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-i?] 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.