Propagation of pendulum waves under deep-seated cord charge blasting in blocky rock mass

The object of the numerical study is the travel of pendulum waves in a blocky medium under nonstationary impact of deep-seated charge blasting on the surface of the expansion chamber. The blocky model is simulated by a two-dimensional lattice of masses connected by elastic springs along the axes and diagonals. The displacements and velocities of the masses at different half-space points are calculated using the finite-difference method.

💡 Research Summary

The paper investigates the transient propagation of pendulum‑type seismic waves generated by a deep‑seated cord‑like explosive charge in a block‑structured rock mass. The authors model the rock mass as a two‑dimensional lattice of identical masses connected by elastic springs along the orthogonal axes and the diagonals. An isotropic configuration is imposed by setting the axial stiffness k1 equal to twice the diagonal stiffness k2 (k1 = 2k2), which ensures that long‑wavelength P‑ and S‑waves travel with the same velocity, mimicking the behavior of a homogeneous elastic medium.

A cylindrical cavity representing the explosion chamber is introduced as a zone without springs, located at depth h + l/2 (l is the lattice spacing). Four point loads of equal magnitude Q0 are applied on the cavity surface, oriented oppositely to preserve symmetry. The temporal loading is a step function multiplied by a sinusoid, Q0 H(t) sin(ωt), where H(t) is the Heaviside function. The governing equations for the displacements u (horizontal) and v (vertical) are derived from Newton’s second law, incorporating forces from the neighboring springs. Free‑surface boundary conditions are applied at the line m = 0, while symmetry conditions are imposed at n = 1, allowing the computation to be restricted to the half‑plane n ≥ 1, m ≤ 0.

The numerical solution employs an explicit finite‑difference scheme. Stability requires the time step τ to satisfy τ ≤ l √(M/2k1), where M is the block mass. The scheme advances the displacement field in time, recording both v and its time derivative (velocity) at each lattice node.

Results are presented for two cavity depths: a shallow case (h = 1 lattice unit) and a deep case (h = 25). At t = 200 (non‑dimensional time), the vertical displacement field shows that in the shallow configuration a Rayleigh surface wave (R‑wave) forms rapidly and dominates the surface response. In contrast, for the deep cavity the R‑wave is barely observable at the same time, indicating a delayed surface manifestation. The cavity also radiates a primary P‑wave, which reflects off the free surface to produce a longitudinal PP‑wave and a shear PS‑wave. The S‑wave emitted directly from the cavity is clearly visible in the shallow case.

Arrival times of the various wave fronts are computed analytically using the known wave speeds (c_R for Rayleigh, c_P for P‑wave, etc.) and verified against the numerical data. The authors examine vertical displacement histories at several observation points (labeled i_A, i_B, i_C) located at different horizontal distances x from the cavity axis. Key observations include:

• The peak vertical displacement in the Rayleigh wave can be several times larger than that of the incident P‑wave when the cavity is shallow (h ≈ 1).
• As h increases, the Rayleigh‑wave amplitude diminishes and eventually reaches a constant value that is lower than the shallow‑case peak.
• For points close to the cavity axis, the PS‑wave may exceed the P‑wave amplitude in the shallow case, but this relationship reverses for deeper cavities.
• The maximum Rayleigh‑wave amplitude |v|_max_R first grows with increasing horizontal distance x, then saturates; the saturation occurs sooner for deeper cavities.
• The horizontal distance at which the Rayleigh‑wave amplitude overtakes the P‑wave amplitude grows linearly with cavity depth.

The study concludes that the blocky nature of the medium introduces high‑frequency oscillations into all wave types (P, R, PP, PS) and that the depth of the explosive source critically controls the relative strength of surface versus body waves. These findings have practical implications for predicting ground motion from underground blasting, assessing seismic hazard in block‑structured rock masses, and designing mitigation measures for infrastructure near such sources. The successful application of an explicit finite‑difference lattice model demonstrates a viable computational framework for complex, non‑stationary loading scenarios in heterogeneous geological media.