In cases where a structure's excitation cannot be altered, the sound radiation from the structure must be minimized by modifying the structure within given design constraints. This dissertation considers minimizing sound radiated by an existing structure with minimal changes to the structure itself. To accomplish this the sound power radiated by a structure was written as a function of the normal surface velocity using the boundary element method. The feasibility of reducing radiated sound power with small patches of constrained layer damping material is proved. Small patches of constrained layer damping material are placed on the structure to effectively reduce radiated sound using reactive structural shearing intensity. The size, shape, and number of patches is explored. Two gradient methods are then used to minimize sound power radiated by a structure as a function of the area covered by constrained layer damping. The method of simulated annealing was used to minimized sound power as a function of damping patch area in cases where gradient methods proved unsuccessful. Reductions in sound power radiated at a single frequency of over 10 dB were achieved by covering just 1.1 percent of the total structural surface area with constrained layer damping material.