Shock Wave Dissipation

Yuan Shen
November 12, 2013

Submitted as coursework for PH240, Stanford University, Fall 2013

Bomb Blasts and Shocks

Fig. 1: Two Processes of Shock Propagation. 0→2 (red arrow) shock front and 2→1 (green curve) unloading. The red curve represents all possible states P2,v2 one can reach by shocking the system.

Instead of storing or saving energy, people sometimes want to enhance energy dissipation. Shock waves from bomb explosion is an example. When a bomb explodes, highly compressed air particles strike human body. Shock waves will then pass through in the body, causing damages to the tissues. [1] Shock waves carry energy just like elastic waves. But they are faster and thus can transport more energy. We will show how the shock energy dissipates into matters and how the protective pads are designed to reduce destructive effects.

Energy Dissipation of Shock Propagation

Shock propagation in materials can be modelled as the two processes shown in Fig. 1: [2]

Shock Front (0→2): As shock waves propagate through matter the shock front will compress materials ahead of it, increasing the pressure and temperature. Fig. 2 shows a simple model of shock wave generated by the impulsive motion of a piston in a cylinder tube. [3] Let us assume that the piston hits and drives the material to the right with a velocity Up. A shock wave appears on the surface and then propagates into the material with some higher velocity Us. Here are the three conservation laws:

  1. Conservation of Mass. For a given time dt, the shock compresses a mass dm=ρ0UsAdt of the material from UsAdt to (Us-Up)Adt, where A is the cross section of the cylinder and ρ0=1v0 is the initial density of the material. Let ρ2 be the density of the compressed material.

    ρ0 Us = ρ2 (Us - Up)
  2. Conservation of Momentum. Driving force of the piston is (P2-P0)A, causing dm to acquire a velocity of Up where P0,2 are the pressures of the system before and after shock front. Then from conservation of momentum

    Up ρ0 Us = P2 - P0
  3. Conservation of Energy. Compressive work done by piston is UpP2Adt which increases the internal energy of the material as well as its kinetic energy.

    UpP2 = ρ0 Us (e2 - e0 + ½Up2)
Fig. 2: Cross-sectional view of SKYDEX padding.
Fig. 3: (a) Twin-hemisphere structure before impact; (b) Twin-hemisphere structure displacing to absorb force during impact.

With some mathematical manipulations, we can eliminate Us,p and get the energy transferred from shock wave to the internal energy of the material.

Δes = e2 - e0 = ½(P2 + P0)(v0 - v2)

Unloading (2→1): After the shock front, the system pressure will be reduced back to P0, which is supposed to be an adiabatic process dS=0. [4]

Δeu = e1 - e2 = ∫ (TdS - Pdv) = - ∫21 Pdv

Putting these two processes together, we have the energy dissipation absorbed by the material during shock propagation.

Δe = Δes + Δeu = ½ (P2 + P0)(v0 - v2) - ∫21 Pdv

Impact Absorbers

To find the numerical values of energy dissipation, we need both the shocked pressure P2 and the equation of states for specific materials. However, the form of the last equation above is enough to guide us in searching for good impact absorbers. To maximize the energy dissipation, is to maximize Δe by increasing Δes and decreasing -Δeu. Therefore, anything with large v0 and small v1,2 would be a good choice. In simple language, those can be highly compressed would be the best impact absorbers. Its also true that the more materials you have (bigger volume), the better energy absorption would be.

Fig. 2 shows a possible design of helmet pads. The pads are comprised of four different material components. Fig. 3 is the impact absorbing layer which dissipates shock wave energy in the way just as we described. The polyurethane hemispheres are chemically bonded together and have a high compression ratio to absorb more energy during impact. They bounce back slowly and may still be able to perform well for a number of subsequent impacts.

© Yuan Shen. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] N. M. Elsayed and J. L. Atkins, Explosion and Blast-Related Injuries: Effects of Explosion and Blast from Military Operations and Acts of Terorism (Academic Press, 2008).

[2] L. P. Orlenko and L. P. Parshev, "Calculation of the Energy of a Shock Wave in Water", J. Appl. Mech. Tech. Phys. 6, 90 (1965).

[3] L. F. Henderson, "General Laws for Propagation of Shock Waves Through Matter", in Handbook of Shock Waves, ed. by G. Ben-Dor, O. Igra and R. Elperin (Academic Press, 2000).

[4] D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley,1999).