The nonlinear propagation of directional spherical waves generated in an unbounded inviscid ideal gas by vibratory motions with small but finite amplitude and moderate frequency of a spherical body is considered. Starting with a regular perturbation expansion for a velocity potential in the near field, a higher‐order problem is investigated in the far field up to the shock formation distance. It is thereby shown that, in the far field concerned, a well‐known simple far‐field equation remains valid for the radial velocity u@B|r
including higher‐order corrections up to O
<−1/ϵ ln ϵ), where r
is a nondimensional radial coordinate and ϵ(≪1) is the expansion parameter of the expansion. A boundary condition appropriate to the equation, which ensures the matching of a far‐field solution with a near‐field solution, can be determined from the near‐field solution obtained by the regular perturbation procedure. As an application of the theory, the third‐order problem is solved for weakly nonlinear acoustic waves radiated by a pulsating sphere. It is further shown that, for weakly nonlinear cylindrical waves with moderate frequency, a similar far‐field equation becomes invalid at the third approximation in the far field up to the shock formation distance.