An inequality for longitudinal and transverse wave attenuation coefficients

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.

6 pages, 2 figures