  In this paper, we rigorously investigate the truncation method for the Cauchy
problem of Helmholtz equations which is widely used to model propagation
phenomena in physical applications. The method is a well-known approach to the
regularization of several types of ill-posed problems, including the model
postulated by Regi\' nska and Regi\' nski \cite{RR06}. Under certain specific
assumptions, we examine the ill-posedness of the non-homogeneous problem by
exploring the representation of solutions based on Fourier mode. Then the
so-called regularized solution is established with respect to a frequency
bounded by an appropriate regularization parameter. Furthermore, we provide a
short analysis of the nonlinear forcing term. The main results show the
stability as well as the strong convergence confirmed by the error estimates in
$L^2$-norm of such regularized solutions. Besides, the regularization
parameters are formulated properly. Finally, some illustrative examples are
provided to corroborate our qualitative analysis.
