Airy方程的一个新解及其在流体线性稳定性分析中的应用

A New Solution to Airy Equation and Its Application to the Analysis of Hydrodynamic Stability

The key step in the analysis of hydrodynamic stability is to solve the Orr-Sommerfeld equation for the uniformly valid asymptotic expansion of its two singular inviscid solutions. Ref.1] defines the expanded function B_h(z,p,q) directly but gives no indication of the relation between B_h.(z,p,q) and Airy equation which is the related equation of the Orr-Sommerfeld equation. Thus, it is unable to explain why the inviscid solutions of Orr-Sommerfeld equation can be expanded in terms of B_k(z,p,q) In this paper a new solution B_k(z) of Airy equation is presented by modifying the Laplace's method, and the generalization of B_k(z) is just the function B_k(z,p,q) used in ref1]. It is shown that B_k(z, p, q) is practically a generalized Airy function.