High accuracy calculating model for sound field simulation with DFT-based FDTD on polar-quaternion-based axis towards craft restoration

In this paper, an approach is proposed to improve the numerical precision at the sound field simulation by the finite-difference time-domain (FDTD) method. The FDTD method is a numerical solution for the wave equation, proposed by R. Courant. Although it has achieved to simulate wave propagation by numerical solution for the wave equation, it causes numerical dispersions due to their approximated approach. Thus, we also studied to improve the precision of the FDTD method and have proposed a calculation model based on spatial spectrum. The improved method allows the FDTD to detach the approximations by DFT-based computation, and it has achieved to reduce numerical dispersion. However, the discrete Fourier transform(DFT)-based computation has caused wrong results, which are the wave spatially-wrapped propagation, due to the periodic extension of the DFT. Therefore, we have proposed a new approach to avoid the problems by employing polar-quaternion for the wave equation. In the approach, the wave-equation is represented in polar-coordinate with quaternion, and the DFT-based partial differential operator is applied on space-time angle (argument) spectrum. The argument domain is originally periodic, and the problems of periodic extension can be avoided by using the proposed approach. The conducted numerical experiments indicated that the approach successfully prevented the FDTD method from causing problems of DFT-based spatial extensions.