This paper is concerned with devising an efficient numerical method for the piezoelectric equations in an unbounded domain, which plays a fundamental role in design and analysis of microacoustic devices with piezoelectric substrate. We make use of the perfectly matched layer method to transform the underlying problem as a surrogate in a bounded domain, which is further solved by a sparse wavelet element method. The latter method can be viewed as a combination of a wavelet element method and a sparse grid method. The numerical results are performed to show the proposed method is very efficient and outperforms the usual finite element method. It can be naturally extended to two/three dimensional problems in an unbounded domain whose boundary consists of line segments or rectangles parallel to coordinate lines or planes.