# Dispersion Curves of Transverse Waves Propagating in Multi-Layered Soils from Experimental Tests in a 100 m Deep Borehole

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. Waves in Infinite Isotropic Elastic Media

#### 2.2. Waves in an Isotropic Elastic Plane

#### 2.3. Empirical Estimate of the Dispersion Law from Acquired Signals

## 3. Experimental Tests

## 4. Results

#### The Effect of Dispersion: Results and Discussion

- A first possibility is that the acquired pulses do not correspond to pure shear waves, but they exhibit coupling between the longitudinal and transverse motion mostly explicated in the x-z plane. The directional nature of the input may force the soil grans to move in the x-z plane, while they do not manifest a significant displacement in the y direction. These waves belong to the class of Lamb waves, characterized by a dispersion curve derived in the second section in Equation (21). Therefore, the leading cause of dispersion may stand in the excitation, which forces the particles to move in a plane, as occurs in Lamb waves which can propagate in plates and spheres.
- It is also plausible that the experimenters measure almost pure shear waves and that the primary source of dispersion stands in the granular nature of the medium. Granular media may exhibit hyperbolic-like dispersive curves [60,61]. Therefore, the constitutive nature of the soil, rather than the boundary conditions may determine the observed behavior. There are copious theoretical and numerical researches on the dispersion of continuum models representative of random granular assemblies [3,4,5,6,7,8,9]. These studies mostly deal with high-order deformation gradients, the constitutive relations descend from the Cosserat theory and the grains interact through Hertz-Mindlin contacts [60,61].

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Time history, (

**b**) Amplitude, phase (

**c**,

**d**) difference of phase of two recorded signal corresponding to a 10 m depth.

**Figure 4.**(

**a**) Time history of the filtered signals, (

**b**) Cross-correlation between the two filtered signals.

**Figure 5.**Profile of the shear waves: the red dots correspond to the time series which yield the maximum correlation, the black dots derive from cross-correlations with lower values.

**Figure 7.**3-D dispersion curves where the phase velocity is function of the signal amplitude in [mV], the wavenumber [m${}^{-1}$] and the depth [m]. The three images (

**a**–

**c**) are different views of the same scatter plot.

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Aloisio, A.; Totani, F.; Alaggio, R.; Totani, G.
Dispersion Curves of Transverse Waves Propagating in Multi-Layered Soils from Experimental Tests in a 100 m Deep Borehole. *Geosciences* **2021**, *11*, 207.
https://doi.org/10.3390/geosciences11050207

**AMA Style**

Aloisio A, Totani F, Alaggio R, Totani G.
Dispersion Curves of Transverse Waves Propagating in Multi-Layered Soils from Experimental Tests in a 100 m Deep Borehole. *Geosciences*. 2021; 11(5):207.
https://doi.org/10.3390/geosciences11050207

**Chicago/Turabian Style**

Aloisio, Angelo, Ferdinando Totani, Rocco Alaggio, and Gianfranco Totani.
2021. "Dispersion Curves of Transverse Waves Propagating in Multi-Layered Soils from Experimental Tests in a 100 m Deep Borehole" *Geosciences* 11, no. 5: 207.
https://doi.org/10.3390/geosciences11050207