A nonstandard wave decomposition to ensure the convergence of Debye series for modeling wave propagation in an immersed anisotropic elastic plate

Abstract

When ultrasonic guided waves in an immersed plate are expressed as Debye series, they are considered as the result of successive reflections from the plate walls. Against all expectations, the Debye series can diverge for any geometry if inhomogeneous waves are involved in the problem. For an anisotropic elastic plate immersed in a fluid, this is the case if the incidence angle is greater than the first critical angle.Physically, this divergence can be explained by the energy coupling between two inhomogeneous waves of same kind of polarization, which are expressed by conjugate wavenumbers. Each of these latter inhomogeneous waves does not transfer energy but a linear combination of them can do it. Mathematically, this is due to the fact that inhomogeneous waves do not constitute a basis orthogonal in the sense of energy, contrarily to homogeneous waves. To avoid that difficulty, an orthogonalization of these inhomogeneous waves is required. Doing so, nonstandard upgoing and downgoing waves in the plate are introduced to ensure the convergence of the new Debye series written in the basis formed by these latter waves. The case of an aluminum plate immersed in water illustrates this study by giving numerical results and a detailed description of the latter nonstandard waves. The different reflection and refraction coefficients at each plate interface are analyzed in terms of Debye series convergence and of distribution of energy fluxes between the waves in the plate. From that investigation, an interesting physical phenomenon is described for one specific pair “angle of incidence/frequency”. For this condition, the quasi-energy brought by the incident harmonic plane wave crosses the plate without any conversion to reflected waves either at the first interface or at the second interface. In this zone, there is a perfect impedance matching between the fluid and the plate.