In the near future, decentralized network systems consisting of a huge number of sensor nodes are expected to play an important role. In such a network, each node should control itself by means of a local interaction algorithm. Although such local interaction algorithms improve system reliability, how to design a local interaction algorithm has become an issue. In this paper, we describe a local interaction algorithm in a partial differential equation (or PDE) and propose a new design method whereby a PDE is derived from the solution we desire. The solution is considered as a pattern of nodes' control values over the network each of which is used to control the node's behavior. As a result, nodes collectively provide network functions such as clustering, collision and congestion avoidance. In this paper, we focus on a periodic pattern comprising sinusoidal waves and derive the PDE whose solution exhibits such a pattern by exploiting the Fourier method.