A method of fundamental solutions (MFS) is proposed and analyzed for the ill-posed problem of finding the wave motion from given lateral Cauchy data in annular domains. A finite difference scheme, known as the Houbolt method, is applied for the time-discretisation rendering a sequence of elliptic systems corresponding to the number of time steps. The solution of the elliptic problems is sought as a linear combination of elements in what is known as a fundamental sequence with source points placed outside of the solution domain. Collocating on the boundary part where Cauchy data is given, a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. Tikhonov regularization is employed to generate a stable solution to the obtained systems of linear equations. It is outlined that the elements in the fundamental sequence constitute a linearly independent and dense set on the boundary of the solution domain in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document}-sense. Numerical results both in two and three-dimensional domains confirm the applicability of the proposed strategy for the considered lateral Cauchy problem for the wave equation both for exact and noisy data.