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We prove a  reducibility  result for a class of quasi-periodically forced linear wave equations on the   $d$  -dimensional torus   ${\mathbb T}^d$   of the form   $$\partial_{tt} v - \Delta v + {\varepsilon} {\cal P}(\omega t)[v] = 0,$$   where the perturbation   ${\cal P}(\omega t)$   is a second order operator of the form   ${\cal P}(\omega t) = - a(\omega t) \Delta - {\cal R}(\omega t)$  , the frequency   $\omega \in {\mathbb R}^\nu$   is in some Borel set of large Lebesgue measure, the function   $a : {\mathbb T}^\nu \to {\mathbb R}$   (independent of the space variable) is sufficiently smooth and   ${\cal R}(\omega t)$   is a time-dependent finite rank operator. This is the first reducibility result for linear wave equations with unbounded perturbations on the higher dimensional torus   ${\mathbb T}^d$  . As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is  reducible .

A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$