A multi-walled, tubular baseball bat has a barrel portion with a mid-section wherein the radial stiffness of the overall barrel wall varies along the barrel length to provide an enlarged sweetspot, improved soft feel and performance, plus unique sounds upon impact. The bat has a frame with a barrel portion of consistent diameter. A secondary member, or members, of tubular form extend internally along the barrel. The secondary member provides the required radial stiffness variation by: 1) variations in the thickness of the wall of the secondary member or by, 2) secondary members with unique geometric external surface profiles or by, 3) the presence of functional air cavities, with or without closed ends, between the main bat frame and the secondary member or members or by, 4) the presence of numerous annular secondary members located side by side less than one-half the length of the barrel portion.