Abstract: In this paper, a class of three-dimensional wave equations with small fractional damping and sine-waving on the boundary is investigated. The boundary of the problem contains small parameters. Using the multi-scale method and the definition and properties of the Riemann-Liouville fractional derivative, the Taylor formula is applied to the original boundary value problem. The zero-order and first-order boundary value problems with respect to small parameters are obtained. Using the method of separating variables, introducing the detuning parameters, and analyzing the solvability conditions of the boundary value problem, the amplitude and phase of the zero-order approximate solution are obtained. Then, the uniformly valid behavior of the solution is illustrated by the theory of differential inequalities. Finally, the difference between the two-dimensional wave solution and the three-dimensional wave solution is analyzed. The changes in the amplitude of the three-dimensional wave with respect to relevant parameters are shown. The three-dimensional fluctuation boundary value problem shows that, when the boundary has small sinusoidal fluctuations and the external force perpendicular to the boundary changes regularly, the wave has an approximate solution. The instantaneous rate of change of the amplitude mode and phase of the solution is determined by the boundary value, the initially mode value and the value of the fractional derivative. It can be found that the amplitudes of the solutions of undamped two-dimensional waves and those of three-dimensional waves are obviously different. The two-dimensional wave only changes the amplitude and phase periodically, but the amplitude mode is constant, and the approximate solution is periodic. The three-dimensional wave changes in both amplitude mode and phase, but small parameters have little effect on its fluctuation.

Key words: multiple scales, fractional derivative, three-dimensional wave equation, the solvable condition