The Hankel transform of first- and second-order tensor fields: definition and use for modeling circularly symmetric leaky waveguides.

Eric Ducasse (1, 2) , Slah Yaacoubi (2, 3, 4)

1 I2M - Institut de Mécanique et d'Ingénierie de Bordeaux
2 UB - Université Sciences et Technologies - Bordeaux 1
3 LMP - Laboratoire de Mécanique Physique
4 LCPC - Laboratoire Central des Ponts et Chaussées

A tensor Hankel transform'' (THT) is defined for vector fields, such as displacement, and second-order tensor fields, such as stress or strain. The THT establishes a bijection between the real space and the wave-vector domain, and, remarkably, cannot be reduced to a scalar transform applied separately to each component. One of the advantages of this approach is that some standard elasticity problems can be concisely rewritten by applying this tensor integral transform coupled with an azimuthal Fourier series expansion. A simple and compact formulation of the boundary conditions is also achieved. Thanks to the THT, we obtain for each azimuthal wavenumber and each azimuthal direction exactly the same wave equation as for a standard 2D model of elastic wave propagation. Thus, waves similar to the standard plane P, SV and SH waves are naturally found. Lastly, the THT is used to calculate the ultrasonic field in an isotropic cylindrical leaky waveguide, the walls of which radiating into a surrounding elastic medium, by using a standard scattering approach.

Mots clés: Circular symmetric waveguide; Cylindrical coordinates; Elastic Wave Scattering; SV and SH waves; Tensor Hankel transform; Fourier-Bessel transform; Hankel transform; 2D Fourier transform

Domaines: Acoustique [physics.class-ph]; Acoustique [physics.class-ph]; Mécanique [physics.med-ph]

Eric Ducasse, Slah Yaacoubi. The Hankel transform of first- and second-order tensor fields: definition and use for modeling circularly symmetric leaky waveguides.. Physics Procedia, 2010, 3 (1), pp.505-514. ⟨10.1016/j.phpro.2010.01.066⟩. ⟨hal-00870570⟩