## 4 Discussion
### Effect of Vertical Track Irregularities on Train Body Accelerations
The values of the train body accelerations and the 30-m-chord track irregularities are also well correlated, as shown in Figure 14.  
This high correlation between the 30-m vertical track irregularity and the vertical acceleration of the train body is due to the relationship between the amplification factor of the chord versus method and the transfer characteristics of vehicle acceleration [27]. The relationship between track irregularities by the chord versus method and absolute track-geometrical errors can be expressed as  
\[\delta_{30m}(\lambda)=H_{c}(\lambda)\;\delta_{abs}(\lambda) \tag{2}\]  
where \(\delta_{30m}(\lambda)\) represents the amplitude of the 30-m-chord track irregularities in the wavelength domain and \(\delta_{abs}\) stands for the absolute track-geometrical errors. Also, the relationship between the train body acceleration and the absolute track-geometrical errors can be defined by the transfer function \(H_{a}(\lambda)\) as  
\[A(\lambda)=H_{a}(\lambda)\;\delta_{abs}(\lambda) \tag{3}\]  
where \(A(\lambda)\) represents the amplitude of the train body acceleration in the wavelength domain. Here, if the waveforms of the above two functions--\(H_{a}(\  
The transfer ratio of the acceleration by track-geometrical error is greatly increased at a critical wavelength. The critical wavelengths are divided into those determined by the natural frequencies of the vehicle suspension system and those determined by the distance between the vehicle bogies regardless of the train suspension system. The critical wavelengths determined by the natural frequency of the vehicle suspension system are proportional to the train speed as follows:  
\[\lambda_{n}=\frac{V}{f_{n}}\quad(n=1,\;2,\;3,\;\cdots) \tag{4}\]  
where \(V\) is the train speed and \(f_{n}\) is the natural frequency of the vehicle suspension system. The critical wavelengths determined by the bogie distance can be obtained by [18; 28] as follows. * Critical wavelengths causing car body bouncing: \[\lambda_{b}=\frac{B}{m}\quad(m=1,\;2,\;3,\;\cdots)\] (5)
* Critical wavelengths causing car body pitching: Figure 11:0 (6) where \(B\) is the center-to-center distance of the bogies. Thus, the first critical wavelengths corresponding to the bogie distance of a Korean KTX high-speed train (18.7 m) are 18.7 m and 37.4 m for bouncing and for pitching, respectively. As can be seen in Figure 15, in the case of application of the 30-m chord at these critical wavelengths, the amplification ratio is close to 1 or more than 1. This is why the 30-m-chord track irregularity and train body acceleration are highly correlated. On the other hand, the critical wavelength related to the natural frequency of the train suspension system depends on the train speed, and so the correlation with the 30-m-chord track irregularity and train body acceleration will vary with the train speed. When applying the 30-m chord, the amplification ratio is larger than 0.8 up to approximately 70 m, and good correlation can be expected. The natural frequencies of the suspension system of the Korean KTX high-speed trains and corresponding wavelengths are shown in Figure 11:1 The natural frequencies of the suspension system of the trains were calculated based on the 2-D mass-spring-damper vehicle model used in the train-track analysis by [PERSON] and [PERSON] [17]. At the train speed of 230 to 300 km/h, the critical wavelengths corresponding to the pitching mode of the  
Figure 11:2 Transfer function of track irregularities ([PERSON] et al. 2012 [27]).