The direct boundary element method is proposed in this thesis to solve acoustic radiation problems as well as to achieve regional noise cancellation in half space with uniform finite impedance over the surface. The boundary integral equation and half space Green's function were derived to accomplish these goals. Those formulations were verified by comparing numerical simulations with theoretical solutions as well as experimental results. In addition, the above formulations were extended to achieve regional noise cancellation in half space by applying the boundary element method;Two methods were investigated to obtain noise cancellation in desired regions. They are the iterative control method and the coupled equation method. A set of Fortran programs including discretizing of geometries, incorporating boundary integral equations, and accommodating the noise cancellation technique were developed. Various problems concerning ill-conditioned matrices in numerical simulation and practical application of noise cancellation technique were discussed as well. Finally, data banks for various configurations of sound sources were set up for quick reference of the locations and driving functions of secondary sources. Thus, noise reduction in a designated area is shown to be feasible;A 6" speaker was used to simulate a noise source with uniform surface velocity. In addition, a ribbed aluminum plate with the dimension 71.12cm x 60cm was used to simulate a noise source with variable surface velocity. Four 10" speakers were used as secondary sources to achieve noise reduction in desired regions at certain frequencies. A multi-channel digital/analog converter was used in order to control desired driving functions for each individual secondary sources. The computer-controlled scanning system including a 2-channel controller, 2-D scanner, and stepping motors was used to place a quarter-inch microphone at certain locations. The acoustic pressure on a 120cm by 120cm plane at various distances above the source plane was measured. A Bruel and Kjaer model 2032 FFT analyzer was used to acquire and process signals from the microphone. The experimental results agreed well with numerical simulations. This indicated that the proposed noise cancellation technique attenuated the acoustic noise level successfully.