Was Hiroshima targeted because its mountains would "rebound the explosion back into the city for more damage"?

Question

In English class, while analyzing the anti-war themes of a book, my teacher brought up something that I had never heard before and honestly significantly doubt the accuracy of. She said that:

Hiroshima was chosen as a target for a nuclear bomb because the mountains around the city would enclose the explosion and rebound it back into the city for more deaths to the population and more damage to the city.

Does this statement have any truth to it?

I've never heard anything like this, and I can't find anything about it online.

Answer 1

The Wikipedia entry on the atomic bombings cites a few sources that confirm this was indeed a criterion, among others (emphasis mine):

In April 1945, Marshall asked Groves to nominate specific targets for bombing for final approval by himself and Stimson. [...]

The Target Committee nominated five targets: Kokura, the site of one of Japan's largest munitions plants; Hiroshima, an embarkation port and industrial center that was the site of a major military headquarters; Yokohama, an urban center for aircraft manufacture, machine tools, docks, electrical equipment and oil refineries; Niigata, a port with industrial facilities including steel and aluminum plants and an oil refinery; and Kyoto, a major industrial center. The target selection was subject to the following criteria:

• The target was larger than 3 mi (4.8 km) in diameter and was an important target in a large city.
• The blast would create effective damage.
• The target was unlikely to be attacked by August 1945.

These cities were largely untouched during the nightly bombing raids and the Army Air Forces agreed to leave them off the target list so accurate assessment of the damage caused by the atomic bombs could be made. Hiroshima was described as "an important army depot and port of embarkation in the middle of an urban industrial area. It is a good radar target and it is such a size that a large part of the city could be extensively damaged. There are adjacent hills which are likely to produce a focusing effect which would considerably increase the blast damage. Due to rivers it is not a good incendiary target."

The exact source cited is a transcript from the Target Committee dated May 10-11. The emphasized quote appears under "6. Status of Targets".

Answer 2

Although I don't know anything about the historical fact of whether Hiroshima was actually chosen because of a conjectured focusing effect by anyone responsible for this decision, I might shed some light on the physical background. From that point of view, the decision might have been influenced by the hope for a significant "focussing effect", but this hope is not well justified by physical fact.

Short answer: Although a detailed treatment of the conditions of this incident would involve several more complicated effects (detailed geometrical model of the terrain around Hiroshima, wave mechanics, ultrasound propagation, temperature effects, etc.) the overall behavior of the energy distribution is roughly covered in large parts by just geometrical optics. Seen from this perspective, the percentage of additional destructive energy due to an alleged "focussing effect" is definitely very low. The main effect of the surrounding mountains is to scatter the pressure wave at a slightly different angle to infinity, but not dominantly back to the target.

Any physicist who is qualified for the matter and is in control of himself, would not ever have recommended using the "focussing effect" to optimize the result because many other variables will have had a greater effect. If a qualified physcist has expressed such a recommendation, it probably might have been biased by social dynamics. Another plausible explanation why the decision could have been influenced by the hope for a "focussing effect", is that a member of the military or a politician thought to know better and wanted to make a name for themselves.

Long answer: For a significant focusing effect to take place, there is simply the need for sufficiently many surface elements to be oriented in such a way that they reflect "beams" of pressure back to the target (angle of incidence = angle of reflection). In the given scenario this means that the mountainsides would have to be almost perpedicular to the ground, which is obviously not the case, as you can see in Google Earth.

A way to derive this result with a little more rigor, is to view the Hiroshima terrain basin as an approximation of a hollow spherical cap which acts as a spherical mirror on the incoming pressure wave of the bomb.

Looking at the terrain in Google Earth I can estimate the average maximum elevation of the mountains surrounding Hiroshima to approximately h = 500 meters. The lateral extension of the Hiroshima basin is around d = 18000 meters in diameter. Using Wikipedia: Spherical Cap one can compute the radius of curvature of this cap to be: r = 81250 meters. According to this site one can calculate the focal length of a spherical mirror to be half the radius of curvature, in our case: f=40625 meters.

The focal length means the following: if a point source of excitation (in our case the Bomb) is exactly in the focal point (i.e. at the focal length distance from the mirror), the reflected wave will be a parallel wave that travels back to infinity (to the stratosphere) without any focussing or defocussing. However if the excitation source is further away from the mirror than that (say at distance s1 from ground), the reflected wave will again be a spherical wave, but this time traveling towards a focus point, the distance (say s2 from ground) of which calculates from the focus equation: 1/f = 1/s1 + 1/s2.

However the nuclear explosion of Hiroshima was ignited at a height of approximately ~580m above ground (Wikipedia) which is not only closer to ground than the focal length of the terrain basin, but it is also a lot closer. For source distances closer to the mirror than the focal length, the above focus equation yields a negative result for s2. What does this mean? Well, it just means that there is a virtual focus point behind the mirror, from where all wave fronts of the reflected wave seem to emerge. And this simply amounts to the reflected wave being scattered instead of being focused!

This first seems to contradict the intuition, because we somehow always expect a hollow spherical mirror to show some amount of focussing. And indeed, if we compare it with the plane mirror, which is also scattering the wave because it has focal length = infinity, we can say that even if the hollow mirror is scattering the wave, it is scattering it less than the plane mirror.

The bottom line is, although the Hiroshima basin is not focussing the reflected wave anywhere in space, let alone the target in the city of Hiroshima, but actually scattering it, it scatters it at least less than if Hiroshima had flat terrain. For the destructive energy at the target this is practically irrelevant however. The wave will just be deflected at a steeper angle towards infinity.

Edit (2018.04.24): Wave mechanics makes this simple consideration a little more complicated in that geometrical optics is not strictly valid anymore then. Anytime a wave front hits a part of the mountain, this location will again be the origin of a spherical wave, spreading in all directions. All of these superpose to give some complicated pattern, usually with some interference (like you see when two stones are thrown into the pond). This is what is called diffraction.

Diffraction doesn't obey the rule "angle of incidence = angle of reflection", i.e. diffracted waves may be cast back even if the inclination angle of the mountains is "not right" and the waves can even move around corners to some extent. So in spite of not showing any geometrical focusing effects, there may have been an additional destructive energy from this effect in the Hiroshima bombing. However, since energy cannot be destroyed nor created during all of this, it is quite simple to estimate an upper bound for the scattered energy.

Suppose E is the original energy that is set free by the bomb. This energy is emmitted through a spherical shock wave and thus is distributed over the surface of a half-sphere of area A = 2*pi*r^2 (the full sphere has area 4*pi*r^2, but the ground cuts it in half). The mountain silhouette as seen from the bomb is a hollow cylinder of height h and radius r, so has a projected surface area of P = 2*pi*r*h. This corresponds to an energy fraction E1 = E*2*pi*r*h/(2*pi*r^2) = E*h/r. If all the energy that hits this virtual cylinder got reflected back to the detonation site, the energy E1 would be what increases destruction due to diffractive effects. So the additional destructive energy percentage is smaller than

E1/E < h/r ~2.8%

In reality it will even be much smaller because part of the energy will get diffracted in all directions instead of just towards ground zero.