Energy dissipation and attenuation under high-loss conditions

The conventional relation Δ = π tan δ between free-vibration logarithmic decrement Δ and loss tangent or internal friction was shown by Parke to fail for high damping conditions. Here it is shown that the relation fails in the opposite direction to that found by Parke when Δ is a spatial decrement derived from attenuation of a plane wave travelling in a passive medium. The following exact relations are found for this case: Δ = 2πα/β = 2π | tan ½δ |, Q -1 = 2(α/β)/| 1 - (α/β) 2 | = | tan δ |, where α and β are the attenuation factor and phase factor of the wave in the material and Q is the quality factor of the material. Frequency response curves are shown for R [identical with] Q -1 /(Δ/π) >or= 1, Q -1 and Δ/π for the cases of both non-resonant absorption of power and resonant absorption. For resonant absorption the phase lag δ may exceed π/2. The deviation between the present results and those of Parke is shown to arise from intrinsic difficulties in precisely defining and measuring period (free vibration) or wavelength (wave motion) under high-loss conditions.