  In this paper, we investigate two hyperbolic flows obtained by adding forcing
terms in direction of the position vector to the hyperbolic mean curvature
flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by
using support function, we reduce it to a hyperbolic
Monge-Amp$\grave{\rm{e}}$re equation successfully, leading to the short-time
existence of the flow by the standard theory of hyperbolic partial differential
equation. If the initial velocity is non-negative and the coefficient function
of the forcing term is non-positive, we also show that there exists a class of
initial velocities such that the solution of the flow exists only on a finite
time interval $[0,T_{max})$, and the solution converges to a point or shocks
and other propagating discontinuities are generated when
$t\rightarrow{T_{max}}$. These generalize the corresponding results in
\cite{klw}. For the second hyperbolic flow, as in \cite{hdl}, we can prove the
system of partial differential equations related to the flow is strictly
hyperbolic, which leads to the short-time existence of the smooth solution of
the flow, and also the uniqueness. We also derive nonlinear wave equations
satisfied by some intrinsic geometric quantities of the evolving hypersurface
under this hyperbolic flow. These generalize the corresponding results in
\cite{hdl}.
