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

The shifted Wave equation on non flat harmonic manifolds