The Cauchy problem for a wave equation with a time periodic coefficient is considered. We prove that if one of the initial data is a compactly supported smooth function and the other initial data is zero, then the energy of the solution of the Cauchy problem grows exponentially. This result is proved by applying the unstable properties of Hill’s equation.
"A remark on parametric resonance for wave equations with a time periodic coefficient." Proc. Japan Acad. Ser. A Math. Sci. 87 (8) 128 - 129, October 2011. https://doi.org/10.3792/pjaa.87.128