Abstract: Only the case in which the parameter e=ka<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 obtained 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 at one end. Three examples of different kinds of holes are given to show the way to solve such two-dimensional or three-dimensional problems.

Key words: Beltrami flow, tensor denotation, symmetry, chaotic phenomena