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Research Papers

# Jumps Across an Outgoing Spherical Shock Wave Front

[+] Author and Article Information
Yukio Sano

38-10 Shibatani-Cho, Takatsuki, Osaka 569-1025, Japan

Tomokazu Sano

Graduate School of Engineering,  Osaka University, Suita, Osaka 565-0871, Japan

J. Appl. Mech 75(4), 041017 (May 14, 2008) (8 pages) doi:10.1115/1.2912942 History: Received May 19, 2007; Revised October 27, 2007; Published May 14, 2008

## Abstract

The shock jump conditions have been used since Rankine published in 1870 and Hugoniot in 1889. However, these conditions, in which the geometrical effect is never included, may not be correctly applied to material responses caused by a spherical wave front. Here, a geometrical effect on jumps in radial particle velocity and radial stress across an outgoing spherical wave front is examined. Two types of jump equations are derived from the conservation laws of mass and momentum. The first equations of Rankine–Hugoniot (RH) type show that the geometrical effect may be neglected at distances of movement of the rear of the wave front that are more than ten times as long as the effective wave front thickness. Furthermore, using four conditions required to satisfy the RH jump conditions, which are contained in the RH type equations, a method is developed to judge the applicability of the RH jump conditions to the jumps. The second type equations for spherical wave fronts of general form are obtained by expressing a volumetric strain wave $ε$ in the wave front by more general wave forms. In the neighborhood of the center of the wave front, for $ε<0.09$, radial particle velocity in the jump in any materials is inversely proportional to the square of a dimensionless distance from the center to the rear, and for $ε<0.04$, radial stress in the jump in some viscous fluids and solids is inversely proportional to the distance. In conclusion, an outgoing spherical wave front attenuates greatly near the center due to the geometrical effect as well as rarefaction waves overtaking from behind, while the geometrical effect is negligible at the specified positions that are distant from the center.

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## Figures

Figure 1

Schematic diagrams of (a) an initial strain wave ε(r,0) with a range 0⩽r⩽r0(0) at time 0 and a strain wave ε(r,t) with a range r1(t)⩽r⩽r0(t) at time t in an inertial coordinate system (r,t); and (b) an initial strain wave ε(ξ,0) with a range 0⩽ξ⩽ξ0(0) at s=0 and a strain wave ε(ξ,s) with a range 0⩽ξ⩽ξ0(s) at s=s in a moving coordinate system (ξ,s), where s[≡r1(t)] is the distance of movement of the rear of an outgoing spherical wave front from t=0 to t=t. The initial volumetric strain ε0 is assumed to be constant along r.

Figure 2

Schematic diagrams of (a) strain waves in outgoing spherical wave fronts ε(r,t) at three times t1, t, and T from t1 to T when the wave front is passing through a fixed position r1; and (b) strain waves ε(h,τ) at τ=τ and τ=1, where τ is the dimensionless time expressed by τ=t∕T, and h is the dimensionless position expressed by h=r∕ζ(τ), where ζ(τ) is the position of the leading edge of the wave front at τ. Three positions r0(t1), r1(T), and r1 are identical, that is, r0(t1)≡r1(T)≡r1.

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