## 1 Introduction  
Passing trains generate ground vibration which propagate through the soil and excite neighbouring buildings. To avoid annoyance of the inhabitants or to protect sensitive equipment it is sometimes necessary to provide mitigation measures against these vibrations. Mitigation measures can be built in the building, in the transmission path through the soil or at the emission. The present contribution focusses on the latter possibility where mainly measures at the railway track are considered. The usual concept of mitigation is to add elastic elements to the track and to increase the mass and the bending stiffness of the track. These measures should reduce the forces that are acting on the soil.  
This topic can be followed back to the 1980s [1] when the Federal Institute of Material Research and Testing (BAM) participated in a German research programme [2] about noise and vibration abutment for urban railways where many track elements had been tested in situ. Standards exist for how to verify the characteristics and the effects of track elements. For example [3], the European research project \"Railway Induced Vibration Abutment Solutions\" (RIVAS) had one special focus on under-sleeper pads under heavy sleepers [4]. Besides laboratory and in situ tests, detailed calculations were performed. The track-soil interaction has been modelled by finite-element boundary-elements, by wavenumber domain methods, and by combined so-called two-and-a-half-dimensional methods [5]. Similar methods have also been used for soft rail pads [6] and under-ballast mats [7, 8]. An overview of the numerical methods can be found in [9]. Many contributionsto track solutions have been published in the series of International Workshops on Railway Noise and Vibration (IWRN), for example [10].  
Several state-of-the-art reports summarize knowledge on mitigation of railway vibration [11; 12; 13; 14; 15; 16]. Some early references already mention a low-frequency reduction that can sometimes be found in measurements [12; 13; 17] and the possible mitigation by a higher bending stiffness [11]. The present article does not aim at another state-of-the art report. The well-known principle of reducing the dynamic axle loads at high frequencies will be demonstrated, but in addition to the state of the art, the main focus is now laid on the low-frequency reduction of the impulses from the passage of static axle loads. A theoretical explanation of this low-frequency mitigation and a model for its evaluation is given, and existing measurements are further studied to find out the rules, for example, for the influence of the speed and the axle sequence of the train which have been overseen in previous works on mitigation measures.  
The frequency-wavenumber method is explained in Section 2. The method takes into account the correct vehicle-track interaction and the correct track-soil interaction. Some theoretical results for elastic elements and an increased bending stiffness of the track are presented in Section 3 where the force transfer of the track and of the vehicle-track interaction are calculated for the high-frequency (dynamic) mitigation, and the force distribution along the track is calculated for the low-frequency (static) mitigation. Measurement results for the ground vibration near isolated and un-isolated tracks are presented in Section 4 for several under-sleeper pads, for under-ballast mats, and for several under-ballast plates and ballast troughs. The interpretation of some experimental observations is given in Section 5 and some recommendations for measurements are derived. The conclusions in Section 6 give the main mitigation effects that were found in the measurements and demonstrate the originality of this article as the new low-frequency reduction of scattered axle impulses and the new method of its calculation by the force distribution of the isolated track.