Theoretical background

This tool computes the target vibration velocity levels per frequency using a distance- and frequency-dependent attenuation model combining geometric spreading and material damping. This document describes the theoretical concepts the tool uses.

The tool computes vibration level at a target location using a combined geometric spreading and material damping model. The attenuation ratio between target and source velocity levels is:

\[\frac{v_b}{v_a} = \left( \frac{r_a}{r_b} \right)^{\gamma} \exp\!\big( \rho_B \pi f (r_a - r_b) \big)\]

where:

• \(v_a\) = vibration velocity at source distance \(r_a\), in [m/s]

• \(v_b\) = vibration velocity at target distance \(r_b\), in [m/s]

• \(f\) = frequency, in [Hz]

• \(\gamma\) = geometric spreading exponent, in [-]

• \(\rho_B\) = Barkan damping parameter, in [s/m]

• \(r_a, r_b\) = source and target distances, in [m]

Wave types and spreading

Ground-borne vibrations in the near-surface are commonly dominated by Rayleigh waves and, depending on layering and frequency, Love waves. Idealised geometric spreading describes how amplitude decays due to energy distribution over expanding wavefronts:

• Spherical spreading (body waves in 3D media): \(A \propto r^{-2}\)

• Cylindrical spreading (surface waves along a line source): \(A \propto r^{-1}\)

• Layered/heterogeneous media may exhibit intermediate or frequency-dependent spreading.

The factor \(\left(\tfrac{r_a}{r_b}\right)^{\gamma}\) captures this phenomenologically, with \(\gamma\) calibrated to site conditions. \(\gamma=0.5\) is used for Rayleigh waves in an ideal half-space because their geometric spreading is weaker than cylindrical or spherical spreading. It reflects the physics of surface-confined energy propagation. If your source is traffic on a road (effectively a line source), \(\gamma≈1\) is common. If you model a point source on the surface in a homogeneous half-space, \(\gamma≈0.5\) is theoretically correct. In reality, soil heterogeneity and damping often push the effective exponent higher (0.7–1.2).

Material damping and frequency dependence

Propagation through soils introduces intrinsic damping (hysteretic losses due to internal friction), scattering (from heterogeneities), and radiation damping (energy leakage). A simple frequency-dependent attenuation is modeled by:

\[A(r_b,f) = A(r_a,f)\,\exp\!\left(-\alpha(f)\,(r_b - r_a)\right),\]

with \(\alpha(f) \propto f\) under many practical assumptions. In the implemented form

\[\exp\!\left(\rho_B \pi f (r_a - r_b)\right),\]

the exponent is negative when \(r_b > r_a\), producing decay that intensifies with both frequency and path length. Choosing \(\rho_B\) so that \(\rho_B\, f\, r\) is dimensionless (e.g., \(\rho_B\) in s/m) ensures physical consistency.

Barkan parameter

The frequency-dependent attenuation term uses the Barkan parameter \(\rho_B\), which controls exponential decay with distance and frequency:

\[\exp\!\big(\rho_B \pi f (r_a - r_b)\big)\]

Two common formulations exist for \(\rho_B\):

1. Basic attenuation model (from quality factor and wave speed):

   \[\rho_B = \frac{1}{Q \, v}\]

   where: - \(Q\) = quality factor, in [-] - \(v\) = wave speed, in [m/s], often shear or Rayleigh wave speed

2. Barkan-based model (includes factor 2 for damping ratio):

   \[\rho_B = \frac{2}{Q \, v}\]

   This version accounts for the relationship between damping ratio \(\xi\) and quality factor:

   \[\xi = \frac{1}{2Q}\]

   and is widely used in soil vibration engineering following Barkan’s approach.