An inequality for longitudinal and transverse wave attenuation coefficients

TL;DR

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Contribution

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Abstract

Total absorption, defined as the net flux of energy out of a bounded region averaged over one cycle for time harmonic motion, must be non-negative when there are no sources of energy within the region. This passivity condition places constraints on the non-dimensional absorption coefficients of longitudinal and transverse waves, $\gamma_L$ and $\gamma_T$, in isotropic linearly viscoelastic materials. Typically, $\gamma_L, \, \gamma_T$ are small, in which case the constraints imply that coefficients of attenuation per unit length, $\alpha_L$, $\alpha_T$, must satisfy the inequality ${\alpha_L}/{\alpha_T} \ge { 4c_{T}^3} / { 3c_{L}^3}$ where $c_L$, $c_T$ are the wave speeds. This inequality, which as far as the author is aware, has not been presented before, provides a relative bound on wave speed in terms of attenuation, or {\it vice versa}. It also serves as a check on the consistency of ultrasonic measurements from the literature, with most but not all of the data considered passing the positive absorption test.