Effect of boundary conditions on the ultrasonic‐beam characteristics of circular disks

The ultrasonic‐beam characteristics of piston (case 1) are first reviewed and then two radially symmetric velocity distributions—a simply supported disk $Ussd [1 − (r/R)2]$ (case 2), and a clamped disk $Ucd[1 − (r/R)2]2$ (case 3)—are used to calculate the axial intensity and directivity of the resulting ultrasonic beam. The oscillations in axial intensity for both cases 2 and 3 are less than that for case 1. The axial intensities are compared both for equal maximum velocities and equal source strengths. According to the Fraunhofer equation, the diverging portion of the beam from a piston begins at approximately 1.6 R²/λ; not at the position of the last axial maximum, approximately R²/λ. The position of the last axial maximum lessens from R²/λ for case 1, to 3/4 R²/λ for case 2, and 9/16 R²/λ for case 3, Directivity calculations show that the conical half‐angle for case 2 is 20% greater than for case 1, and for case 3, it is 60% greater than case 1. The first side lobes for cases 2 and 3 are 7 and 13 dB down, respectively, from the first side lobe of case 1. The second side lobes for cases 2 and 3 are 9 and 18 dB down, respectively, from the second side lobe of case 1.