  In this paper, we study the Cauchy problem for a wave equation with general
strong damping $-\mu(|D|)\Delta u_t$ motivated by [Tao, Anal. PDE (2009)] and
[Ebert-Girardi-Reissig, Math. Ann. (2020)]. By employing energy methods in the
Fourier space and WKB analysis, we derive decay estimates for solutions under a
large class of $\mu(|D|)$. In particularly, a threshold
$\lim\nolimits_{|\xi|\to\infty}\mu(|\xi|)=\infty$ is discovered for the
regularity-loss phenomenon, where $\mu(|\xi|)$ denotes the symbol of
$\mu(|D|)$. Furthermore, we investigate different asymptotic profiles of
solution with additionally $L^1$ initial data, where some refined estimates in
the sense of enhanced decay rate and reduced regularity are found. The derived
results almost cover the known results with sufficiently small loss.
