Fluid mud often exists in coastal areas with an interface separating it from its upper water layer. When a surface wave propagates over a bed covered with water and fluid mud, it will cause an interfacial wave of the mud-water interface, which damps the surface wave and results in mass transport of fluid mud. Most researches about wave attenuation and mass transport of fluid mud are based on the assumption that the mud-water interface is unbroken. This assumption excludes the breaking interfacial waves that are known as an important mechanism responsible for mass and momentum transport between the two fluids. When the surface wave is long, its velocity field, which also serves as basic flows, may be susceptible to the Kelvin-Helmholtz (K-H) instability if the shears at the interface are strong enough. In the present paper, the critical conditions for the K-H instability to occur for the mud-water interface is investigated via linear stability analysis and numerical simulation. It is found that, for a K-H instability to occur, the Stokes boundary layer thickness induced by a surface wave must be large enough to penetrate the fluid mud layer and produce a strong shear at the interface. Meanwhile, a critical condition is found for a long surface wave to cause breakup of mud-water interface through K-H instability. This is practically instructive for waterway and harbor construction and protection because it predicts that a thicker mud layer is harder to be taken away by a surface wave. (C) 2019 Author(s).