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In this article, we study the hyperbolic Anderson model in dimension 1,driven by a time-independent rough noise, i.e. the noise associated with thefractional Brownian motion of Hurst index $H \in (1/4,1/2)$. We prove that,with appropriate normalization and centering, the spatial integral of thesolution converges in distribution to the standard normal distribution, and weestimate the speed of this convergence in the total variation distance. We alsoprove the corresponding functional limit result. Our method is based on aversion of the second-order Gaussian Poincar\'e inequality developed recentlyin [27], and relies on delicate moment estimates for the increments of thefirst and second Malliavin derivatives of the solution. These estimates areobtained using a connection with the wave equation with delta initial velocity,a method which is different than the one used in [27] for the parabolicAnderson model.

Hyperbolic Anderson model with time-independent rough noise: Gaussian fluctuations