Analysis of two types of harmonic waves in a Zener viscoelastic material

The governing equations of the Zener viscoelastic model in 1d, in terms of either the displacement or the stress, are found to involve two d'Alembertians. Two types of waves are investigated: (i) Spatially attenuated and temporally periodic waves (SATP), and (ii) Temporally attenuated and spatially periodic waves (TASP). In the SATP waves, the attenuation increases monotonically with the frequency ω at a decreasing rate and then saturates to a constant value, while the phase velocity initially increases and then saturates to a constant value at high frequencies. On the other hand, the group velocity initially increases monotonically, reaches a maximum value, and then decreases before saturating to a constant value. For a given set of material constants, the group velocity is always greater than the phase velocity. The TASP waves are studied in terms of the wave number k where the characteristic ODE for the time function is a cubic polynomial. To obtain d'Alembert solutions, additional conditions need to be satisfied.