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P. Borejko, F. Ziegler:
"A New Wave-Theoretical Model of the Wave Impeding Block";
Talk: The 2nd International Symposium on Environmental Vibrations: Prediction, Monitoring, Mitigation and Evaluation, Okayama, Japan; 2005-09-20 - 2005-09-22.

An important environmental issue in the design of high-speed railway systems is the transmission of vibrations from a moving train through the track structure and the surrounding soils to the nearby residential or industrial structures. These vibrations propagate as progressive waves through the soil medium, and they are then received by the nearby structures in the form of ground-borne micro-vibrations, thus annoying the people living alongside the railways and causing vibration-sensitive devices located inside buildings to malfunction (Hung and Yang [1]).

The propagation of vibrations from the source (the track) into the surrounding soils can significantly be impeded by the installation of an artificial stiff plate (an artificial bedrock) at a certain depth below the source (Takemiya and Fujiwara [2]), the so-called wave impeding block (WIB). The reduction of ground-borne vibrations by the WIB can theoretically be explained by means of the modal solution for the wave field generated by a harmonic source in a layer of soil over a hard bedrock, expressed by an infinite sum of normal modes. The n-th vibration mode contributing to the solution will not propagate as a wave along the layer when the excitation frequency is less than the cutoff frequency fn given by fn = [(n - ½)c]/(2h), where c is the characteristic wave speed (the P-wave speed or the S-wave speed in the layer) and h is the depth of the layer (Ewing et al. [3]). When the first mode alone suffices to give a good approximation to the total wave field in the soil medium, one can see that the WIB installed at a depth h below the source impedes locally the spreading of vibration frequencies which are less than the frequency f1 given by f1 = c/(4h).

The purpose of this paper is to present a new wave-theoretical model offering a lucid explanation of the impeding effect of the WIB on the propagation of vibrations in the soil medium. This model consists of a half-space of soil that is simultaneously excited by two localized sources, where the secondary source (the WIB) is buried directly below the primary surface source (the track). The problem is to determine the time record of the vibrations received at a surface point (the structure site). The present exact solution [derived by applying the ray-integral method (Pao and Gajewski [4])] for this problem is expressed by a sum of two wave fields (including the contributions from the P-waves, the S-waves, the head S-waves, and the Rayleigh surface waves), where the first is excited by the surface source and the second is excited by the buried source. One can see that, when the second field is ignored in the solution (i.e., the buried source is absent), the entire record of the received vibrations is considerably magnified.

References

[1] H.H. Hung and Y.B. Yang, "A review of researches on ground-borne vibrations with emphasis on those induced by trains", Proc Natl. Sci. Counc. ROC(A) 25, 1-16 (2001).
[2] H. Takemiya and A. Fujiwara, "Wave propagation/impediment in a stratum and wave impeding block (WIB) measured for SSI response reduction", Soil Dyn. Earthquake Eng. 13, 49-61 (1994).
[3] W.M. Ewing, W.S. Jardetzky and F. Press, Elastic waves in layered media, McGraw-Hill, New York (1957).
[4] Y.H. Pao and R.R. Gajewski, "The generalized ray theory and transient responses of layered elastic solids", Physical Acoustics 13, 184-265, (1977).