Acoustic formulation of elastic guided wave propagation and scattering in curved tubular structures

Recently, the use of guided wave technology in conjunction with tomographic techniques has provided the possibility of obtaining point-by-point maps of corrosion or erosion depth over the entire volume of a pipeline section between two ring arrays of ultrasonic transducers. However, current research has focused on straight pipes and little work has been done on pipe bends and other curved tubular structures which are also the most susceptible to developing damage. Tomography of curved tubes is challenging because of the complexity and computational cost of the 3-D elastic model required to accurately describe guided wave propagation. Based on the definition of travel-time-preserving orthogonal parametric representations of curved tubes, this paper demonstrates that guided wave propagation and scattering can be approximated by an equivalent 2-D acoustic model which is inhomogeneous and elliptically anisotropic. Numerical methods to solve the full wave equation and predict ray paths and travel times are introduced and applied to the case of a bend. Particular emphasis is given to the shortest-path ray tracing method, which is applied to the 2-D model to compute ray paths and predict travel times of the fundamental flexural mode, A₀, propagating across a curved pipe. Good agreement is found between predictions and experiments performed on a 220-mmdiameter (8-in-diameter) (D) pipe with 1.5D bend radius. The 2-D model also reveals the existence of an acoustic lensing effect which leads to a focusing phenomenon also confirmed by the experiments. The computational efficiency of the 2-D model makes it ideally suited for tomographic algorithms.