###### Abstract  
This paper proposes a time-shifting boundary element method in the time domain to calculate the radiating pressures of an arbitrary object pulsating at eigenfrequencies of the interior (i.e., interior resonance frequencies). In this paper, the frequency shifting is time-step-dependent and could be viewed as an iterative, or relaxation, technique for the solution of the problem. The proposed method avoids numerical problems due to the internal resonance frequency by initializing the iteration with each scaled frequency. The scaled frequency is approximately equal to the  
methods. The wave-vector-time domain method is only suitable for simple objects. When using the Kirchhoff integral equation method, the error increases with forward marching time. [PERSON] and [PERSON] [6] distributed the interior surface source to solve the singularity and non-unique solutions of the conventional boundary integral equation. The interior source surface is selected through an error estimate of the numerical integration. In [PERSON] and [PERSON] [7], a general form of the hypersingular BIE was developed for 3-D acoustic wave problems, containing at most weakly singular integrals. The weakly singular form of the hypersingular BIE for exterior acoustics problems is highly efficient and competitive. The convolution quadrature method shown in [PERSON] [8] takes into account the time dependency in spatial discretizations for linear time-invariant non-homogeneous evolution equations and is more efficient than the Galerkin or collocation time approximations. [PERSON] [9] regularized the hypersingular integral in the Helmholtz integral equation. The computations were carried out by the standard Gaussian quadrature procedure without any special quadrature scheme or excessive quadrature points near singular points. [PERSON] and [PERSON] [10] showed that the smoothness requirement is a serious issue associated with the hypersingular BIE and proposed an improved weakly singular form of the hypersingular BIE. This new form involves only tangential derivatives of the density function, and its discretization is thus easier to perform. [PERSON] et al. [11] derived the non-hypersingular boundary integral equation for problems involved in the composite Helmholtz integral equation at the interior resonance frequency. The Burton-Miller-type integral equation was derived by [PERSON] et al. [12] to stabilize numerical computing at the interior resonance frequency. [PERSON] and [PERSON] [13] proposed a robust iterating BEM in the time domain to solve scattered waves for an arbitrary object without considering the interior resonance frequency. [PERSON] et al. [14] employed the convolution quadrature method for discretization in time and the Galerkin boundary element method for space discretization. An a priori cut-off strategy was presented to efficiently reduce errors in numerical integration due to discretization. [PERSON] et al. [15] included a controlling factor in the boundary integral equation for the time-domain BEM to stabilize numerical computing at the interior resonance frequency. [PERSON] and [PERSON] [16] suppressed low-order fictitious internal modes, which cause non-unique solutions within the effective frequency range of boundary element calculations, by the newly formulated time-domain CHIEF method. The fast multipole method, FMM-BEM, as developed by [PERSON] et al. [17], has verified that the drawback of fully populated system matrices in BEM could be improved. [PERSON] and [PERSON] [18] regularized the hypersingular Helmholtz integral ([PERSON] and [PERSON] formulation) according to the identities from the associated Laplace equation. In the scheme, weakly singular integrals are computed. The time-domain equivalent source method (TESM) was developed by [PERSON] et al. [19] to solve radiating waves. The TESM was formulated as a time averaging scheme to reduce instability due to errors amplified exponentially with time.