The study of wallpaper patterns has traditionally been carried out as a branch of discrete geometry, which uses a fruitful interplay between geometry and algebra to classify patterns. But if one is interested in generating repeat patterns, the field of analysis has much to offer. In this article, we use ideas from PDEs and Fourier analysis to produce recipes for creating interesting wallpaper patterns with any desired symmetry group. As a side benefit, our approach enables us to show what it would be like for an infinite planar membrane to vibrate like a wallpaper drum.