An integral equation relates the single frequency vertical distribution of ambient ocean noise to the measured array output when the noise power at the array arises from the superposition of uncorrelated plane waves. This integral equation has an infinite number of solutions. However, from the physical viewpoint, there is one solution, referred to as the “principal solution,” which is the most plausible representation of the true noise field. The “principal solution,” is precisely defined, and then classical Hilbert space theory is used to determine this solution from a finite number of array measurements. The number of measurements required depends only on the interelement spacings of the array. In selecting this finite number of steering angles to make the array measurements, any choice that yields independent samples for a certain finite sequence of trigometric functions is sufficient. Not surprisingly, these measurements contain all the information available in the observed distribution, the array output, and an explicit expression is given for the observed distribution. Special array configurations are exhibited for which the “principal solution” is a partial sum in the Fourier expansion of the true noise field.