Sensitivity of Ultrasonic Wave Velocity Estimation Using the Christoffel Equation for Wood Non-Destructive Characterization

Luis Espinosa, Loïc Brancheriau, Flavio Prieto, Philippe Lasaygues


To perform a non-destructive evaluation of wood, the Christoffel equation is frequently used to describe the relationship between the ultrasonic wave velocity and the mechanical parameters. In the context of acoustical tomography imaging of standing trees, the key contribution of this numerical study is to determine the influence of mechanical parameters of the wood radial-tangential plane on the wave velocity computation using the Christoffel equation. Mechanical parameters from six species were selected. A sensitivity analysis was carried out by increasing and decreasing every parameter by a given percentage, and then by computing the variation of velocity for a set of wave direction of propagations. The evolution of the wave velocity, according to the direction of propagation, depended on the considered species; there was a difference between the softwoods and the hardwoods. The sensitivity analysis showed a bigger influence of the Young’s moduli, followed by the Poisson’s ratio, and finally by the shear modulus. However, these last two parameters cannot be neglected when using the Christoffel equation to solve the inverse problem of standing tree tomography. A proposed solution involves determining the propagation paths using the Young’s moduli as variables and then inversing the set of equations in accordance with the overall parameters.


Wood; Ultrasonic; Wave propagation; Orthotropy; Christoffel equation

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Welcome to BioResources! This online, peer-reviewed journal is devoted to the science and engineering of biomaterials and chemicals from lignocellulosic sources for new end uses and new capabilities. The editors of BioResources would be very happy to assist you during the process of submitting or reviewing articles. Please note that logging in is required in order to submit or review articles. Martin A. Hubbe, (919) 513-3022,; Lucian A. Lucia, (919) 515-7707, URLs:; ISSN: 1930-2126