Theory and numerical simulation for a design of broadband constant beam pattern transducer

The theory and numerical simulations for the constant beam pattern (CBP) broadband transducer design are introduced. For a large spherical transducer, if the normal direction radial velocity distribution on the surface of the sphere transducer is a single Legendre polynomial, the far field angular beam pattern shows the same Legendre polynomial distribution due to a spherical Hankel function asymptotic approximation. Arbitrary radial velocity shading functions, including the equal sidelobe classic Dolph-Chebyshev shading, can be expanded by the Legendre series per Sturm-Liouville theory, such that the converging acoustic beam pattern displays the same angular shape as the original shading function itself for the broadband CBP transducer.