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[en] Thermodynamically consistent fractional Burgers constitutive models for viscoelastic media, divided into two classes according to model behavior in stress relaxation and creep tests near the initial time instant, are coupled with the equation of motion and strain forming the fractional Burgers wave equations. The Cauchy problem is solved for both classes of Burgers models using an integral transform method, and an analytical solution is obtained as a convolution of the solution kernels and initial data. The form of the solution kernel is found to be dependent on model parameters, while its support properties imply infinite wave propagation speed for the first class and finite speed for the second class. Spatial profiles corresponding to the initial Dirac delta displacement with zero initial velocity display features which are not expected in wave propagation behavior.