In order to find the electric field on the concave sphere, a infinite series of point charge pairs {q_i, -q_i} are placed symmetrically with coordinates {X_i, -X_i} if we choose the x direction to be perpendicular to the plane of the transducer and the origin to be situated at its center. The pair {q_i, -q_i+1} are image charges to the plane. It can be proved that, and (where a > H, the center thickness of the transducer). Thus, it can be proved that the configuration of the point charge pairs satisfies the Laplace's equation and the boundary conditions. Consequently, the electric field intensity at the point {x, y} on the sphere can be given by:

where y is the distance from the axis. This formula can be calculated by a computer and some numerical results can be given. By numerical calculation it can be found that the formula (1) can be expressed in a simple function with high accuracy if the radius of the concave sphere R >> H or R >> d, the diameter of the piezocomposite. That read

(2)

where is the unit vector pointing to the center of the sphere. Applying the approximation of the weak electromechanical coupling and formula (1) or (2), the characteristics of the sound field the pulse echo and the frequency spectrum of this transducer can be investigated numerically. The numerical results are consistent with the experimental by IAAS. Finally, some laboratory results of this wideband transducer made of nonhomogeneous thickness p-5 material can be given in this paper.