E-Journal of Advanced Maintenance Vol.2 (2010/2011) 181-190
Japan Society of Maintenology

Efficient Transduction of Circumferential Lamb Waves by a Pair of Line
Focus Type Noncontact Air-coupled Ultrasonic Transducers and its
Application for Accurate Measurement of Pipe Wall Thickness
Hideo NISHINO1,*, Kenichi YOSHIDA1, Tadashi ASANO1, Yuta TANIGUCHI1,
Hitoshi OGAWA2, Masakazu TAKAHASHI3, Yukio OGURA3
1The University of Tokushima, 2-1 Minami-josanjima, Tokushima-city, Tokushima 770-8506, Japan
Tokushima Prefectural Industrial Technology Center, 11-2 Saikai, Saiga-town, Tokushima-city,
Tokushima 770-8021, Japan
3
Japan Probe Co. Ltd., 1-1-14 Nakamura-town, Minami-ward, Yokohama-city, Kanagawa 232-0033, Japan
2

ABSTRACT
Efficient transduction of a circumferential (C-) Lamb wave by a pair of noncontact air-coupled ultrasonictransducers (NAUTs) was presented. A line focus type (LFT-) of the NAUT was employed for the efficienttransduction of the C-Lamb wave, which was designed so as to take an almost same angle of the incident
longitudinal beam on the circular pipe surface. Outstanding characteristics of the LFT-NAUT for the C-Lamb wave
transduction were theoretically and experimentally shown in comparison to those of the conventional plane type
(CPT-) NAUT. The efficiency of the C-Lamb wave generated and detected by the LFT-NAUT took about 20 times
higher than that by the CPT-NAUT. As for an application of the C-Lamb wave generated by the LFT-NAUT, anovel method of an accurate estimation of pipe wall thickness was introduced and evaluated with a theoretical model.
It was confirmed that the maximum error between the experiments and the theoretical model was at around 10 μm.
KEYWORDS

ARTICLE INFORMATION

ultrasonic, nondestructive evaluation, pipe wall thinning, Lamb wave,
circumferential Lamb wave, guided wave, health monitoring, noncontact
air-coupled ultrasonic transducer

Article history:
Received 22 November 2010
Accepted 3 February 2011

1.

Introduction

Noncontact air-coupled ultrasonic transducer (NAUT) is one of the powerful tools for monitoring
the condition of various materials because of its noncontacting nature. Recently, highly sensitive NAUT
[1-3] consisted of piezoelectric pillars embedded in a polymer matrix (so-called composite transducer)
have been exploited with their optimized electronics. Because of the condition for the transverse
resonance [4] of a plate specimen, effective transduction of the Lamb waves rather than the bulk acoustic
waves is easily realized [3] by the NAUT. Thus, the noncontact generation of the Lamb wave by the
NAUT is a useful tool to evaluate plate specimens because of its noncontacting nature. Therefore, many
applications for plate specimens using the NAUT have been presented. Measurements of anisotropy [5],thickness [6-9] and adhesive quality [10] of plates have been proposed and carried out using the NAUT.
A circumferential (C-) Lamb wave [11-18] propagating along the circumference of the pipe has
also been generated [19] by the NAUT for a wall thickness measurement of piping. In general, an
efficient transduction of the Lamb or the C-Lamb wave can be achieved in accordance with thecritical angle incidence of the longitudinal wave from the air to the surface of the solid specimen. The
efficient condition [3] of the critical angle incidence is easily achieved for the Lamb wave
transduction; however, it is very difficult for the C-Lamb wave transduction to meet the efficientcondition because of the curved surface of the pipe. Namely, the incident angle of the plane wave tothe circular surface of the pipe varies depending on the position of the incidence. To solve thisproblem, this paper proposes a novel and efficient transduction method for the C-Lamb wave using
line focus type (LFT-) NAUTs. The method can maintain almost constant incident angle to the*Corresponding author, Tel: 81-88-656-7357, Fax: 81-88-656-9062, E-mail: nishino@me.tokushima-u.ac.jp

ISSN-1883-9894/10/11 © 2010/2011 – JSM and the authors. All rights reserved.

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness

circular surface so as to generate and detect the C-Lamb wave effectively. The principle andcharacteristics of the method were presented in this paper.
On the other hand, the authors proposed and evaluated [19, 20] an accurate measurement of apipe wall thickness using the C-Lamb wave generated by the conventional plane type (CPT-) NAUT
having a flat aperture. The principle of the measurement is fundamentally based on the precise
detection of an angular wave number of the C-Lamb wave depending on the wall thickness of the pipe.
In this paper, measurements of the wall thicknesses were carried out with a pair of the LFT-NAUTs
according to the principle. After describing briefly the principle of the accurate estimation of the pipe
wall thickness, the experimental verifications were shown. The experimental estimations of the wall
thicknesses agreed excellently with both the actual thicknesses measured by the thickness-meter andthe calculation results of the mathematical model based on the theory described. It was confirmed thatthe maximum error of the experiments to theoretical calculations was around 10 μm.

2.

Characteristics of line focus type NAUT for C-Lamb wave transduction

In this section, important characteristics for the generation of the C-Lamb wave using the criticalangle incidence of the longitudinal wave in the air to a circular surface using the LFT- and
CPT-NAUT were presented.
The relationship between the pipe specimen (outer radius: R) and the flat aperture of the
CPT-NAUT is shown in Fig. 1(a). It is very easy to understand that the incident angle i of thelongitudinal wave from the flat aperture to the pipe surface is completely equal to the angular position
ϕ of the pipe.
On the other hand, the relationship between the pipe specimen and the concave aperture of the
LFT-NAUT (focal length: RF) is shown in Fig. 1(b). The angular position of the pipe is also described
as a function of ϕ. The focal point of the LFT-NAUT assumes to be at x1 on the x-axis. The angle of abeam tracing of the longitudinal wave from the centerline of the LFT-NAUT is describe as a functionof θ as depicted in Fig. 1(b). The incident angle i to the pipe surface and the phase velocity vp on thepipe surface with respect to the LFT-NAUT are shown in Eqs. (1) and (2), respectively.
i = tanvp =−1⎛ x1 cos ϕ ⎞⎜⎝ R − x sin ϕ ⎟⎠ ,
1

(1)

,

(2)

vair
sin i

where vair is the longitudinal velocity of the air. The propagation distance dCPT of the longitudinal
wave between the pipe surface and the flat aperture and the propagation distance dLFT between thepipe surface and the concave aperture, respectively, are
dCPT = D − R cos ϕ ,d LFT = RF −(3)

R + x1 − 2Rx1 sin ϕ ,
2

2

(4)

where D is the distance between the flat aperture and the x-axis, as shown in Fig. 1(a). Since the phaseadvance of the longitudinal wave propagation from the aperture to the pipe surface is as same as thephase on the pipe surface. Thus the phases pCPT and pLFT on the pipe surface are
pCPT = kair dCPT ,

(5)
(6)

p LFT = kair d LFT ,where kair is the wave number of the longitudinal wave in the air. The phase on the pipe surface aswell as the incident angle is one of the most important parameters to generate the C-Lamb wave
effectively. That is, the C-Lamb waves are effectively generated when the phase on the pipe surface
equals those of the C-Lamb waves.

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness

(c)
(a)
(b)
Fig. 1. Coordinate systems and variables used in the calculations for the CPT-NAUT (a),the LFT-NAUT (b) and the circular axis on the pipe surface (c).

(a)(b)
Fig. 2. Dispersion relations of the phase velocity (a) and angular wave number (b)of the 114.1 mm outer diameter and 3 mm thick aluminum pipe.

In the following experimental verifications, we focused on the transduction in the CL2 mode of the
C-Lamb wave at the frequency range around 310 kHz propagating along the 114.1 mm outer diameterand 3 mm thick aluminum (Al) pipe. Figures 2(a) and 2(b) show the dispersion relations of the phasevelocity and angular wave number of the Al pipe, respectively. The CL1 and CL2 modes are very similar
to the A0 and S0 modes of the Lamb wave generated in a plate specimen, respectively. Figure 3 shows
the incident angle regarding to both the CPT- and LFT-NAUT as a function of the location on thecircular axis along the outer circumference of the pipe (see Fig. 1(c)). The origin of the circular axis set
at the intersection of the pipe surface and the centerline of the each transducer, as shown in Fig. 1(c).
That is, the origin of the circular axis is where the angular position ϕ takes ϕ0 as shown in Fig. 1(c). The
parameters used in the calculations are summarized in Table 1. Since the x1 set to be 8.47 mm, theincident angle and the phase velocity at the origin of the circular axis are 8.53˚ and 2290 m/s,
respectively. This value of the phase velocity equals the phase velocity of the CL2 mode of the C-Lambwave at 310 kHz, as shown in Fig. 2. As shown in Fig. 3, the incident angle for the LFT-NAUT takes analmost same value at around 8.53˚ (from 8.22˚ to 8.53˚ in the range of ±15 mm) while the incident angle
for the CPT-NAUT is almost linearly and considerably changed as a function of the location on thecircular axis. In the Lamb and C-Lamb waves generation by the critical angle method, an accuracy of
the angle of incidence is of extremely importance especially in the air-coupled transductions [3].

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness

Outer radius of pipe
R
(mm)
57.1

Table 1 Parameters used in the calculations
Focal length
Focal point on x-axis
RF
x1
(mm)
(mm)
110

8.47

Velocity of the air
vair
(m/s)
340

Fig. 3. Incident angle as a function of the location on the circular axis. The incident angle of the
LFT-NAUT is almost constant in the range of ±15 mm.

Fig. 4. Phase velocity as a function of the location on the circular axis. The phase velocity of the
LFT-NAUT is almost constant in the range of ±15 mm.

The phase velocity on the pipe surface calculated by eq. (2) is shown in Fig. 4. As for the
CPT-NAUT, the phase velocity is drastically changed as a function of the location on the pipe surface.
In contrast to the case of the CPT-NAUT, the phase velocity in the LFT-NAUT is also takes an almost
constant value around 2290 m/s (from 2290 to 2370 m/s in the range of ±15 mm).
Spatial distributions of the phases on the circular axis for the CPT- and LFT-NAUT are shown in
Figs. 5(a) and 5(b), respectively, while the phase on the apertures equals 0˚. As for the LFT-NAUT
depicted in Fig. 5(b), the spatial distribution is almost linearly distributed and equal to that of the CL2mode C-Lamb wave at the frequency of 310 kHz in the range of ±15 mm on the circular axis. In
contrast to this, as for the CPT-AUT, the spatial distribution varies nonlinearly and equals that of theCL2 mode C-Lamb wave only at around the origin of the circular axis.
As shown in the above explanations for the LFT- and CPT-NAUT, the characteristics of the
LFT-NAUT are quite optimized for the transduction of the C-Lamb waves. The following chapter
shows the experimental verifications of the LFT-NAUT for the C-Lamb wave transduction and its
application.

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness(a)(b)
Fig. 5. Spatial phase distributions on the circular axis for the CPT-NAUT (a) and the LFT-NAUT (b).
The spatial phase distribution generated by the LFT-NAUT is almost linearly allocated on the circularaxis and is as almost same as that of the CL2 mode of the C-Lamb wave.

3.

Experimental verification

3.1.

Experimental apparatus

An illustration of the experimental setup is depicted in Fig. 6. A face-to-face setting of a pair of
the LFT-NAUTs as the transmitter and receiver, respectively, was used for the transduction of the
C-Lamb waves. The two transducers were located at their confocal positions. A pair of the
CPT-NAUT was also used for comparison. The aperture sizes of both the LFT-NAUT and the
CPT-NAUT are 20 x 60 mm2 and 14 x 20 mm2, respectively. The center frequencies of the
LFT-NAUT and the CPT-NAUT are 310 and 340 kHz, respectively, which are depending on theelectric capacitances of both transducers mainly related to their aperture sizes. Namely, the aperturearea of the LFT-NAUT is 4.3 times larger than that of the CPT-NAUT. 10- and 130-cycle tone-burst
signals were generated (Tektronix AFG3102) and amplified (NF BA4825) to 20 V peak-to-peak for
the LFT-NAUT and to 200 V for the CPT-NAUT and fed into the transmitters to generate the CL2mode C-Lamb waves. The difference of the input voltages between the two transducers is dependingon the sensitivities for the transduction of the C-Lamb wave between them. The received signals forthe both transducers were amplified (NF 9311 and NF3628) 90 dB. A digital oscilloscope (Tektronix
DPO7054) was employed to observe and store the signals detected. 300 times averaging sequence inthe digital oscilloscope and the 10 kHz width 48dB/Oct band-pass filter were used to improve thesignal to noise ratio (S/N). 114.1 mm outer diameter and 3 mm thick Al pipes were used. 10 different
wall thinnings (up to around 1.0 mm) were introduced to the inner surface of the Al pipes by a lathe. A
universal length-measuring machine (Tsugami T-ULM500) having 1 μm nominal accuracy was used tomeasure the wall thicknesses of all the Al pipes. Table 2 shows the averaged wall thickness losses of all
the specimens used in the experiments. The details of the specimens were described in ref. [20].

Fig. 6. Schematic illustration of the experimental apparatus of the two NAUTs,
the Al pipe specimen and the C-Lamb wave circling along the pipe

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness
Table 2
Sample No.
Loss of wall(mm)
Std. dev.
(mm)

1Wall thickness losses and their standard deviations
2

3

4

5

6

7

8

9

10

0.000 0.098 0.254 0.398 0.439 0.544 0.641 0.764 0.876 1.016
0.025 0.019 0.032 0.020 0.029 0.032 0.027 0.053 0.025 0.035

To generate and detect the CL2 mode of the C-Lamb wave effectively, the procedure for setting
the pair of the transducers to the pipe specimen was introduced as follows:(1) Locate the pair of the LFT-NAUTs at a confocal face-to-face position so as to take the maximum amplitude
without the pipe specimen. As for the pair of the CPT-NAUTs, achieve only the face-to-face location.
(2) Insert the pipe specimen between the pair of the transducers.
(3) Search the optimum location of the pipe specimen for the effective transduction so as to take themaximum amplitude of the CL2 mode of the C-Lamb wave generated. It can be obtained while theangle of incidence takes precisely the critical angle for the transductions.
3.2.

Verification for the transduction efficiency

Time domain signals of the CL2 mode of the C-Lamb wave generated and detected by the pairs ofboth the CPT-NAUTs and the LFT-NAUTs are shown in Figs. 7(a) and 7(b), respectively. The 10-cycle
toneburst signals were used for the generations in the both cases. The center frequencies of the toneburst
signals were 340 kHz for the CPT-NAUT and 310 kHz for the LFT-NAUT, which were as same as thecenter frequencies of the transducers, respectively. The three wave packets could be seen in the bothcases. The first wave packet (0th lap wave) was the wave that was generated by the transmitter, and was
detected by the receiver after propagating roughly half along the circumference of the pipe (see Fig. 6).
The second and third wave packets (1st and 2nd lap waves) in Figs. 7(a) and 7(b) were the waves thathave propagated roughly 1.5 and 2.5 turns along the circumference of the pipe, respectively.
The amplitudes of the first wave packets generated and detected by the CPT- and LFT-NAUT
were 0.37 and 0.75 V peak-to-peak, respectively. Therefore, the multiplication of the transduction
efficiency of both the transmission and reception of the CL2 mode for the LFT-NAUT was almost 20
times larger than that for the CPT-NAUT because the amplitude of the input signals for the
CPT-NAUT is 10 times larger than that for the LFT-NAUT as described in the previous section. On
the other hand, the aperture area of the LFT-NAUT is 4.3 times larger than that of the CPT-NAUT,
this means that the efficiency per a unit aperture-area of the LFT-NAUT is 4.3 times larger than thatof the other transducer. Anyway, not the CPT-NAUT but the LFT-NAUT has taken the ability tomaintain the optimized angle of incidence on wider surface of the pipe for much efficient transduction.
Therefore, it can be estimated that the transduction efficiency of the CPT-NAUT won't be increased
with an increase of the aperture area; conversely, that of the LFT-NAUT will be increased with it. It
was clearly confirmed that the transduction efficiency of the CL2 mode of the C-Lamb wave for the
LFT-NAUT was much higher than that for the CPT-NAUT at the respective center frequencies.

(a)(b)
Fig. 7. Typical time domain signals generated and detected by the CPT-NAUT (a) and the LFT-NAUT (b).

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness

3.3.

Experimental estimation of the pipe wall thinning using a pair of LFT-NAUTs

An accurate estimation method [19, 20] of a pipe wall thickness by detecting a minute differenceof an angular wave number of the C-Lamb wave is briefly described. Figure 8 shows the angularwave number as a function of the frequency for four different wall thinnings at around the center
frequency 310 kHz of the LFT-NAUT. The angular wave number is the least increased with anincrease of the wall thickness loss. Thus, it is too difficult to detect such a small difference of theangular wave number using a conventional time-of-flight method. On one hand, in the present method,
the generation and detection of the C-Lamb wave was realized in the same manner described in thesection 3.1. That is to say, the C-Lamb wave circling along the circumference of the pipe was
efficiently transmitted and received by a pair of the LFT-NAUT. On the other hand, for the accurate
detection of the angular wave number, a tone burst signal having a large number of cycles(130-cycles) is used so as to superpose the C-Lamb wave itself on its own propagation orbit. In thissetting, the amplitude of the superposed regions changes considerably owing to the change of the
wave number of the C-Lamb wave. The amplitude A of the superposed region isnA=∑ A exp ( 2π imp ) ,

(7)

n

m=0

where n, An and p are number of the superposition, amplitude of each circling wave and angular wave
number of the C-Lamb wave, respectively. Figure 9 shows calculated amplitudes (n=1) for various
wall thinnings as a function of frequency. The amplitude peaks shifted on the frequency axis with anincrease of wall thinning.
The typical time domain signals of the CL2 mode of the C-Lamb waves propagating in the pipe
without any wall thickness loss are shown in Fig. 10, which were generated by 130 cycle tone burst
signals. In contrast to the time domain signals generated by 10-cycle tone burst signals (Fig. 8), it
was confirmed that all the wave packets were extremely long and were superposed onto themselves.
Therefore, it was clearly confirmed that the amplitude of the superposed region (after 100 μs) was
drastically changed owing to the frequency as shown in Fig. 10. This phenomenon was the result of
the relative phase differences between the 0th, 1st and 2nd lap waves owing to the change of thefrequency. The normalized amplitude as a function of frequency for the wall thickness losses of
0.000, 0.098, 0.254 and 0.398 mm is shown in Fig. 11. The amplitude at the mid time of the eachsuperposed region was measured. The mid times of the 1st and 2nd superposed regions were 180
and 300 μs, which were determined by eye using the time domain signals of 310 kHz (the center
frequency of the LFT-NAUT) as shown in Fig. 10. In comparison to the theoretical results shown in
Fig. 9, it could be found that the peak amplitudes changed with a change of the frequency becauseof the center frequency (310 kHz) of the LFT-NAUT as shown in Fig. 11. In contrast to this, thepeak frequencies were almost same as those estimated by the theoretical calculations as shown in
Fig. 9. The peak frequency in Fig. 11 as a function of wall thickness loss is summarized in Fig.12.
Red circles and blue dotted lines are the experimental results and the theoretical calculations,
respectively. To determine the experimental peak frequency of the characteristic-curve in Fig. 11, a
least square fitting to the Gaussian-curve was applied to each region around the peak. The
theoretical calculations (the blue doted lines in Fig. 12) were carried out on the basis of ref. [13]with the longitudinal and transverse wave velocities (vl = 6400 m/s, vt = 3040 m/s) of Al material. It
was obviously confirmed that the experimental results agreed excellently with the theoretical
calculations. The maximum error between the experiments and theory was around 10 μm, which
was as almost same as that obtained by the CPT-NAUT [19, 20]. This result means that theaccuracy of the method for the wall thickness estimation is not improved although the transduction
efficiency of the LFT-NAUT is increased extremely. On the other hand, the robustness of themethod using the LFT-NAUT must be increased so much in actual noisy situations because of its
efficient sensitivity.

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness

Fig. 8. Dispersion relation of the angular wave number of the C-Lamb wave fordifferent wall thinnings at around the center frequency of the LFT-NAUT.

Fig. 9. Calculated normalized amplitude of the superposed region(0th and 1st laps) as a function of the frequency

Fig. 10. Time domain signals of the CL2 mode generated by 130-cycle tone bursts for different center
frequencies. The amplitude of the superposed regions changed owing to the frequency.

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H. Nishino, et al./ Efficient Transduction of Circumferential Lamb Waves bya Pair of Line Focus Type Noncontact Air-coupled Ultrasonic Transducers andits Application for Accurate Measurement of Pipe Wall Thickness
0.6

Amplitude (V)0.5

0.000 mm loss
0.098 mm loss
0.254 mm loss
0.398 mm loss

0.4
0.3
0.2
0.1
0.0
300 302 304 306 308 310 312 314 316 318 320
Frequency (kHz)Fig. 11. Amplitude of the superposed region (0th and 1st laps) as a function of the frequency

Fig. 12. Peak frequency as a function of wall thickness loss. Red circles and blue dotted lines indicate
experimental results and theoretical estimation.

4. Conclusion
This paper describes a novel and extremely efficient method for the transduction of thecircumferential (C-) Lamb wave using a pair of the line focus type (LFT-) noncontact air-coupled
ultrasonic transducers (NAUTs) and its application for a wall thickness measurement of a pipe.
Comparing to the conventional plane type (CPT-) NAUT, the outstanding characteristics of the
LFT-NAUT for the highly efficient transductions were presented as follows:(1) The incident angle of the longitudinal wave in the air to the circular pipe surface is almost constantas a function of the circular position.
(2) Corresponding to the constant angle of the incidence, the phase velocity on the circular surface ofthe pipe is also almost constant for the efficient transduction.
(3) The spatial distribution of the phase on the circular surface emerged by the incidence of thelongitudinal wave in the air is also almost constant for the efficient transduction.
(4) In the experimental verifications, the transduction efficiency of the LFT-NAUT was 20-times
larger than that of the CPT-NAUT. The highly efficient transduction was clearly confirmed.

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As for the verification of wall thickness measurements using the LFT-NAUT, the accuracy of theresults was around 10 μm. The highly accurate estimation of the wall thickness together with theextremely high transduction efficiency has been achieved in noncontact manner using the
LFT-NAUT.

Acknowledgement
This work was partially supported by an academic grant from the Condition Monitoring
Technology (CMT) division of the Japan Society of Maintenology (JSM).

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