Resonant amplified seismic response within a hill or mountain

Abstract

We show theoretically what is meant by the term '(surface shape) resonance' in connection with the seismic response of a protuberance (emerging from flat ground) such as a hill or mountain of arbitrary shape. We address the specific problem of cylindrical protuberances of rectangular shape submitted to a SH plane wave. We find that the principal (i.e., qualitative) characteristics of the seismic response of a mountain are quite similar to those of a hill, and that the occurrence of significative amplification of the displacement field within these structures is due to the coupling of the incident wave to (surface shape) resonances.

Parametric study of the seismic response of a hill or mountain

“…Our parametric study is justified by the fact that a protuberance (e.g., dike, hill or mountain) gives rise to a large variety of seismic responses and that it has not, until now, been possible to identify the principal features of the incident wave, the protuberance, and the underground (to which the protuberance is connected), that condition these responses. In [29] we showed why these, often amplified, responses are largely the result of the coupling of the incident wave to (surface shape) resonances, but, we still do not know how these resonances (i.e., their frequencies of occurrence and amplitudes) depend on the various parameters of the wave-structure interaction.…”

“…Moreover, our previous contribution [29] showed that the seismic response of each configuration, at a given frequency, tends to be spatially non-uniform, so that this is the cause of another computational difficulty (i.e., the determination of the field at a great number of points in a domain that includes the protuberance and a portion of the underground). But, all in all, these voluminous computations are not out of reach of what modern digital computers can do in a reasonable length of time.…”

“…In each of these figures: the upper left-hand, lower left-hand and lower right-hand panels depict T (0, h; f R ), T (−w/2, h; f R ) and T (w/2, h; f R ) as a function of θ i , whereas the upper right-hand panel depicts 1/D(f R ) as a function of θ i , f R being the indicated resonance frequency and D(f ) the determinant of the linear matrix equation by which we compute the seismic response (as indicated in [29], the positions of the minima of D(f ) define the frequencies of resonance). Red, blue and black curves relate to the real part, the imaginary part and the absolute value of a complex function.…”

“…The question we address in fig. 7 is how large must the approximation order M of the aforementioned matrix equation (see [29]) be for the seismic response to be correctly predicted in the absence of material damping, this being the most severe case (compared to the cases in the presence of material damping) for the prediction of resonant response. u [2] (0,h) u [2] (−w/2,h) u [2] (0,h) u [2] (−w/2,h) u [2] (w/2,h)…”

“…This contribution deals with a seismic scattering problem identical to the one studied in our two recent contributions [28,29] so that all the theoretical and numerical details can be found therein and will not be repeated here. However, it is necessary to state what precisely is the seismic scattering problem we are here interested in.…”