# How strong was the shockwave from the Tianjin explosion?

Short answer: Within 100 meters of the explosion I expect buildings to be completely destroyed, while windows within 2-3 kilometers are likely broken.

Long answer: An explosion occurred last night in Tianjin, China. Authorities aren’t quite sure what caused it yet, but images and video of the blast are abundant on the internet. A compilation of cellphone videos has been posted by the New York Times, which includes the explosion and deafening blast, but also some heartbreaking screams. Here’s a gif of one video:

The shaking at the end is due to the arrival of the shockwave. Light travels faster than sound, so you’ll always see something before you hear it- sort of like how lightning always precedes thunder. You can hear it in the video, linked to in the caption above. In fact, in the same way that you can tell the distance to a lightning flash by counting the seconds until the thunder, you can tell the distance to the explosion in many of the recordings by counting how many seconds are between the explosion and the arrival of the shockwave. Unfortunately, explosion shockwaves tend to travel faster than the speed of sound, so it’s hard to say exactly how fast that shockwave was moving. In the video above the delay is about 6 seconds so I estimate that the guy was at least 2 kilometers away, which is over a mile.

Shockwaves can be just as dangerous as the explosion itself. Close to the explosion the shockwave can be lethal and can cause buildings to collapse. People can also be injured at greater distances by broken windows and shattered glass. If you ever see an explosion like this, cover your ears and get away from windows.

I want to know how strong the shockwave from the Tianjin explosion was, and to calculate that I need to know how strong the explosion was. The incident in Tianjin actually consisted of two separate explosions, which were both registered by seismometers. The Chinese Earthquake Administration reports that the explosions registered with magnitudes of 2.3 and 2.9, and were equivalent to the detonation of 3 tonnes and 21 tonnes of TNT respectively [1].

The TNT equivalent is a useful number, because it lets us apply what we know about nuclear weapons to the explosions in Tianjin. This is because the explosive yield of nuclear bombs is often discussed in TNT equivalents, so much of the physics for describing the shockwave can be applied.

Specifically, we’re interested in the Sardovsky equation – given the equivalent mass of TNT that’s exploding and the distance from the explosion, it can calculate the overpressure of the shockwave when it passes. The shockwave compresses the air like a sound wave, resulting in a temporary increase in air pressure as the wave passes, which is ultimately what does the damage. The equation is:

`    ΔP = A * (M)1/3 / R + B * (M)2/3 / R2 + C * M / R3`

where M is the equivalent mass of TNT and R is the distance from the explosion. The coefficients A, B, and C, depend on the atmospheric conditions – is the explosion on the surface of the earth, or is it in midair? In the case that the explosion is on the surface of the earth, under standard conditions, we find a formula like:

`    ΔP = 0.95 * (M)1/3 / R + 3.9 * (M)2/3 / R2 + 13.0 * M / R3`

where M is in kilograms and R is in meters, while the overpressure is in atms [2].

At a distance of 100 meters, I find that the pressure of the shockwave is greater than 83% of standard atmospheric pressure, amounting to 12 PSI. At a distance of 200 meters, the overpressure is 23% of atmosphere, amounting to 3.4 PSI. At 1000 meters, the overpressure is 3% of atmosphere, which is 0.44 PSI.

What kind of damage can these pressures do at these distances? I found an old FEMA document about shockwave damage from car and truck bombs, which have yields similar to the explosion in Tianjin [3]. That document has a table describing the levels of damage that can be caused by shockwave overpressures. At overpressures above 10 PSI, buildings are completely destroyed. For overpressures of 2-5 PSI concrete walls can be toppled, as well as wood framed buildings. At the lowest end, glass breaks due to overpressures of 0.15 and 0.22 PSI.

This tells me that there is likely serious destruction within 100 meters of the explosion, with buildings sustaining damage out to 200-300 meters. The highest risk for window breakage and glass-related injures is likely within 3 kilometers. The fact that the window didn’t break in the video above, which I estimated was taken from greater than 2 kilometers away, makes me slightly more confident in these numbers.

I don’t feel very proud of this calculation, but there it is. The physics seems sterile and doesn’t capture the sadness of the injury and loss of life. But maybe we can learn something for the future – can seismic data be used to predict the scale of destruction and extent of injuries in the area surrounding an explosion? Could this be a useful tool for emergency responders? I’d like to believe it can.