Blast Wave Overpressure | mysimulator.uk

A blast wave is a self-similar expanding shock

An intense point-source energy release - a chemical explosion, in idealised form - deposits energy into a small volume so fast that the surrounding air is driven outward as a sharp, supersonic shock front: a near-discontinuous jump in pressure, density and temperature racing outward through undisturbed air ahead of it. In the earliest, strong-shock phase, the blast radius depends on only two physical quantities - the deposited energy E and the ambient air density rho - plus time, and dimensional analysis alone (no detailed shock physics needed) fixes how the radius must grow.

live demo - an expanding shock front from a point-source blast● LIVE

Sedov-Taylor scaling

Leonid Sedov and Geoffrey Taylor independently derived, in the 1940s, that for a strong point-source blast in a uniform atmosphere the shock radius R must scale as:

R(t) = ξ₀ · (E t² / ρ)^(1/5)

E   : energy released
ρ : ambient air density
t   : time since detonation
ξ₀ : dimensionless constant (≈1.0-1.15, from the detailed self-similar solution)

// radius grows as t^(2/5) - fast at first, decelerating steadily
// as the same fixed energy is spread over an ever-larger shock shell

This self-similar solution is remarkable because it needs no knowledge of the explosive's chemistry, only the total energy released - which is precisely how Taylor famously estimated the yield of the first atomic bomb test in 1945 from nothing but publicly released time-lapse photographs of the fireball radius over time, matching the t^(2/5) growth curve to back out E.

Overpressure: the quantity that actually does damage

Blast damage to structures and to people is caused not by the shock's velocity but by overpressure - the amount by which local air pressure exceeds normal atmospheric pressure as the shock front and the following blast wind pass a given point. Overpressure is highest right at the shock front and decays with distance from the source; a useful, widely used empirical approximation (Kingery-Bulmash-type curves, or the simpler Sachs-scaled forms) relates peak overpressure to the scaled distance Z = R / E^(1/3), a form that follows from the same dimensional reasoning used in TNT-equivalent scaling.

The Friedlander waveform

At a fixed observer location, overpressure does not simply spike and vanish - it rises almost instantaneously as the shock front arrives, then decays, and characteristically overshoots into a longer, weaker negative phase (below-ambient pressure, a partial vacuum) before returning to ambient. This time history is well modelled by the Friedlander waveform:

P(t) = P₀ · (1 - t/t₀) · e^(-b t/t₀)      for 0 ≤ t

P₀ : peak overpressure at shock arrival (t=0 here)
t₀ : positive-phase duration
b   : decay shape parameter

// (1 - t/t₀) crosses zero at t=t₀, producing the negative phase
// the exponential term sets how sharply the initial peak decays

The near-instant rise followed by exponential-ish decay and a negative-phase undershoot is the reason blast injuries and structural damage patterns look different from a simple sustained overpressure: the impulse - the time-integral of overpressure over the positive phase - matters as much as the peak value for how much net momentum a structure or a person's chest wall actually receives.

Damage thresholds and structural design

Blast-resistant structural engineering works from tabulated overpressure thresholds: roughly 1-2 psi (7-14 kPa) causes window breakage, 3-5 psi begins to damage typical unreinforced masonry and wood-frame structures, and above roughly 10-15 psi even reinforced structures suffer serious damage. Because peak overpressure falls off steeply with scaled distance (roughly as an inverse-cube-ish power at moderate range, before flattening at very long range), the standoff distance from a potential blast source is by far the single most effective design lever - which is exactly why blast-safety design (vehicle checkpoints, quarry blast zones, industrial explosive storage) leans so heavily on minimum standoff distances rather than trying to armour every nearby structure against a worst-case near-field overpressure.

Frequently asked questions

Why does the blast radius grow as time to the two-fifths power instead of linearly?

Because Sedov-Taylor scaling shows that for a strong point-source blast, the shock radius depends only on the energy released, the ambient air density and time, and dimensional analysis fixes the only combination that has units of length as R proportional to (E t squared / rho) to the one-fifth power. Physically, the same fixed energy is being spread across an ever-larger shock shell, so the front continuously decelerates rather than moving at constant speed.

What's the difference between the shock front's speed and its overpressure?

Shock speed describes how fast the pressure discontinuity itself is racing outward through the air; overpressure describes how much the local air pressure at a point exceeds normal atmospheric pressure as that front and the following blast wind pass. Structural and injury damage correlate with overpressure and its time-integrated impulse, not directly with how fast the shock front happens to be moving.

Why does overpressure go briefly negative after the initial spike?

The Friedlander waveform captures a real physical effect: after the sharp positive-overpressure spike at shock arrival, the disturbed air rebounds and overshoots into a longer, weaker below-ambient negative phase before pressure returns to normal. This negative phase is typically longer in duration but much lower in peak magnitude than the initial positive spike.

Try it live

Everything above runs in your browser — open Blast Wave Overpressure and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Blast Wave Overpressure simulation

Blast Waves: Sedov-Taylor Scaling and the Friedlander Overpressure Curve