The minimum sound pressure is an important aspect of noise control. This means that minimal or no noise at a location will be experienced at a certain frequency. In many engineering applications, it is desirable to compute and assign the frequency corresponding to the minimum sound pressure. This paper presents three novel methods for the prediction of frequencies corresponding to the minima of radiated sound pressure. Two of them are developed for determining zero sound pressure frequencies, which correspond to a response close to zero. They are based on the application of linear matrix algebra methods in conjunction with the fundamental definitions for the existence of local minima. The other is developed to solve for frequencies of the minimum response points corresponding to a zero slope in the frequency response function curve by using the dichotomy method. In addition, an inverse structural modification for the assignment of the zero sound pressure frequency and antiresonant frequencies is presented. At these frequencies, the modification causes the selected location to experience the minimum sound pressure, while the other locations selected on the structure do not vibrate. Numerical examples of a simply supported plate in air and water are analyzed to demonstrate the effectiveness and accuracy of the proposed approaches.