YU Xiaotao, PENG Linhui,, and YU Gaokun

1) School of Information Science and Engineering, Ocean University of China, Qingdao 266100, P. R. China

2) State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing 100190, P. R. China

Extracting the Subsonic Anti-Symmetric Lamb Wave from a Submerged Thin Spherical Shell Backscattering Through Iterative Time Reversal

YU Xiaotao1),2), PENG Linhui1),2),*, and YU Gaokun1)

1) School of Information Science and Engineering, Ocean University of China, Qingdao 266100, P. R. China

2) State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing 100190, P. R. China

The extraction of the weakly excited anti-symmetric Lamb wave from a submerged thin spherical shell backscattering is very difficult if the carrier frequency of the incident short tone burst is not at its frequency of greatest enhancement. Based on a single channel iterative time reversal technique, a method for isolating the subsonic anti-symmetric Lamb wave is proposed in this paper. The approach does not depend on the form function of a thin shell and any other priori knowledge, and it is also robust in the presence of some stochastic noise. Both theoretical and numerical results show that the subsonic anti-symmetric Lamb wave can be identified, even when the carrier frequency of the incident short tone burst is away from the frequency of greatest enhancement. The phenomenon may also be observed even in the case that the subsonic anti-symmetric Lamb wave is submerged in the noise, other than the case with the Signal to Noise Ratio being less than 10 dB, when the amplitude of the noise is comparable with the specular wave. In this paper, each iteration process contains a traditional transmission and time reversal transmission steps. The two steps can automatically compensate the time delay of the subsonic anti-symmetric Lamb wave relative to the specular wave and within-mode dispersion in the forward wave propagation.

backscattering; Lamb wave; time reversal; frequency of greatest enhancement; SNR

1 Introduction

For a short tone burst incidence, the subsonic antisymmetric Lamb wave backscattered by a submerged airfilled thin spherical shell can be relatively enhanced in a midfrequency range. For the example of a 5% thick stainless steel shell emphasized in the present study, this enhancement is evident for ka from about 10 to 25. The frequency of greatest enhancement has a strong dependence on the radius and thickness of the shell, so the extraction of the subsonic anti-symmetric Lamb wave is of significance to estimate geometric parameters of a shell.

This midfrequency enhancement phenomenon was first observed by Gaunaurd and Werby (1987) in their study of the Lamb and Franz waves scattered by spherical shells in water. It is found that the form function of a shell has a resonance peak (greatest enhancement) in a midfrequency range, and the frequency of greatest enhancement decreases with the shell thickness. Through the analysis of the backscattering amplitude of a thin spherical shell, Sammelmannet al. (1989) has shown that the dispersion curve of the lowest order, anti-symmetric Lamb wave bifurcates to two different curves at a critical frequency. The two different dispersion curves can be differentiated by phase speed,i.e., one is supersonic and the other is subsonic. The subsonic anti-symmetric Lamb wave contributes to the midfrequency enhancement over a certain broad frequency range. Moreover, the frequency-thickness productkhof greatest enhancement is found to be roughly constant. The backscattering amplitude excited by tone bursts is also calculated by the modified ray approximation (Zhanget al., 1992), and it is illustrated that the amplitude of the anti-symmetric Lamb wave is relatively large compared with the specular wave in the midfrequency range, which can be used to identify the frequency of greatest enhancement. Using these properties, Liet al. (2005) estimated the radius and thicknesses of spherical shells. This midfrequency enhancement phenomenon has also been studied by some other authors (Kaduchak and Marston, 1993a, b; Kaduchaket al., 1995).

The frequency of greatest enhancement in the above investigations is obtained by calculating the form function of a shell. However, in practice, it is not easy to extract the form function from the echoes of a shell. The antisymmetric Lamb wave was experimentally isolated from the scattering responseviaa single channel, iterative time reversal by Yinget al. (2009), Waters (2009) and Waterset al. (2009). Power-iterated single-channel time-reversal has been extended to employ Lanczos iterations (Waters and Barbone, 2010, 2012) which possess a superior convergence in comparison to the standard power-iterated technique (Waterset al., 2012). It is shown that the anti-symmetric Lamb can be identified from the echoes for the carrier frequency at the frequency of greatest enhancement. However, for the carrier frequency away from the frequency of great enhancement, the anti-symmetric Lamb wave is so weekly excited that it can hardly be identified, especially in the presence of noise.

In this paper, based on the single channel, iterative time reversal technique and the application of time reversal process in structural health monitoring (Parket al., 2007, 2009; Jun and Lee, 2012), both theoretical and numerical results show that the anti-symmetric Lamb wave can be identified, even when the carrier frequency of the incident short tone burst is away from the frequency of greatest enhancement. First, the midfrequency enhancement phenomena of the thin spherical shell is introduced in Section 2, and a dominant resonance of the echo, the subsonic anti-symmetric Lamb wave, is observed in both time and frequency domain. In Section 3, the frequency of the dominant resonance is isolated through an improved single channel, iterative time reversal technique, and in the case that the noise is present. Then the validity of the improved iteration steps in compensating the time delay relative to the specular wave and within-mode dispersion of the anti-symmetric Lamb is analyzed in Section 4. The paper is concluded in Section 5.

2 Midfrequency Enhancement of the Thin Spherical Shells

For the purpose of interpreting the frequency drift and converging through the iterative time reversal, the midfrequency enhancement phenomena of thin spherical shells is first investigated in this section. The dominant resonance of the echo is also discussed in both the frequency and time domain.

The steady state of the far-field (r>>ka2) backscattering can be written as the real part of

wherepincis the amplitude of the incident plane wave,ais the radius of the spherical shell,x=kais the dimensionless frequency, andk=ω/c,ωis the angular frequency of the incident signal,kandcare the wave number and sound speed in the surrounding water, respectively. The backscattering form functionf(x) is

where the functionsBn(x) andDn(x) are 5×5 determinants given by Kargl and Marston (1989). The material parameters of the spherical shell and surrounding water are displayed in Table 1. Fig.1 shows the calculated backscattering form function of shells with different thicknesses and a same 1 m radius (below the same). The ratio between the radius,a, and thickness,h, of the shell is denoted byh/a. And for eachh/a, an apparent broad peak, which is dominated by the lowest order, subsonic, antisymmetric Lamb wave, appears in a midfrequency range. Fig.1 also displays the result that the enhanced midfrequency range varies approximately asa/h, suggesting the following inverse problem for estimating theh/a.

In order to display the midfrequency enhancement phenomenon in the time domain, the echoes excited by tone bursts with different carrier frequenciesk0a=14, 21, 28 and 66.2 are calculated withh/a=0.05 as shown in Fig.2. It should be noted that the echoes presented in this paper are calculated by OASES&SCATT model which is based on the wavenumber integration and virtual source method(Schmidt, 2004; Lee and Schmidt, 2003). Consistent with the frequency of greatest enhancement obtained from the backscattering form function, the lowest order subsonic anti-symmetric Lamb waves,a0-, are relatively enhanced atk0a=21 compared with the specular wave. In Fig.2d, the echo excited by a 4-cycle sine tone at a much higher carrier frequencyk0a=66.2 is presented, and thea0-wave is so weakly excited that it can hardly be isolated.

Table 1 Material parameters of the spherical shell and surrounding water

Fig.1 The backscattering form function of spherical shells with different h/a.

Fig.2 Echoes of the spherical shell with different k0a.

The first arrival is the specular wave, and the second arrival is the subsonic anti-symmetric Lamb wave (a0-).

3 Isolation of the Subsonic Anti-Symmetric Lamb Wave

In this section, an improved single channel, iterative time reversal technique is used to enhance the echo from a thin spherical shell. The single-channel iterative time reversal technique was previously described by Pautetet al. (2005). Based on the steps used by Waterset al. (2012), the procedure of the single channel, iterative time reversal technique is improved as followed:

1) Excite a submerged air-filled spherical shell with a short tone burst.

2) Truncate a portion of the echo using a time-domain window.

3) Reverse the temporal truncated echo and transmit the new time reversed signal.

4) Truncate the focused signal received after step (3) as the tone burst of step (1).

5) Repeat steps (2)–(4) iteratively.

Different from the process used by Waterset al.(2012), a much longer time-domain window is used in step (2), and the step (4) is added. The extraction process contains one initial traditional transmission and multiple time reversal transmissions. The two steps can compensate the time delay between the specular reflection and thea0-wave and within-mode dispersion during propagating in the shell, and enhance the capability of the iteration time reversal.

3.1 The Theoretical Process of the Iterative Time Reversal

The theoretical process of the single channel, iterative time reversal is performed in the case that the transfer function of thea0- wave is obtained from a ray approximation, and the echo contains only the specular wave and first receiveda0- wave.

The ray approximation (Zhanget al., 1992) for themtha0-contribution to the steady state form function for backscattering is

wheremis the circumnavigation index,ηis a propagation-related phase shift,clis the phase velocity along the outer surface of the shell,wis the angular frequency of thea0- wave,xandcare defined as in Eq. (1). Its magnitude is simply related to the radiation damping parameterβ(w) for thea0- wave.

In this paper, the ray approximation of thea0-wave is simplified as

k2andldenote respectively the wavenumber and propagating distance in the shell.

The echo is supposed to contain only the specular wave and first receiveda0-wave. Their transfer function in the shell can be written as

wherecgis the group velocity of thea0-wave propagating in the shell.

Because the specular wave does not penetrate into the shell, the transfer function can be described aswithout amplitude and within-mode dispersion. The transfer function of thea0- wave isA2(w) and2ejkldenote the amplitude and within-mode dispersion function during propagating in the shell.

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For the plane wave incidence, neglecting the spherical spreading damping factorr-1, the steady state of the scattered pressure can be written as follows

Then, relaunch the conjugated pressure, and the second received pressure can be written to

Thes/sanda0-/a0- superpose each other as the focused signal as shown in Eq. (7). After iterative time reversal operation, the focused signal is of the form

Then the amplitude ratio of thea0- wave relative to the specular wave is of the form

Eq. (9) shows that the amplitude ratio increases with the iterations increasing, which manifests that the focused signal is more and more dominated by thea0- wave. Therefore, the frequency of the focused signal may shift and converge to the frequency of greatest enhancement at which thea0- wave can be maximizedly excited.

3.2 Numerical Simulations

The predictions of Eq. (9) are confirmed through the following numerical simulations for the above iteration process with OASES&SCATT when thea0- wave is weakly excited.

In this section, a 4-cycle sine tone burst with a much higher carrier frequencyk0a=66.2 and with the excited Lamb waves being weakly excited is employed. Fig.3 shows the echo with its wigner-ville distribution (WVD) where two light spots (cut across by vertical lines) are found after the specular wave. Compared with the arriving time of thea0- wave obtained from Fig.2, they are identified as them=0 andm=1a0- waves. In the spectrum of the focused signal, a weak enhancement peak (cut across by a horizontal line) is found nearka=21. With the frequency of greatest enhancementka=21 lying on the side lobe of the incident spectrum far away from the carrier frequency, the exciteda0- waves contribute little to the energy of the echo. The second received echo after time reversal is shown in Fig.4, where the focused signal is primarily contributed by the specular wave.

Fig.3 WVD of the echo excited by k0a=66.2.

Fig.4 The received echo after time reversal.

Fig.5 spectra of the focused signals after iterative time reversal.

Eq. (13), which is an increasing function versusnas Eq. (9), shows that the focused signal may be more and more dominated by thea0- wave with the frequency drifting and converging to the frequency of greatest enhancement. Eq. (10) displays the result when the noise can be coherently superposed during the iterative time reversal process. However, the noise is not coherently amplif i ed as rapidly as the coherent target response in practice, which enhances the validity of this method.

The derivation of Eq. (9) and Eq. (13) indicates that the focused signal should be identified from the noise in the time domain, which limits the application of the method at a weaker Signal to Noise Ratio (SNR).

TheSNRis defined asSNR=10log(Es/En), whereEscident short tone burst when thea0- waves are maximizedly excited. However, the iterationnshould be balanced considering the amplitude dispersion during the iterative time reversal which makes thea0- wave expanse in the time domain. It is apparent that the frequency drift from the carrier frequency to the frequency of greatest enhancement is not reversible during the iteration, which supports that the phenomenon may also be observed in the case that some other Lamb waves are excited in the echo.

3.3 The Effect of Noise

In the presence of noise whose transfer function isNe-jkr, the focused signal received after iterative time reversal can be written as

Then the amplitude ratio of thea0-wave relative to the specular wave is of the form

Compared with Eq. (9), the effect of the noise is performed in Eq. (11). The expressionN(A1+1)/Pincmay be neglected if the amplitude of the noise is very low relative to amplitude of the incident plane wave when Eq.(11) can be simplified as Eq. (9). After a constant,a, is introduced for compensating the scattering loss of the signal, Eq. (11) can be modified toandEndenote the maximum amplitudes of the signal and the noise in the truncated time window. The results of numerical calculation of the echoes without noise and with theSNRat 30 dB, 20 dB and 10 dB are shown in Fig.6. Figs.6a–d show that thea0- wave is gradually submerged in the noise. The spectra of the focused signals after iterative time reversal are given in Fig.7, where the frequency of the greatest enhancement can hardly be identified atSNR=10 dB. Numerical simulations demonstrate that the focused signal is completely submerged in the noise for much more iterations. Therefore, the critical value ofSNRmay be 10 dB under which thea0- wave cannot be isolated.

Fig.6 Echoes from the spherical shell in the presence of stochastic noise.

Fig.7 Spectra of the focused signal after iterative time reversal operation.

4 Discussions

In Section 3, a drift to the frequency of greatest enhancement is observed through the improved single channel, iterative time reversal technique. As discussed in Section 2, the resonance is dominated by the subsonic anti-symmetric Lamb wave in a thin spherical shell.

In this paper, each iteration process contains a traditional transmission and time reversal transmission steps, which is different from the process used by Waterset al.(2012). The two steps can compensate the time delay factor, etjω-Δ, of thea0- wave relative to the specular wave. Fig.8a shows the numerical results of the echoes excited by a 4-cycle, narrow band sine tone burst with a carrier frequencyk0a=21. Fig.8b displays the second received response after time reversal containings/s,a0-/s,a0-/a0- ands/a0-. After time reversal, thea0-/a0- ands/sare received at the same time and superpose each other as the focused signal. Therefore, thea0- wave once weakly excited may be picked up along with the focused signal instead of searching through the whole time window which is usually not easy to perform.

Eq. (7) also shows that the within-mode dispersion factor,2ejklin the forward wave propagation process is automatically compensated during the time reversal process, resulting in only amplitude dispersion at the end. Numerical results shown in Fig.9a shows that the different frequencies are received at different times bending the bright stripe consistent with the dispersion curve shown by Zhanget al. (1992). However, after time reversal, the bent bright stripe straightens without narrowinga0- in the time domain for the amplitude dispersion shown in Fig.9b.

Fig.8 (a) The echo of the spherical shell; (b) The second received echo after time reversal.

Fig.9 The wigner-ville distribution (WVD) of the echoes. (a) before time reversal; (b) after time reversal.

5 Conclusions

A feasible method for extracting the frequency of greatest enhancement dominated by thea0- wave is presented in this paper. Not requiring the form function and any other priori knowledge, the extraction method is based on a single channel, iterative time reversal process. Both theoretical and numerical simulation results show that the frequency of the focused signal shifts and converges to the frequency of greatest enhancement with the single channel, iterative time reversal process. The phenomenon can be used to identify the frequency of greatest enhancement, which is consistent with the result obtained from the form function.

The effect of the stochastic noise is also studied in this paper. An obvious defect is that the validity of this method is bounded by the absoluteSNRof the system. Because the iterative time reversal process amplif i es the coherent target response and does not act to decrease the level of the noise. Therefore, further study in enhancing the validity of this method at much lowerSNRis necessary.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (46976019) and the open project of the State Key Laboratory of Acoustics, Chinese Academy of Sciences (SKLA201202). The authors acknowledge their funds to support the research.

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(Edited by Xie Jun)

(Received October 12, 2012; revised February 27, 2013; accepted March 13, 2014)

? Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2014

* Corresponding author. E-mail: penglh@ouc.edu.cn