  The aim of this paper is to derive higher order energy estimates for
solutions to the Cauchy problem for damped wave models with time-dependent
propagation speed and dissipation. The model of interest is \begin{equation*}
u_{tt}-\lambda^2(t)\omega^2(t)\Delta u +\rho(t)\omega(t)u_t=0, \quad
u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x). \end{equation*} The coefficients
$\lambda=\lambda(t)$ and $\rho=\rho(t)$ are shape functions and
$\omega=\omega(t)$ is an oscillating function. If $\omega(t)\equiv1$ and
$\rho(t)u_t$ is an "effective" dissipation term, then $L^2-L^2$ energy
estimates are proved in [2]. In contrast, the main goal of the present paper is
to generalize the previous results to coefficients including an oscillating
function in the time-dependent coefficients. We will explain how the interplay
between the shape functions and oscillating behavior of the coefficient will
influence energy estimates.
