Reducing unwanted noise radiating from a vibrating structure continues to be a major challenge in engineering design. In this paper, high impedance structures characterized with uniquely identifiable resonance modes are considered for sound power reduction using an enclosing surface. An optimization strategy for the thin enclosure design is presented. A close-fitting and equipment-mounted enclosure is attached to the vibrating structure with translational springs. To model the coupled system, a new compact and symmetric system matrix analysis using component modal synthesis and fluid displacement potential to deal with the coupled system is developed. This leads to a new application of optimization to the coupled system to reduce the radiated sound power. Numerical optimizations are performed for a plate enclosure. It is shown that large reductions in radiated sound power are possible with optimally positioned and rated attachment springs.