Propagation of a long wave in a curved duct. II: Applications of matched expansion to long wave propagation through a hole with variable cross- section (Q1057745)

[For part I see the authors, ibid. 4, 79-91 (1983; Zbl 0542.76095).] Only the case in which the parameter \(\epsilon =ka\ll 1\) is considered in this paper, where k is the wave number and a is the characteristic radius of the cross-section of the hole. The general asymptotic expansion of the complex velocity potential of a long wave propagating in the hole with variable cross-section is obained by regular perturbation: The methods of matched asymptotic expansion are employed to calculate the reflection coefficients, scattering coefficients and radiation coefficients at the open ends of the hole when a long wave propagates through it, which may be open at both ends or only one end. Three examples of different kinds of holes are given to show the way to solve such two-dimensional or three-dimensional problems.