  In this paper, we are interested in the Cauchy problem for the viscoelastic
damped wave equation with memory of type I. By applying WKB analysis and
Fourier analysis, we explain the memory's influence on dissipative structures
and asymptotic profiles of solutions to the model with weighted $L^1$ initial
data. Furthermore, concerning standard energy and the solution itself, we
establish singular limit relations between the Moore-Gibson-Thompson equation
with memory and the viscoelastic damped wave equation with memory.
