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One of the main ingredients of existing multiscale numerical methods for homogenization problems is an accurate description of the coarse scale quantities, e.g., the homogenized coefficient via local microscopic computations. Typical multiscale frameworks use local problems that suffer from the so-called resonance or cell-boundary error, dominating the all other errors in multiscale computations. Previously, the second order wave equation was used as a local problem to eliminate such an error. Although this approach eliminates the resonance error totally, the computational cost of the method is known to increase with increasing wave speed. In this paper, the possibility of integrating perfectly matched layers to the local wave equation is explored. In particular, questions in relation with accuracy and reduced computational costs are addressed. Numerical simulations are provided in a simplified one-dimensional setting to illustrate the ideas.