It was shown in 1966 by MacDonald that the conventional relation Λ=πtan δ between spatial logarithmic decrement and loss tangent fails for high damping conditions and that the exact relation reads Λ = 2πtan δ/2. This means that if one calculates tan δ from the spatial logarithmic decrement and then uses the conventional relation, a value is found for tan δ which is too small. Moreover, in 1966 Parke showed that the same conventional relation Λ = πtan δ between the free-vibration logarithmic decrement Λ and the loss tangent, fails in the opposite direction, under conditions of high damping.
It is the aim of this paper to show that: (i) the relation deduced by MacDonald is also valid for free vibrations provided that the proper, i.e. complex, frequency is used; (ii) the relations given by MacDonald can be deduced in a very simple way by making use of the concept of characteristic impedance.
Moreover, our analysis of the energy dissipation in the material showed that in the relations for the components of the relevant complex elastic modulus, M1 and M2, where M* = M1+iM2, in highly attenuating materials the velocity of energy transport c1 is a more proper parameter than c the velocity of propagation.