ACKNOWLEDGEMENTS
The authors would like to thank Dr Ole Nielsen for his advice on lumped element modelling of mechanical andacousitcal systems and for key insights in the modelling approach, Professor Jim Woodhouse for his advice on
vibration analysis, Mr John Hazelwood for assistance in the laboratory, Mr Anthony Luckett for CAD work andthe 3D printing of sensors, and the Engineering and Physical Sciences Research Council for funding this work.

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