  We solve the shifted wave equation \begin{align*} \frac{\partial^2}{\partial
t^2}\varphi(x,t)=(\Delta_x+\rho^2)\varphi(x,t) \end{align*} on a non compact
simply connected harmonic manifold with mean curvature of the horospheres
$2\rho>0$. We give an explicit representation of the solution as the inverse
dual Abel transform of the spherical means of there initial conditions using
the local injectivity of the Abel transform and symmetry properties of the
spherical mean value operator. Furthermore we investigate the wave equation
using the Fourier transform on harmonic manifolds of rank one. Additionally we
show an analogous of the classical Paley-Wiener theorem and use it to show an
asymptotic Huygens principle as well as asymptotic equidistribution of the
energy of a solution of the shifted wave equation under assumptions on the
$\mathbf{c}$-function.
