Decomposition of the unsteady wave patterns for Bessho form translating-pulsating source green function

In order to interpret the physical feature of Bessho form translating-pulsating source Green function, the phase function is extracted from the integral representation and stationary-phase analysis is carried out in this paper. The complex characteristics of the integral variable and segmentation of the integral intervals are discussed in m complex plane. In θ space, the interval [−π/2+φ, −π/2+φ-iɛ] is dominant in the near-field flow, and there is a one-to-one correspondence between the real intervals in m space and the unsteady wave patterns in far field. If 4τ>1 (τ is the Brard number), there are three kinds of propagation wave patterns such as ring-fan wave pattern, fan wave pattern and inner V wave pattern, and if 0<4τ<1, a ring wave pattern, an outer V and inner V wave pattern are presented in far field. The ring-fan or ring wave pattern corresponds to the interval [−π+α, −π/2+φ] for integral terms about k 2, and the fan or outer V wave pattern and inner V wave pattern correspond to [−π+α, −π/2) and (−π/2, −π/2+φ] respectively for terms about k 1. Numerical result shows that it is beneficial to decompose the unsteady wave patterns under the condition of τ≠0 by converting the integral variable θ to m. In addition, the constant-phase curve equations are derived when the source is performing only pulsating or translating.