Determination of frequency dependent ultrasound absorption bymeans of radiation force based power measurements
Tina A. FUHRMANN(1) , Konrad MEHLE(1) , David WALTSCHEW(1) , Klaus-V. JENDERKA(1)(1)

University of Applied Sciences Merseburg, Department of Engineering and Natural Sciences, Germany,tina.fuhrmann@hs-merseburg.de

Abstract
Frequency dependent acoustic absorption of propagating ultrasound waves is a key quantity in acoustic
material characterization. The standard approaches to determine the amplitude absorption coefficient
either analyze the amplitude decay of subsequent rear surface reflections of a sample with known thickness (materials with low acoustic absorption) or the insertion loss of samples with different thicknesses(materials with higher acoustic absorption). Our new approach uses the total temporal averaged ultrasonic power determined by means of a Radiation Force Balance, by comparing the power values of
excited bursts (number of cycles > 100) with and without the sample. The required information about
the thickness and speed of sound of the sample, as well as consequently the amplitude reflection andtransmission coefficients, were determined in advance within the same measurement setup. The effectsof half-wave layers on the reflection and transmission coefficients are incorporated in the analyzing
routines. Additionally, the frequencies of the resulting peaks of transmitted power can be used forrefinement of the samples thickness or speed-of-sound measurements. Determining ultrasound absorption by means of Radiation Force based power measurements takes advantage of the low uncertaintyof the method, but requires an extended time frame for the measurements.
Keywords: Absorption, Attenuation, Speed of Sound, Radiation Force Balance, Acoustic Material
Properties

1

INTRODUCTION

The knowledge of acoustic material properties is important to develop new and improve existing ultrasoundtechnologies, for instance clamp-on sensors in the medical field. They are used in diagnostics e.g. forblood flow measurements or bubble detection. The sensors are directly connected to the blood-carrying
tubes and therefore the measurement accuracy, as well as the feasibility of quantitative measurements
in general, strongly depends on the knowledge of the tube properties and the correct configuration of
the sensors on the basis of these properties. Hence the tubes properties must be measured accurately inadvance.
Material quantities of interest usually are the speed of sound, impedance, and attenuation or absorption,
as well as the thickness and density. A compromise has to be made between accuracy, and time and effort
spent on the measurement. To access the attenuation, two methods are common [1]: The first is the single
sample through transmission method where the amplitude decay of subsequent rear surface reflections isanalyzed. The sample’s thickness must be known and its absorption must be smaller to allow multiple
reflections. The second is the double sample through transmission method which uses the insertion loss ofat least two samples with different thicknesses and is used for materials with higher acoustic absorption.
To obtain small uncertainties, big efforts have to be made concerning the measurement setup (placement
and alignment of equipment), the experimental execution and to meet the narrow conditions under which
all the assumptions are valid. Those conditions are the use of very short pulses, a narrow sound field
geometry, and the exact determination of the acoustic impedance to calculate the transmission and985

reflection coefficients. It also requires the exact determination of the sample’s speed of sound and density.
In our approach all the above mentioned quantities - absorption, thickness, density, speed of sound,and impedance of the sample - are measured very accurately and combined in only one measurement
setup. For the absorption measurement a radiation force balance (RFB) is used. Very short pulses
are replaced by very long ones, which results in a higher ultrasonic power available for measurement andtherefore more accurate readings. It also implies the usage of a total transmission and reflection coefficient,
respectively, that takes all transmitted and reflected echoes into account. In ultrasound metrology thelowest uncertainties (about 3 %) can be achieved for measurement of the total temporal averaged ultrasonicpower.
Our samples are made of PMMA (polymethacrylate) because it is often used as buffer rod, for ultrasoundand radiation therapy phantoms, as well as for comparison measurements. Its ultrasound properties canbe found in literature [2, 3, 4].

2

THEORY

2.1 Density Measurement
For the density measurement the buoyancy of the sample in water is used and therefore requires only
a balance. With the measured mass of the sample in air mS and in water mS,w as well as the known
density of water at a given temperature %w from [5] the density %S of the sample can be calculated:%s =m S · %w
mS − mS,w(1)

This results from the equilibrium of buoyancy force from the sample FS,b = %w VS g and the weight force
Fw,g = mW g, with g being the constant of gravitational acceleration. Then immediately
VS =

mS
mw
mS − mS,w
=
=
%S
%w
%w

(2)

follows, which finally results in equation 1.
2.2 Speed of sound and Thickness Measurement
To measure the speed of sound and thickness of the sample a pulse-echo method is used [6].

Figure 1. Measurement of thickness and speed of sound of the sample. Here dS is the thickness of thesample, z1 and z2 the ultrasound path lengths in front and behind the sample, t1 and t2 the runtimesof ultrasound to the front and back wall of the sample, and tS and tw the runtimes to the ultrasoundreflector with and without the sample in the ultrasound path, respectively.
The experimental setup and runtimes of interest are shown in figure 1. They are defined by
z1
dS
z2
z1
(3)
+2·
+2·
(5)
tw = 2 ·
cw
cw
cw
cw
z1
dS
z1
dS
z2
t2 = 2 ·+2·(4)tS = 2 ·
+2·
+2·
(6)
cw
cS
cw
cS
cw
and for the calculations the temperature dependent speed of sound from water published in [7] is used.
The runtimes are given through measurement and the unknown variables are dS and cS as well as z1 andz2 :t1 = 2 ·
cS = cw ·986

tw − tS+1
t2 − t1(7)

dS = cS ·t2 − t12(8)

2.3 Absorption Measurement
The absorption measurement is done in transmitting mode. A large absorbing target, which is connected
to a balance, absorbs almost the whole acoustic power that reaches the target. The front face of theultrasound transducer, the sample’s, and the absorber’s surfaces are aligned parallel.
With our measurement setup (section 3) we measure, strictly speaking, the attenuation within the probe.
However, the following conditions allow the assumption that attenuation is only due to absorption andtherefore the terms can be used equivalently in this paper:• The front and back walls of our samples are parallel to each other and perpendicular to the centerline of ultrasound propagation. Therefore the incident and emerging ultrasound angles are 90◦ andno energy is reflected or scattered out of the absorbing target due to geometrical conditions of thesample.
• The material examined in this investigation (PMMA) is in a good approximation homogeneous andfree from scatterers. Therefore no ultrasound energy is scattered out of the target area due toscattering within the sample.

Figure 2. The sample is shown in yellow with its thickness dS . For the theoretical derivation of thereflection and transmission coefficients RS and TS with extended (infinite) ultrasound pulses the indices0 and 1 are used for water and the sample, respectively.
To derive the (complex) amplitude transmission coefficient we start by taking into account the firsttransmission through the sample [8]. The ultrasound wave enters the sample (transmission coefficientT01 ), passes through it and emerges on the opposite side (T10 ). Considering the advance of the phase
eikdS and the absorption e−αdS , the total transmission coefficient is given by
TS = T01 T10 · eikS dS e−αdS(9)If k has an imaginary part, then the factor eikS dS also adds attenuation to the transmission coefficient.
Next, also the reflection at the lower boundary R10 (figure 2), the inner passage through the sample, thereflection at the upper boundary (R10 ) and another passage through the sample are included:
TS = T01 T10 · eikS dS e−αdS + T01 R10 R10 T10 · e3ikS dS e−3αdS(10)Note that possible phase jumps are represented by the sign of the reflection coefficient. When considering
an infinite amount of reflections within the probes, we finally we come to the formula
TS = T01 T10 · eikS dS e−αdS+ T01 T10 · eikS dS e−αdS · R10 R10 · e2ikS dS e−2αdS+ T01 T10 · eikS dS e−αdS · R10 R10 · e2ikS dS e−2αdS

2

+ ...
= T01 T10 · eikS dS e−αdS · lim

m→∞

m
X

2
R10
· e2ikS dS e−2αdSl,l ∈ N0

(11)

l=0

m
P
l
1
= 1−x
for all
This is an infinite series in the form Sl = lim
xl which is given by Sl = lim 1−x
m→∞ l=0l→∞ 1−x22ikS dS −2αdS|x| < 1. The term R10 · eeis always smaller than 1 because each factor is smaller than 1
and |e2ikS dS | ≤ 1, respectively. Therefore the transmission and, analogously, the reflection coefficients are987

given by
TS =T01 T10 · eikS dS e−αdS
2 · e2ikS dS e−2αdS
1 − R10

RS = R01 +(12)

T01 R10 T10 · e2ikS dS e−2αdS
2 · e2ikS dS e−2αdS
1 − R10(13)

With the known densities and speeds of sound from water (%w [5],cw [7]) and the sample (%S ,cS ; see sections
2.1 and 2.2) the transmission and reflection coefficients for the water-sample border (T01 , R01 ) can becalculated from the acoustic impedances (Z = % c):
T01 =

2ZS
ZS + Zw

(14)

R01 =

ZS − Zw
ZS + Zw(15)

With the radiation force balance, the ultrasonic power P received at an absorbing target is determinedthrough measurement of the mass “jumps” P = cW · g · ∆m when turning the ultrasound transducer on.
Therefore the absorption can be obtained by numerically fitting the equationsrPScw (ϑ) · ∆mS
T01 T10 · eikS dS e−αdS(16)==2 · e2ikS dS e−2αdSPecw (ϑ) · ∆me
1 − R10with PS and Pe being the measured ultrasound powers at the radiation force balance with and without
the sample in the ultrasound path, cW (ϑ) the temperature dependent speed of sound of water, and ∆mSand ∆me the mass differences when turning on the ultrasound transducer with and without the sample
in the ultrasound path. For α we use the general formula
α = a · fb

(17)

with a and b being constants and f being the frequency.
2.4 Half-wave layers
In the previous section the transmission and reflection coefficients for infinite ultrasound waves were
derived. Since the wave has an infinite length it is important to add the phases correctly. This is doneby the term e2ikS dS and immediately leads to effects that provide further information.
S
First it predicts half-wave layers at sample thicknesses dS = mλ= mπ2kS (with m = 1, 2, ...). From theequations 12 and 13 it can be easily seen that (by neglecting the absorption) the transmission coefficient
TS = 1 and RS = 0 (using the relations eikS dS = cos(kS dS ) + i sin(kS dS ), T01 = 1 + R01 and R01 = R10 ).
Taking the absorption into account it still holds that the transmission coefficient has a maximum at thoseS
frequencies where the sample thickness corresponds to half a wavelength (f = mc
2dS ) and the reflection
coefficient a minimum.
It secondly means, that an oscillation of the ultrasonic power is present in the spectrum that is only
dependent on the speed of sound of the sample and its thickness. It can therefore be used to refine oneof those measurements, whichever is less accurate. A detailed analysis and proof of concept are beyond
the scope of this paper and will be included in detail in future work.

3

EXPERIMENTAL SETUP, CALCULATION, AND MATERIAL

In figure 3 the experimental setup is shown: measurement of absorption/attenuation - reflector open,
measurement of speed of sound and sample thickness - reflector closed.
Absorption measurement
To determine the ultrasonic power a radiation force balance (RFB) method as described in [9] is used. The
setup is equal to the RFB setup in [10] used by PTB and has an uncertainty of about 3% at 5 MHz. The
ultrasound transducer (Olympus, diameter 5 MHz, 12.7 mm (0.5 inch), bandwidth 3.14 MHz) is oriented
upward and run by a generator (Tektronix AFG3101) and amplifier (121CR rf power labs, 50 dB) in
transmit-mode sending 2 µs bursts at a single frequency. The target (HAM A, Precision Acoustics) is
placed 10 cm above the transducer covering the whole cross section of the ultrasonic field and absorbs
99% of the received ultrasound energy. The mass difference is measured with a balance (AX105 Delta
Range from Mettler-Toledo, ∆m = 0.1 mg). The experiments are run and analyzed with MATLAB 2019b.
Temperature was controlled with the thermometer GMH 3710 from Greisinger electronic.

988

Figure 3. Measurement of attenuation. 1..absorbing target, 2..sample, 3..laterally displaceable reflector,
4..sample holder, 5..transducer
Absorption calculation
A numerical fit of equation 16, using equations 12 to 17, was used to calculate the coefficients a and b of
the absorption α = a · f b . Using MATLAB 2019b, the sum of the squared residuals was minimized. The
increments of a and b where 0.001 and 0.01, respectively. Since a frequency exponent very close to 1 wasfound, the fit model for the hydrophone measurements was changed to α = c + a · f .
Speed of sound and thickness measurement
A pulser-receiver (Panametrics Model 5800) runs the US-transducer and emits bursts of three cycles.
The received signal is send to an oscilloscope (Keysight InfiniiVision DSO-X 3034T) and read out by
MATLAB. The reflector can be moved along the x-direction into the sound field.
Hydrophone measurement
The US transducer is placed downwards in a degassed waterbath (temperature and oxygen measurementwith Hach HQ 40d). The hydrophone (ONDA HGL-0400 S/N 1380, effective diameter 0.4 mm) is placedin the distance of 15 cm from the transducer. The transducer is run with a function generator (Agilent 33
220 A) and the hydrophone signal is acquired with an oscilloscope (PicoScope PS5244A) and processedby the ONDA software Sonic Software for AIMS III, Ver. 5.2.4.0.
Samples
For this paper different PMMA discs of thicknesses 2 mm, 3 mm, 5 mm, 7.5 mm, and 10 mm were used.
PMMA is a suitable plastic for comparative measurements and for verifying the new measurement configuration.

4

RESULTS

4.1 PMMA data
At least three independent measurements were made at room temperature for each mentioned property.
The results for density, sample thickness and speed of sound measurements of the PMMA discs are shownin table 1. In addition, the calculated impedances, uncertainties, and mean values are presented.
4.2 Power Measurement and Absorption
The ultrasonic power was measured at room temperature once per sample with the radiation force balancemethod as a proof of concept. The results are shown in figure 4. More experiments, as well as adetailed uncertainty consideration, will be subject to a following paper. For comparison reasons, theultrasound peak-to-peak pressure was measured three times per sample with a hydrophone and is also989

Table 1. The thicknesses were measured with a caliper († ) and the ultrasound method (‡ ). Density,
thickness2 , and speed of sound of the PMMA discs were calculated with equations 1, 8, and 7. In thelast row the average values x plus-minus the standard deviations σ are shown.
samplePMMA 2PMMA 3
PMMA 5PMMA 7.5
PMMA 10x
uncertainty

thickness†
(mm)
2.00
3.00
5.02
7.51
10.01

thickness‡
(mm)
1.996
2.987
5.009
7.513
10.00

mass
(kg)
6.674
9.970
16.78
25.11
33.44

≤ 0.1%≤ 0.2 %

0.001 gdensity(kg/m3 )
1185
1186
1186
1186
1186
1186
1 kg/m3 = 0.8h(a) Radiation Force Balance

speed of sound
(m/s)
2732
2740
2755
2754
2739
2744
≤ 0.4 %

impedance(Mrayl)
3.240
3.249
3.267
3.266
3.248
3.254
≤ 0.4 %(b) Hydrophone

Figure 4. Ultrasonic power measured with the radiation force balance (a) and hydrophone (b) of PMMA
discs of different thicknesses. Reference measurements in water without PMMA samples. Absorption
values are listed in table 2.
shown in figure 4. The method used is not suitable for very small absorptions and therefore the inaccurate
measurements at 2 mm and 3 mm are evaluated.
Table 2. Absorption of PMMA samples at room temperature measured with the radiation force balance(α = a · f b ) as well as a hydrophone (α = c + a · f ). Values at 5 MHz are calculated. See also figure 4.
samplePMMA 2PMMA 3
PMMA 5PMMA 7.5
PMMA 10averageuncertainty

Radiation Force Balanceabα(5 MHz)(dB/(cm MHzb ))(dB/cm)
1.18
1.008
5.98
1.19
1.001
5.96
1.20
1.000
6.00
1.17
1.002
5.87
1.18
1.000
5.90
1.18
1.002
5.94
≤ 1%
≤ 0.4 % ≤ 1.1 %Hydrophoneac(dB/(cm MHz))(dB/cm)
n.a.
n.a.
0.80
1.033
0.69
1.649
0.71
1.477
0.73
1.39
0.06 = 8 %0.32 = 23 %α(5 MHz)(dB/cm)

5.03
5.12
5.03
5.06
≈ 24.4 %The calculated values for absorption parameters are shown in table 2 and figure 5. The standard deviationof the parameters was in all cases larger than the fit error and was therefore taken as uncertainty of the990

(a) Radiation Force Balance(b) Hydrophone

Figure 5. Absorption of PMMA discs of different thicknesses as a function of frequency. The absorption
was calculated with a numerical fit (equation 16) using the measured powers in case of the Radiation
Force Balance and not with a fit of calculated absorption values at specific frequencies. Therefore nomarkers are shown in (a) but only the fit curves.
parameters.

5

DISCUSSION

The measured data corresponds to literature values from PMMA as shown in table 3.
Within the bandwidth of the transducer, the power showed the expected behavior (figure 4): The power
decreases at the borders of the bandwidth frequencies and it also decreases with thicker PMMA discs due
to more absorption. The half-wave layer effects could be seen in the radiation force balance method as
predicted in equation 12. They are stronger the thinner the PMMA samples are due to less absorption.
This is a direct result from equation 12. The frequency exponent of the absorption is, as expected, close
to 1 since PMMA is a solid plastic without plasticizers.
Uncertainties are not mentioned in the publications. The main source of uncertainty in our experiments is
at present the exchange of the sample between the reference and a sample measurement and subsequently
slightly different placements and alignments of the accessories. For the hydrophone measurement evensmall alignment errors of the sample lead to diffraction of the sound field and refraction out of thehydrophone’s sensitive area. In case of the radiation force balance this error is negligible since thedistance from the sample to the target is small and since it covers the whole ultrasound field.
The calculated absorption of the PMMA discs is similar for the radiation force balance method and thehydrophone measurement. The differences are probably due to the above mentioned errors. As otherresearch shows, the uncertainties are lower with the radiation force balance method. This is a result ofTable 3. Comparison of obtained PMMA data with literature. Shown are the density %, speed of soundc, impedance Z, the frequency dependent absorption coefficient α and the absorption at 5 MHz. ∗ values
are calculated from the published data.
Carlson, 2003 [2]% (kg/m3 )
1192

c (m/s)2750

Z (Mrayl)
3.28

α (dB/(cm MHz))
1.06∗

Bloomfield, 2000 [3]Cheeke, 2012 [4]RFB method

1.18 · 103
1.18 · 103
1186

2.74 · 103
2.61 · 103
2744

3.23
3.08
3.254

1.1
2.48∗
1.18

α(5 MHz) (dB/cm)
60.84 Np/m
=5.28 dB/cm∗
5.5∗
12.4
5.94

991

the large absorbing target (vs. a small receiving area when using a hydrophone) and due to the usageof long pulses and therefore more received power (vs. short pulses for the hydrophone measurement).
Half-wave-layer effects can only be seen with the radiation force balance measurement when using longpulses.
Our research was done as a proof of concept and will be refined in the future.

6

CONCLUSIONSIn this proof-of-concept study we have shown that the radiation force balance method is suitable to measure the attenuation coefficient of a sample very accurately. The measurement of the sample’s thickness,density, speed of sound, and absorption can be realized in the same measurement setup. The reported
absorption values were obtained at room temperature in the frequency range 2 MHz to 8 MHz and compared with hydrophone measurements. The radiation force balance method uses very long pulses to use alarger ultrasound power for measurement and to take all transmitted echoes into account for attenuation
calculations. The method seems suitable for all materials that partially transmit ultrasound and will be
extended to other materials in the future.

ACKNOWLEDGMENTS
We want to thank the Investitionsbank Sachsen-Anhalt and the European Regional Development Fund(ERDF) to give financial support to the research and development project CaPS (Project-No. 1704/00064).

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[6] Hsu, D. K.; Hughes, M. S. Simultaneous ultrasonic velocity and sample thickness measurement andapplication in composites. JASA, 92 (2), 1992, 669-675[7] Bilaniuk, N.; Wong, G. S. K. Speed of sound in pure water as a function of temperature. JASA, 93(3), 1993, 1609-1612.
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