## 1 Introduction  
Irregularities are the main source if excitation of train-induced ground vibrations, which are in the frequency range of 4 to 100 Hz (maximal 250 Hz). Irregularities can be divided into _vertical_, horizontal, rotational and gauge errors; vehicle and _track_ irregularities; and _low-frequency_ alignment errors and high-frequency roughness errors, where the areas of interest of the present article are underlined. The irregularities are presented as wavelength or wavenumber spectra. Some normative alignment spectra from US, China, France, and Germany are presented and compared in [1]. Measurement examples of track errors are given, for example, in [2, 3]. The track errors in the present article are based on own axle-box measurements at a ground vibration measurement site [4]. The article is focused on the mid-frequency range of 8 to 30 Hz, which is usually the dominant ground vibration in the mid and far field. In ground vibration analysis, different numerical models of the railway track and the soil have been established: the wavenumber-domain models [5, 6, 7, 8], finite/infinite-element models [9, 10, 11, 12] (SBFEM), boundary-element models [13, 14], and 2.5D coupled wavenumber-domain (finite or boundary-element) models [5, 15, 16, 17]. The present contribution is based on frequency-wavenumber methods and their approximations; see Appendix A. The additional method of superposition of axle pulses was also used in [18] for trans-Rayleightrains that are moving faster than the waves in the soil; however, these are not included in the present contribution. When the theoretical models are compared with measurements, the geometric irregularities at the wheel-rail contact are used as the excitation in most cases. Moreover, the irregularities from the US standard [2], which are quite high, are often used. For some model verifications, the irregularities are measured [5; 8; 17; 19] mainly by hand-driven trolleys, which are suitable for roughness and for which the results must be extrapolated to low frequencies. In [19], a low-frequency method was used in addition to the high-frequency roughness measurement and the combined results showed an almost unique curve. Axel-box measurements [4; 5] provide results for a wide frequency range and include track and vehicle irregularities that can be distinguished by their different frequency characteristics [20]. In all these cases, the inertial forces of the wheelset running over the irregularities are understood as the excitation of the ground vibration. In addition to track and vehicle irregularities, the following irregularities can be distinguished: rail unevenness or roughness [21] and wheel unevenness or roughness, which are both at high frequencies above 50 Hz and are not covered in this article; low order wheel out-of-roundness between 10 and 50 Hz [22]; geometric track alignment or level errors, geometric track support errors and track support stiffness variations usually below 30 Hz [23; 24; 25]; and deterministic track stiffness variation with sleeper-distance frequency above 50 Hz [26]. All these irregularities generate dynamic axle loads when an axle (the unsprung mass) passes over these irregularities [27]. An additional effect follows from the passage of a static load over a track with an irregular track support (irregular ballast, irregular soil, irregular sleeper contact). The randomly varying track support stiffness was analysed experimentally in [28; 29; 30], and it is known that the track support can strongly vary up to hanging sleepers. In the case of varying track stiffness, the ground vibration is generated in a different way by the irregular transfer of the static load to the ground. The structure of this article is as follows. First (Section 2), the importance of the mid-frequency ground vibration component is illustrated by measurements at two sites.