Simple model of shock-wave attenuation in snow

Abstract

AbstractA simple momentum model, assuming that snow compacts along a prescribed pressure–density curve, is used to calculate the pressure attenuation of shock waves in snow. Four shock-loading situations are examined: instantaneously applied pressure impulses for one-dimensional, cylindrical and spherical shock-wave geometries, and a one-dimensional pressure impulse of finite duration. Calculations show that for an instantaneously applied impulse the pressure attenuation for one-dimensional, cylindrical and spherical shock waves is determined by the pressure density (P–ρ) compaction curve of snow. The maximum attenuation for a one-dimensional shock wave is proportional to (Xf–X0)−1.5 for the multi-stage (P–ρ) curve and (Xf–X0)−2 when compaction occurs in a single step (single-stage compaction), where (Xf–X0) is the shock-wave propagation distance. Cylindrical waves have a maximum attenutation that varies from (R–R0)−2 for single-stage compaction and (R–R0)−1.5 for multi-stage compaction, when (R – R0) ≪ R0, where R is the propagation radius and R0 is the interior radius over which a pressure impulse is applied, to R−4 when (R – R0) ≫ R0 Spherical waves have a maximum attenuation that varies from (R – R0)−2 for single-stage compaction and (R – R0)−1.5 for multi-stage compaction to R−6 when 〈R – R0〉 ≫ R0.The shock-wave pressure in snow for a finite-duration pressure impulse is determined by the pressure impulse versus time profile during the time interval of the impulse. After the pressure impulse ends, shock-wave pressure attenuation is the same as for an instantaneously applied pressure impulse containing the same total momentum. Pressure attenuation near a shock-wave source, where the duration of the shock wave is relatively short, is greater than for a shock wave farther from a source where the shock wave has a relatively long duration. Shock-wave attenuation in snow can be delayed or reduced by increasing the duration of a finite-duration pressure impulse. A sufficiently long-duration impulse may result in no shock-wave pressure attenuation in a shallow snow cover.