  The diffusive-viscous wave equation (DVWE) is widely used in seismic
exploration since it can explain frequency-dependent seismic reflections in a
reservoir with hydrocarbons. Most of the existing numerical approximations for
the DVWE are based on domain truncation with ad hoc boundary conditions.
However, this would generate artificial reflections as well as truncation
errors. To this end, we directly consider the DVWE in unbounded domains. We
first show the existence, uniqueness, and regularity of the solution of the
DVWE. We then develop a Hermite spectral Galerkin scheme and derive the
corresponding error estimate showing that the Hermite spectral Galerkin
approximation delivers a spectral rate of convergence provided sufficiently
smooth solutions. Several numerical experiments with constant and discontinuous
coefficients are provided to verify the theoretical result and to demonstrate
the effectiveness of the proposed method. In particular, We verify the error
estimate for both smooth and non-smooth source terms and initial conditions. In
view of the error estimate and the regularity result, we show the sharpness of
the convergence rate in terms of the regularity of the source term. We also
show that the artificial reflection does not occur by using the present method.
