This paper investigates a thin sensor used to detect the position of an object in front of an ultrasonic transducer using changes in the radiation impedance. The sensor consists of a rectangular plate and a piezoelectric transducer, and the configuration is determined based on the results of a finite element analysis simulation. Stripe flexural vibration modes are generated on the plate, radiating sound waves into the air between the plate and the object. The radiation angle of these sound waves is dependent on the driving frequency, resulting in a change in the sound field and the electrical admittance characteristics. The sensing performance is examined using two resonant vibration modes. The sensor can determine the position of an object uniquely within a two-dimensional area, and the lower resonant mode gave a wider measurable range. The sensitivity is improved six-fold over that of our conventional sensor using the same sensing mechanism.