The simulation model (SM) of dolphin click source originally presented by [Dubrovskiy and Giro, ‘‘Echolocation in Bats & Dolphins,’’ Chap. 10, 59–63 (2003)] was further developed by Gladilin, Dubrovskiy, Mohl, and Walberg, Acoust. Phys. 50, 463-468 (2004). The SM describes movements of a solid sphere frozen in a compliant ring. A jet of air in a duct connecting air cavities brings the sphere into self-oscillations. The SM allows calculating the waveform and sound pressure of the generated clicks. Here the mathematical and simulation model (MSM) of click production in dolphins is further developed. The MSM describes radiation of clicks by bodies comprising of waterlike tissue with different shapes (a ring, a blunted cone) pulsating and (or) oscillating under action of muscles tension. Calculations are made for the energy efficient case when acting force is consecutively applied not to the entire body but to its thin layer, which resulted in action of the progressing wave array. Directivity patterns of such a waterlike compliant body in main directions are similar to that of a rigid body with the same shape and dimensions. Comparisons are made of MSM predictions with properties of an actual radiation field of dolphins (in clicks wave forms, sound pressure levels, and directivity patterns) and dimensions of biological structures supposedly responsible for click production in dolphins.