On the behavior of the Helmholtz equation least-squares method solutions for acoustic radiation and reconstruction

This thesis considers reconstruction of time-harmonic acoustic field from arbitrarily shaped vibrating bodies by the Helmholtz equation least-squares (HELS) method. HELS is a series expansion method developed for near-field acoustical holography, in which certain families of particular solutions to the Helmholtz equation are used for approximation of acoustic quantities. The expansion coefficients are found by minimizing the least-squares error between the assumed-form solution and field pressure data with constraints imposed to ensure stability of a solution. A numerical study of HELS including discretization, regularization, and the choice of expansion functions is carried out.;It is found that reconstruction problem may be well conditioned for small standoff distances and the constraints may be omitted. However, in this case HELS depends on the validity of the Rayleigh hypothesis. Namely, if the Rayleigh hypothesis is invalid, the number of measurements in HELS must be increased. On the other hand, regularization allows the number of measurements to be reduced.;Regularization by quadratic constraints relies on the knowledge of the exact solution's norm and rate of convergence of the approximating sequence. These quantities are not readily available in practice. Hence, a number of direct regularization methods with regularization parameter selected by error-free parameter-choice methods are considered.;The choices of expansion functions (localized spherical waves, distributed spherical waves, and distributed point sources) are compared. It is found that distributed spherical waves may lead to a better accuracy of reconstruction. Positioning of the auxiliary sources is examined. A second layer of measurements may be used to determine successful regularization strategies.;Numerically generated data in 2D and experimental data in 3D are used in examples. The main conclusion of the thesis is that HELS with the right choice of expansion functions and regularization can be an effective tool in reconstruction of sound radiation.