Wave propagation in thermoelastic media with relaxation time

In this manuscript, we investigate the propagation of plane time-harmonic waves in an infinite linear thermoelastic medium, assuming a known wavelength. For the thermodynamic response, we adopt the Lord–Shulman (L-S) theory of generalized thermoelasticity. Our analysis reveals two uncoupled, undamped transverse waves, along with possibly two to three coupled, temporally damped, and dispersive longitudinal waves. The speed of transverse waves remains unaffected by thermal effects, whereas longitudinal waves exhibit thermal dependence. Indeed, we observe a time-damped longitudinal quasi-elastic wave, with its amplitude exponentially decreasing to zero as time approaches infinity. Furthermore, there may be two possible longitudinal quasi-thermal waves that are damped in time with different decreasing rates or a single plane harmonic in time longitudinal quasi-thermal wave. We also examine surface wave propagation in a semi-infinite thermoelastic medium, considering a stress-free, heat-exchanging half-space boundary. The explicit forms of the dispersion relation and the secular equation are derived. Numerical computations for both plane and surface waves are conducted using a specific model, with graphical representations of the results and corresponding analytical discussions. Finally, we present generalized observations regarding the surface wave profile.