The exact partial wave series for the scattering by a sphere centered on an ideal Bessel beam was recently given by Marston ["Scattering of a Bessel beam by a sphere," J. Acoust. Soc. Am. 121, 753-758 (2007)]. That series is applied here to solid elastic spheres in water and to an empty spherical shell in water. The examples are selected to illustrate the effect of varying the beam's conical angle so as to modify the coupling to specific resonances in the response of each type of sphere considered. The backscattering may be reduced or increased depending on properties of the resonance and of the specular contribution. Changing the conical angle is equivalent to changing the beamwidth. Some applications of the Van de Hulst localization principle to the interpretation of the partial wave series and to the interpretation of the scattering dependence on the beam's conical angle are discussed. Some potential applications to the analysis of the scattering by spheres of more general axisymmetric beams are noted.