Detection of cracks in a thin air-filled hollow cylinder by application of the DORT method to elastic components of the echo E. Kerbrat * , D. Clorennec, C. Prada, D. Royer, D. Cassereau, M. Fink Laboratoire Ondes et Acoustique, ESPCI, Universit e Paris 7, CNRS, 10, rue Vauquelin, 75231 Paris Cedex 05, France Abstract The DORT method (D ecomposition de l’Op erateur de Retournement Temporel in French) is a scattering analysis technique which uses arrays of transducers. This method is efficient for detection of selective focusing on point-like scatterers. It has been also applied to analyze the scattering by an air-filled cylindrical steel shell immersed in water. It was shown that the diagonalization of the time reversal operator allows us to separate the different elastic components of the scattered field. Here, we apply the method to detect flaws in hollow cylinders. In this case, the dominant components are the three circumferential waves (A 0 ,A 1 and S 0 Lamb modes). Each Lamb mode corresponds to an invariant of the time reversal operator. The dispersion curves of these waves are calculated from these invariants. Resonance frequencies of the shell are deduced from the frequency dependence of the eigenvalues of the time reversal operator. It is shown that the presence of a crack (0.2 mm in depth) affects significantly the eigenvalue dis- tribution of the time reversal operator. Thus, the DORT method offers a new means for detecting defects in a shell. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: DORT method; Lamb waves; Nondestructive evaluation; Hollow cylinder; Crack 1. Introduction We are interested in detecting cracks in a thin air- filled hollow cylinder, which supports the propagation of Lamb waves. The technique used is the DORT (French acronym for Decomposition of the Time Re- versal Operator) method [1]. This method requires ul- trasonic transducer arrays working in transmit and receive modes. It has already been applied to different problems such as flaw detection in solids [2] or a water waveguide [3] and separation of Lamb modes propa- gating in a thin air-filled hollow cylinder free of any defects [4]. Concerning flaw detection in solids, we have studied the detection of flat bottom holes in titanium alloy samples that contains a high speckle due to mi- crostructure contribution. We initially review the DORT method principle. The results concerning propagation in a thin air-filled hollow cylinder in an undamaged region are then compared to those obtained with a crack on the same shell. 2. The DORT method The DORT method has been described in several papers [5]. This method is derived from the theo- retical analysis of time reversal mirrors. It is based on a matrix formalism that describes a transmit–receive process performed with an array of L transducers. Such an array insonifying a given medium can be consid- ered as a linear and time invariant input–output system. It is characterized by L  L inter-element impulse re- sponses k lm ðtÞ defined as the output of the 1th element when the input of the mth element is a delta impulse. At a given frequency, the L  L transfer matrix KðxÞ¼ ½K lm ðxÞ 1<l;m<L can be calculated by the Fourier trans- form of the responses k lm ðtÞ. The time reversal operator, defined as K  ðxÞKðxÞ, can be diagonalized. Each ei- genvector represents an invariant of the time reversal process. It was shown that in our case [4] the number of significant eigenvalues provides information regarding the number of Lamb modes. Furthermore, each pair of Ultrasonics 40 (2002) 715–720 www.elsevier.com/locate/ultras * Corresponding author. Tel.: +33-1-40-79-44-52; fax: +33-1-40-79- 44-68. E-mail address: estelle.kerbrat@espci.fr (E. Kerbrat). 0041-624X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0041-624X(02)00199-3