The continuous integrals and integral equations which form the theory of Nearfield Acoustic Holography for planar and odd-shaped source boundary surfaces are reviewed, and the approximations necessary to reduce these to a set of finite and discrete operations are developed. These equations represent the solution of the Helmholtz equation with specified boundary conditions by Green's function methods.;Two computational methods for reconstructing planar source boundary fields from planar holograms are developed. The first method is an approximation of the continuous solution method which the convolution theorem of Fourier Transforms provides. This method has a high sensitivity to noise; it is shown how this problem is partially alleviated by the use of a high spatial frequency filter.;As an alternative to this reconstruction method, a conjugate gradient descent method is developed based on the finite and discrete propagation method discussed. Although this reconstruction method requires more time than the first method presented, it is relatively insensitive to noise, and it extends the feasible reconstruction range, or distance between hologram and boundary.;The reduction to finite and discrete form, by a Finite Element technique, of the relationship between the Dirichlet and Neumann boundary conditions for an odd shaped surface is reviewed. To develop a unique relationship, a knowledge of the boundary surface's characteristic frequencies is essential, and a method for detecting these frequencies for an odd-shaped boundary is presented. A comparison of the characteristic frequencies predicted by this method and those given by theory is presented for a spherical surface.;In analyzing the reduction to discrete form, of the propagation of planar holograms (a record of the radiated field over a plane above the boundary plane), four new methods for representing the Green's functions numerically are developed. The relationship of two earlier methods to these new forms is established. The results of numerical testing which demonstrate the effectiveness of the new representations are presented.;A technique for reconstructing the odd-shaped surface boundary conditions from a hologram of general two-dimensional shape is developed. Results from a numerical study which support this technique are presented. Two cases are considered in this study: reconstruction of a spherical boundary from a planar hologram, and reconstruction of a spherical boundary from a concentric, spherical hologram.
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