A new variant of nonlinear wave equation is obtained for a solitary nonlinear cylindrical radial displacement wave to which an approximate problem-solving method tested against the problem on a plane longitudinal wave is applied. A numerical analysis is carried out for the wave initial profile in the form of the Macdonald function. About 30 variants of initial parameters are studied numerically, namely, three variants of materials (aluminum, copper, steel), three variants of the wave bottom, three variants of initial maximal amplitude, and one variant that unites two harmonics. For every variant a set of nine two-dimensional graphs “wave shape–travel distance” that shows the level of wave distortion is plotted. This profile turns out to be unusual from the point of view of profile shape. The profile shapes considered before (sinusoidal, bell-shaped, in the form of Whittaker and Hankel functions) changed substantially forming a plateau on the top and then two humps, manifesting in this way strong distortion. The shape of the Macdonald function is hardly distorted, the profile bottom increases slightly and only the profile maximum value increases considerably, changing the steepness of the profile left part.