AFAIK, even smaller (Mw 4.2) earthquakes in Japan can trigger tsunamis. So was the atom bomb used in Dark Knight Rises was not this powerful?
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First it was 4.2 on Richter scale. 4.2 MW is just unit of energy.
I don't think the bomb would have caused a tsunami though it may have caused huge waves but they wouldn't have reached Gotham.
The Bomb was capable of destroying 6 mile radius area. So assuming that bomb was dropped 6 miles away from Gotham in the ocean, the water would have absorbed much energy and say the effect would have been reduced by an additional 1 or 2 miles. (The energy is constant, the effect of exploding in water doesn't make a difference)
Also I would like to quote from article http://www.geologyinmotion.com/2011/03/how-much-power-in-japanese-tsunami-of.html
"This means that the power of the tsunami along that shoreline was about 1.3 *10^12, or 1.3 petawatts."
1 petawatt is 10^9 MW or (1000000000MW). (Easily many times the bombs dropped in Hisroshima and Nagasaki).
And this was the energy of the waves at the coastline in japan, Just imagine the energy at the point where it was created.
Although I accept the effect of nuclear explosion under water would have been very serious as it would have caused the entire water to get contaminated by nuclear radiation. All marine life would have died in many mile radii and still thousands of people would have been affected.
The pressure of the bomb, even if detonated in mid-air, would displace the water much like dropping a bowling ball in your full bathtub. Gotham would be crushed by the "splash" let alone the tsunami.
It wouldn't cause a tsunami, but still fairly large and destructive waves at short ranges should be expected unless "the bat" went up to a high altitude.
http://en.wikipedia.org/wiki/Ivy_Mike The first test of a fusion explosion produced around 10 megatons of energy, was not an underwater explosion but still produced 20 feet waves that "stripped the test islands clean of vegetation".
According to numerous resources, nuclear detonation cause rapid waves but that waves couldn't enough to propagate hazardous tsunami waves. Tsunami are huge ocean waves caused by natural forces like underwater earthquakes.
According to tsunami website,
There have been several underwater atomic tests.
The BAKER shot in Bikini Atoll in 1946, the WIGWAM shot off California in 1955, the UMBRELLA shot and the WAHOO shot at Enewetak in 1958.
BAKER was a 20 kiloton atomic bomb detonated 200 feet below the surface in a shallow lagoon. It generated a 90 foot wave 1000 feet from the detonation. However, that wave was only 6 feet high 22000 feet from the detonation.
WIGWAM was a 30 kiloton atomic bomb detonated 2000 feet below the surface. The official report on WIGWAM does not mention any significant waves from the explosion.
UMBRELLA was a 8 kiloton atomic bomb detonated 150 feet below the surface. WAHOO was a 9 kiloton atomic bomb detonated 500 feet below the surface. The official reports on UMBRELLA and WAHOO do not mention any significant waves from the detonation.
The difference between the WIGWAM and WAHOO deep water tests is that the bubble created by the explosion broadens and can collapse before reaching the surface thus limiting the wave generated at the surface.
The following link show the conclusions of atomic testing on the damage effects on ships:
One conclusion is that underwater atomic explosions do damage at twice the distance as air or surface explosions. The damage is not so much due to waves but to the blast wave being transmitted through water to the ship hull and secondary effects like hogging.
One thing to note about all the underwater shots: fallout was limited by the water and in the case of the deep underwater explosions, little fallout was released in the air.
Compare the above tests to the 15 kiloton Hiroshima bomb. Severe damage to builds occurred within a 1 mile radius and firestorm was started with a radius of about 2 miles. The bomb did severe damage to a greater area than could be expected to be done with an underwater detonation.
Finally, there is no comparison between the energy released by an atomic bomb to the energy released by an earthquake. The recent earthquake in Japan was the equivalent of 6.7 trillion tons of TNT. That's more about 1000 times the explosive power of all the atomic weapons in the world or about 17,000 Hiroshima bombs.
FYI, check out this youtube video that has a nuclear bomb explosion at sea - bikini atolls, that didn't cause tsunami.
Finally in IMDB faq,