The equations of motion for the longitudinal (i.e. in-plane) wave response of an elastic-viscoelastic-elastic laminated beam containing a generalized shear Voigt viscoelastic core are derived along with closed form expressions for the complex flexural wavenumbers. Comparisons with experiment are made in a mechanical mobility format. The existence of this form of damping is supported through the agreement observed between theory and high resolution sine wave dwell measurements of the driving point mechanical mobility of two damped laminates. The generalized shear Voigt damping core is shown to have a significant effect on the damping performance for anti-resonance modes of the laminate with free-free boundary constraints within the spectral band governed by out-of-phase, in-plane motion between the base and constraining layers. This spectral band for shear Voigt damping is determined from consideration of the limiting case of a lumped parameter mass-spring-mass model. In addition, the resonance modes of the free-free laminate showed a waveguide behavior where the damping is equivalent to the internal damping of the base structure. Finally, this analysis coupled with experimental results show the shear Voigt damping mechanism is dominant in elastic-viscoelastic laminates with relatively soft damping cores and/or stiff constraining layers.