This article describes an approach to array design that results in the construction of transducers producing precisely controlled radiation patterns. The method of dense random arrays employs an array of equally weighted elements, randomly distributed with a density distribution that matches a desired shading. As the spatial density of elements increases, the theoretical radiation pattern approaches that of the ideal shading. If the ideal shading is real (e.g., for a symmetric beam), all elements in the random array will be either in phase or in phase opposition, and the array will require a single amplifier coupled with a phase inverter. If the ideal shading is both real and positive (e.g., Gaussian or Blackman–Harris shading), all elements will be in phase and will require only one amplifier. This approach was used to design a linear array to generate a 2° wide Gaussian beam. The elements had a minimum spacing of one-quarter of a wavelength and were placed using a Monte Carlo approach. A theoretical sidelobe level of −21.8 dB was achieved, with good agreement in the central beam (2.4° beamwidth predicted). A 500-kHz acoustic transducer with this element distribution was built using 110-μm-thick polyvinylidene fluoride (PVDF). Experiments reveal a radiation pattern very close to the predicted pattern, with −20.8-dB sidelobes.