Prędkość Propagacji Fal: Zasady i Wzory

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.

Sedov-Taylor scaling

Leonid Sednov 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

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

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.

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

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.

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