The two-dimensional resonance of sediment-filled valleys

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Abstract
This paper presents a numerical study of the response of relatively embanked sediment-filled valleys to incident plane SH, SV, and P waves, in the two-dimensional case. The Aki-Larner technique used here is shown to be reliable even for steep interface slopes. Numerical results show the existence and the importance in such valleys of specific two-dimensional resonance patterns, which may be classified in three categories: the antiplane shear modes, corresponding to SH motion; the in-plane shear modes (SV); and the in-plane bulk modes (P). Each of them is characterized by the consistency of the peak frequencies, and the in-phase motion (modulo 180°) across the whole valley. They induce a very large amplification, even in the case of significant damping (up to 4 times the corresponding one-dimensional prediction), a very long duration of motion, and large differential motion. The characteristics of the fundamental modes are in good qualitative agreement with the relevant experimental observations. The existence of this two-dimensional resonance is controlled by the shape ratio (thickness to half-width ratio) and the velocity contrast: a quantitative relationship is proposed in the SH case. The frequency of these resonance modes, for each pattern, is shown to depend only on two parameters: the one-dimensional resonance frequency at the valley center and the shape ratio. A simple model of a soft rectangular inclusion is shown to provide satisfactory quantitative formulas to estimate the fundamental resonance frequencies of any valley. As to the amplitude of this two-dimensional resonance, the general trends of its dependence on the different valley parameters (shape ratio, velocity contrast, Poisson ratio, damping) and on the incident wave field characteristics (wave type, incidence angle) are indicated. An important result, however, for earthquake engineering purposes is that both the two-dimensional resonant frequencies and amplification values differ a lot from their “classical” one-dimensional estimates.