_{1}Waves by Arbitrarily Arranged Cavities in Saturated Soils

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Based on Biot’s saturated soil wave theory, using wave function expansion method, theoretical solutions of multiple scattering of plain P_{1} waves are achieved by rows of cavities as barrier with arbitrarily arranged cavities in saturated soil. Undetermined complex coefficients after wave function expansion are obtained by cavities-soil stress and displacement free boundary conditions. Numerical examples are used to investigate variation of dimensionless displacement amplitude at the back and force of cavities barrier under P_{1} wave incident, and it is also discussed that the main parameters influenced isolation effect such as scattering orders, separation of cavities, distances between cavity rows, numbers of cavities, and arrangement of barriers. The results clearly demonstrate optimum design proposals with rows of cavities: with the multiple scattering order increases, the displacement amplitude tends to converge and the deviation caused by subsequent scattering cannot be neglected; it will obtain higher calculation accuracy when the order of scattering is truncated at

With the accelerated process of urbanization, cities have been expanding continuously. Artificial vibration caused by large-scale construction is increasingly frequent, whether the mechanical vibration during the construction process or the traffic load during running period will do harm to adjacent constructions, underground pipelines, tunnels, and important equipment. It will also affect people’s production and life. Therefore, the approach to deal with decreasing artificial vibration and controlling vibration pollution has become one of the most important research topics in soil dynamics.

As a representative discontinuous barrier of vibration isolation, rows of cavities not only can be convenience to construction and supporting, but also are more economical than rows of piles as barrier (Figure

The sketch of artificial isolation vibration by cavities as discontinuous barrier.

Since the calculation of cavities as barrier is usually regarded as the same origin of piles as barrier, many literatures are concentrated on pile rows. Liao and Sangery [_{1} wave incident in saturated soil as well.

Researches specialized in cavities as barrier are stemmed from 1970s. Woods et al. [

On the other hand, isolation vibration is derived from the propagation of elastic waves, which contains multiple scattering and diffraction process. It is usually taken into account in the field of acoustics and electromagnetism. The

However, most of the studies mentioned above are still aimed at single-phase medium, which does not conform to real physical process of solid-liquid coupling scattering between soil skeleton and fluid in saturated soil. Literatures are rarely focused on elastic waves propagation in a number of arbitrarily arranged and arbitrary radius cavities, and much less attention is paid to coherent scattering law of elastic waves encountering multiple irregular cavities in saturated medium with the point of view of multiple scattering. Therefore, based on Biot’s poroelastic theory [_{1} wave incident when it confronts with arbitrarily arranged and arbitrary radius cavities in saturated soil. The normalized displacement amplitude behind the cavity barrier is numerically analyzed, which has certain engineering guiding significance for vibration isolation.

Assuming a hollow cylinder embedded in saturated medium with infinite length (Figure _{1}, P_{2}, and SV waves’ scattering as depicted in Figure _{s}, _{s}; rest can be done as the same manner; for example, the relative coordinate of_{os}, _{os}, the relative coordinates of cavity

The propagation of elastic waves subjected to a hollow cylinder.

2D model of plain P_{1} waves’ multiple scattering by arbitrarily arranged and arbitrary radius cavity barrier in saturated soil.

The physical process of multiple scattering could be described as follows: an arbitrary subject of scatterer is named as _{1} waves _{1} waves, the total wave field of scatterer

Based on the modified Biot’s wave function [_{1}, compressional slow wave P_{2}, and shear wave S. In consideration of incident P_{2} wave’s rapid attenuation in saturated soil and occupying small quantity energy of elastic waves, the P_{2} wave is neglected in propagation of elastic waves induced by artificial vibration, while the effects of wave P_{1} are only considered.

The incident P waves’ potential function is expanded into a series of Fourier-Bessel series in cylindrical coordinate system such as_{1}, whose subscript 1 indicates P_{1} wave and subscript

For the sake of differentiating coupled scattering waves that emerged at cylindrical heterogeneity under incident P waves, scattering waves consisting of P_{1}, P_{2}, and SV wave are written as the series forms of displacement potential functions as follows:_{1} and P_{2}; subscripts 1 and 2 of the wave number _{1} and P_{2} wave too;

The boundary conditions of the first order of scattering are

Assuming the cavity-soil interface is permeable, the fluid stress is free:

Substituting the equilibrium equation of Biot’s poroelastic theory under the cylindrical coordinate system, the stress of soil skeleton and fluid can be expressed by potential functions such as

Substituting (

Simplifying (

For the

The fluid at interface also meets the stress-free condition as follows:

As the coordinates depicted above, it is required to transfer the origin of the elementary wave functions so as to implement the boundary conditions.

Therefore, the iterative relation of the

Assuming that the surrounding soil is a homogenous and saturated medium and the depth of cavity is infinite, it turns out to be a total space two-phase problem. There is a series of time-harmonic plane P_{1} waves propagating perpendicularly to the group of cavities. The angle of incident P_{1} wave is

Calculation model of equally spaced and unidiameter parallel cavity barriers in saturated soil.

Linear type (single-row)

Hexagon type (double-row)

When the barrier is hexagonally arranged, it becomes a double-row barrier, and the row-distance is

The selected soil physical and mechanical parameters are listed in Table

Biot’s parameters in saturated soil.

| ^{−2 }Pa) | ^{3}) | | | ^{3}) | | | |
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7.556 | 2.61 | 1000 | 0.94 | 7.407 | 2204.5 | 0.45 | 0.8 | 0.6 |

The incident wave frequency is treated as dimensionless while doing numerical calculation:

The influences of scattering order, cavity spacing, row-distance, and configuration of arrangement are discussed as below.

On one hand, a comparison with the existing theoretical result [

Midline dimensionless displacement amplitude

In elastic media

In saturated soil

On the other hand, the variation curves of midline dimensionless displacement amplitude behind the single-row cavity barrier when scattering orders

The property of displacement amplitude

Dimensionless displacement amplitude

The contour of dimensionless displacement amplitude

Dimensionless displacement amplitude

Figure

Dimensionless displacement amplitude

In order to investigate the multiple scattering properties and characteristics of elastic waves by cavity rows as barrier with different arrangement in in saturated soil, a double-row rectangular arrangement (counterpoint) is implemented in Figure _{s} catity-spacing and 2.5_{s} row-distance two-row barrier. Compared with the displacement amplitude in Figure

Calculation model of equally spaced and unidiameter parallel cavity barrier in saturated soil (counterpoint). Hexagon type.

Dimensionless displacement amplitude

On the other hand, the influence of screening effectiveness with the variation of cavity number is also significant. Midline displacement amplitude behind the barrier with different number of cavities is showed in Figure

Midline dimensionless displacement amplitude

Based on the Biot’s poroelastic theory and multiple scattering theory, the plane P_{1} waves’ multiple scattering by arbitrarily arranged and arbitrary radius cavity barrier in saturated soil is derived and solved. The screening effectiveness of P_{1} waves by single and multirow cavities has been calculated in numerical analysis, which concludes that

The variable

The authors declare that they have no conflicts of interest.

The work described in this paper was supported by the National Natural Science Foundation of China (Grant no. 51408549) and Zhejiang Provincial Natural Science Foundation of China (Grant nos. LY18E080024, LY16E090005, and LQ15E080008).