## 3 Numerical Results  
### A Sphere Pulsating at Internal Resonance Frequencies  
Figure 4 shows a pulsating sphere with a radius (\(R\)) of 1 m in an infinite air domain. The mean density of the air is 1.21 kg/m\({}^{3}\), and the speed of sound is 343 m/s. Figure 4: Sphere in an infinite air domain. Figure 3: The iterating process for computing the radiating pressure. Internal Eigenfrequency Problem  
The computing conditions are listed in Table 1, where K is the wave number. The pulsating frequencies are at eigenfrequency and high frequencies, i.e., KR = \(\pi\)\(\sim\)5. The normal velocity amplitude is assumed to be 1. It is noted that when KR is equal to \(\pi\), the frequency 171.5 Hz is the first internal eigenfrequency. The number of elements is dependent on the factor \(l/\lambda\). \(l\) is the maximum side length of the elements, and \(\lambda\) is the wavelength. The recommended value of \(l/\lambda\) is 0.065-0.069 according to the numerical tests carried out in this section. Moreover, the selection of the total number of time steps (N) can refer to the ratio of \(\Delta t\) and the wave period. In this paper, this ratio (\(\Delta t/\) wave period) is 0.11-0.14, and robust convergence can be attained. Figure 5a indicates that the iterating process is divergent at the first internal eigenfrequency without shifting the time step (\(F\) = 0 in Equation (4)). In Figure 5, the label on the X axis represents the \(I\)th iteration carried out. On the Y axis, the label indicates the maximum absolute difference between two successive loops. P(\(t_{n}\))\({}_{m,l}\) is the radiating pressure of the \(m\)th element at the \(n\)th time step during the \(I\)th iteration. Figure 5b shows that the present method with shifting of the time step (\(F\) = 0.35 in Equation (4)) can stabilize the numerical schemes at the first internal eigenfrequency and attain convergent results. The time-shifting factor--the \(F\) value in Equation (4)--for the first internal eigenfrequency is suggested to be 0.35 in Figure 5b. Figure 6 elaborates more on the convergence plot shown in Figure 5b. Figure 6 shows the iterating process of one element at different time steps for different iterating loops. Figure 6 indicates that a convergent result is attained after about 20 iterations. It was found that the results calculated by the 20 th and 21 st iterating loops were almost the same. In other words, the radiating pressures at different time steps were almost consistent for the 20 th and 21 st iterating loops, and convergence was achieved. Table 2 gives a comparison between the analytical and numerical solutions. The difference between the amplitudes of numerical and exact solutions defines the error value. The radiating pressure calculated by the present method with shifting time (\(F\) = 0.35) is 392 Pa, and the analytical solution is 395.48 Pa. The error is therefore below 1%. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**KR**} & **Frequency** & **Total Time Steps** & **No. of** & \\  & **(Hz)** & **per Period (N)** & **Elements** & **/**\(\lambda\)** & **\(\Delta t\)/Wave Period** \\ \hline \(\pi\) & 171.5 & 8 & 732 & 0.065 & 0.14 \\ \hline
4 & 218.4 & 10 & 1148 & 0.067 & 0.11 \\ \hline
5 & 273.0 & 10 & 1652 & 0.069 & 0.11 \\ \hline \hline \end{tabular} \(l\): the max. side length of the element; \(\lambda\): the wavelength; KR: the production of the wave number and the sphere diameter.  
\end{table} 
Table 1: The selected frequencies and computing conditions for the pulsating sphere.