Revisiting Manne et al. (2000): A Reformulation and Alternative Interpretation Under the Modified Internal-Energy Theory of Second-Sound

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Mathematics and Natural Sciences


Returning to Manne et al. (2000), wherein a class of hyperbolic reaction–diffusion-acoustic models is considered, we show that the inexact transport equation derived by these authors becomes the exact equation for the propagation of second-sound (i.e., thermal waves) in a rigid solid when reformulated under the modified internal energy model, one of several formulations of what has come to be known as hyperbolic heat conduction theory. We then compare/contrast this PDE with/to its exact version, which is based on Cattaneo’s equation for the thermal flux, in the context of a signaling problem involving weak discontinuities of second-sound propagating in a solid rod. Our analysis also reveals a connection between the modified internal energy model and what has been termed “inertial theory”, a competing theory of second-sound in solids, and suggests a generalization of the quantity known as the “thermasy”. We conclude with a discussion on the application of the present findings to a well known, widely applicable, class of nonlinear wave equations.

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Wave Motion

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