Physics Behind the Dambuster’s Bouncing Bomb

Introduction
The Dambuster Raid and the bouncing bomb played a key role in WW2. The aim was to disrupt industrial productivity of Germany. The Raid was also a well-publicised success story when Britain was stretched to breaking point during the war. The bomb was used to destroy the Mohne and the Edersee Dams and flood the Ruhr Valley, thereby destroying a large proportion of the Germans manufacturing power; thus having the desired knock on effects for the German war effort.

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The bomb was initially conceived by Dr Barnes Wallis in April 1942 in a paper he wrote called ‘spherical bomb – surface torpedo’[1]. The concept was then taken up by Air Chief Marshal the Hon Sir Ralph Cochrane of the Royal Air Force a strong advocate of precision bombing. Also important to bringing the concept to fruition was Air Marshal Arthur Harris commander of Bomber Command. Through these influential commanders Wallis’ idea was brought to a committee and it given the go ahead.
Wallis faced a range of practical issues such as: the size-to-weight ratio of the bomb with the ability of the aircraft to physically lift and deliver it; how much backspin was needed to be imparted to the bomb in order for it to have a controlled and accurate flight; speed of flight; height from which to drop it and the velocity of the aircraft at point of delivery. All these factors needed to be understood and overcome in order that the bomb could be delivered to the optimum point on the dam wall and then detonated. In answering this question this study will consider four key factors: the weapon design, the delivery of the weapon, the detonation and how all of these had a great enough affect to destroy the dam.
It is useful at this point to qualify the definition of the bouncing bomb. The use of bouncing to describe the Operation Chastise bomb is loose. The physics of bouncing by definition requires a level of elasticity which as the object hits a solid, a fluid or a powder results in a permanent or non-permanent change in the objects form (elasticity). This doesn’t occur with the ‘bouncing’ bomb. It is better to define the Chastise Bomb as ricocheting but for the purpose of this study the phrase bouncing bomb will be used[2].
Weapon Design
This was where the bomb started. A key area which needed to be addressed was the shape of the bomb. This had a major role in providing a reliable and successful bomb. This section explains the reasoning behind the cylindrical shape and how this affected the bombs delivery.
The shape of the bomb was a key issue. Wallis’ initial trials used spherical models, so that identical contact with the water would be made throughout its flight; however the bounces were often too unpredictable due to release and water surface conditions. Therefore, to achieve greater stability Wallis experimented with a cylindrical bomb. This negated the unpredictability but did not stabilise the issues of trajectory and keeping it level. He realised that by using backspin these problems could be overcome. Backspin was also a key aspect in the delivery of the weapon to the detonation point. This vital aspect will be further discussed in this study in the delivery method section.
Once at the point of detonation the bomb was required to explode underwater. On explosion a shock wave would be created, enough to destabilise the dam wall. The weight of water would then provide the breach. Wallis started with trying to find the correct measurements for the amount of explosive needed to breech the dam. He used a model on a scale of 1/17 of the real thing. He then used 100g of gelignite 1.2m away from the wall giving the same effect as a 10 tonne bomb 60m away from the dam. This had no effect. He continued his trials until he achieved 150g of explosive 0.3m away from the dam, which meant that he had to use 13 tonnes of explosive 15m away from the dam. When scaled back up, this would need 18 tonnes of casing which would give a 31 tonne bomb to match the effect required. This was a significantly larger bomb that could be dropped by the aircraft to be used. It was clear that he would have to find a different method. He reduced the mass to 4.3 tonnes and would use multiple bombs to breech the dam[3].
The final dimensions of the bomb were 60 inches long and 50 inches wide[4]. This is roughly 1.52m in length and 1.27m in width, with a final weight of 9,250[5]. See figure 1.
Delivery Method
His next problem was working out speed of the bombs, how far above the surface they needed to be dropped, the distance from the dam and the best way to control the skips of the bomb. His first trials were conducted in his garden at home. He fired marbles across a bucket of water to see whether it would bounce off the surface. It worked and he could control the skip by adjusting the catapult. He now needed to discover if he could control the bomb when it was using multiple skips. For this he needed a slightly larger apparatus and used a huge ship tank at Teddington.
Starting with a spherical bomb, he tested different size-to-weight ratios and by using backspin he could control the bounces. This also helped the bomb to sink in a predictable manner when it reached the wall. Here he had success, however Wallis found the flight of the bomb was often unpredictable. He found if he increased the mass significantly it became more stable however for reasons already stated a larger bomb was impractical.
Wallis had realised that stability could be achieved by using a cylindrical casing and imparting backspin. This would keep the barrel on its axis and stop it from tilting and therefore follow its correct trajectory. Much like a child’s spinning-top toy, the more backspin you gave the bomb the harder it would be to knock it off its axis, this is angular momentum (this is explained in the paragraph below). He tested the idea in the tank trying out the different revolutions. He also found that by varying the size-to-weight ratio of the cylinders he could keep a 5 ton barrel level on the water and then get it to spin down the dam once it hit the water[6]. Also by rapidly spinning the device backwards this would counteract the forward velocity of the aircraft. Wallis calculated how many bounces would be required before reaching the dam. This calculation needed to include the drop distance from the dam, the elevation of the aircraft and its forward velocity. Importantly with each bounce the bomb would slow due to the viscosity of the water and the drag effect that it had. Using this equation Wallis was able to calculate the speed of the spin to ensure that the bomb had slowed down to almost zero velocity by the time it reached the dam[7]. He measured that the cylinder would need to be going at 450 to 500 revolutions per minute2 in order to achieve this effect.
Angular momentum has the same role as linear momentum but in rotation. The equation for angular momentum is. The equation for linear momentum is “”. In the equation for angular momentum the ‘I’ replaces the ‘m’ and the ‘ω’ replaces the ‘v’. The ‘I’ is the moment of inertia which is an objects reluctance to change its state of rotational motion[8]. The equation for the moment of inertia changes with the different shapes it is acting on. For a cylinder the moment of inertia is. This meant that by increasing the mass and the radius the moment of inertia will increase making it more stable. However Wallis was restricted by the size of the planes and their ability to carry a heavy bomb. So he used the largest diameter as possible and then put the majority of the weight of the bomb as close to the edge of the cylinder as possible. This way it would have the same effect as a flywheel giving the barrel lots of momentum. The ‘ω’ is the angular velocity which is how quick the cylinder is rotating its unit is rad s-1. The equation for ω is which shows as you increase the frequency then the ω will increase by a considerable amount. When you put the moment of inertia and angular velocity together you get the angular momentum of a rotating object. It also shows you that by increasing the angular velocity makes it much more difficult to knock the barrel off its axis. Going back to the spinning top the faster you spin it the more difficult it becomes to knock it over. This is what gave the bouncing bomb a clean flight and made sure that it remained on course and didn’t tilt off its axis.
The backspin had a secondary effect. By dropping the bomb without backspin the device would naturally receive a turning effect through the horizontal axis in the opposite direction; the net result of this would be that the bomb would not slow in a uniform or predictable manner and therefore likely skip out over the dam rather than slowing and dropping down the inside face. Forward spinning the bomb would have a similar effect to that experienced by a bicycle wheel being rolled at a curb. It wants to keep going[9].
There is a third effect achieved by imparting backspin. This is the key relationship that Wallis would have been aware of and used to calculate speed, height and turning effect. This effect is the Kuttas Lift Theorem or the Kutta–Joukowski Theorem. Developed by German Martin Wihelm Kutta and Russian Nikolai Zhukovsky (Joukowski), in the early twentieth century, the theorem demonstrates the aerodynamic relationship between lift, speed of a rotating cylinder and density of the substance it is moving through (air or fluid)[10]. This theorem sometimes known as the Magnus effect when applied to the conditions of the Dam Buster raid allowed the bomb to ‘crawl’ down the face of the dam wall. The water surrounding the cylinder in conjunction with the back rotation caused striking hydrodynamic forces that pulled the bomb back towards the wall[11]. As seen in figure 2.
All three of these effects were identified, quantified, understood and overcome by Wallis, through his thorough trials and experiments and his deep knowledge of physics.
Detonation Mechanism
The aim of Operation Chastise was to blow up the dam; the easiest way to do this would be to blow the explosive charge on the water side of the dam at the optimum depth. This would make the most of the explosive power. With the weight of the water behind the explosion, it would increase the affect of the force of the bomb. This weight would pressure the dam to breaking through whatever weaknesses had been caused by the initial force of the bomb. The bomb contained three hydrostatic pistols which measured the water pressure as the bomb sank, the bomb would then detonate at a depth of 30 feet. It also had a time fuse that would detonate after 90 seconds as a backup. This was reasonably well developed technology drawn from the experiences of the First World War naval fighting and the ongoing anti-submarine war effort. In essence the hydrostatic pressure, used in the hydrodynamic pistol, increases uniformly according to the simplified equation of P = p g h (where P is hydrostatic pressure, p is the fluid density (kg/m3), g is gravity and h is height of the water)[12]; the change in the hydrostatic pressure would trigger the hydrodynamic pistol to explode at a depth of 30 feet (9.14m)[13].
Target Effect
Once delivered to the detonation point against the dam wall at the correct depth the weapon exploded. This maximised the benefits of the “bubble pulse” effect typical of underwater explosions, greatly increasing its effectiveness of the explosion and the pressure. The dam wasn’t going to fall by just using the explosive power of the TNT and RDX applied to the external wall of the dam, but by using the pressure of 30 foot (9.14m) of water pressing down on the explosion. The initial force exerted by the exploding bomb was meant to weaken the dam; the water would do the rest. Compared to air water has a significantly higher density than air. Water has a higher quotient of inertia than air. Although this makes water more difficult to move it does mean that it is an excellent conductor of shock waves from an explosion. The damage achieved by these shock waves will be amplified by the subsequent physical movement of water and by the repeated secondary shockwaves or “bubble pulse”[14]. The small seemingly insignificant cracks formed by the bomb would then be exploited by the water forcing the gaps to get larger until the point where the dam couldn’t hold it any longer. The dam then crumbled.
The equation for pressure is (P=pressure, F=force, A=area) this can be rearranged to give this shows us that the pressure will make a huge difference to the force of the explosion.
Summary
In summarising this study of the physics behind the dam buster raid it is important to recognise the breadth of Barnes Wallis’s experimentation and trials. He overcame the issues of weapon design: its explosive effect and detonation method and issues of casing; the delivery method in terms of speed, height and skip effect; the detonation method; and then the weapon effect on the target. A clear understanding physics and a deep understanding of fluid mechanics, hydrodynamic pressure and the crucial consequence of Magnus Effect were essential for Wallis’s concept to succeed.
Bibliography
1 – http://en.wikipedia.org/wiki/Bouncing_bomb – I used wikipedia just to gain some background knowledge and to use in my introduction.
2 – Johnson, W. (1998). “Ricochet of non-spinning projectiles, mainly from water Part I: Some historical contributions”. International Journal of Impact Engineering (UK: Elsevier) – this was from the same Wikipedia page but the extract was taken from this paper written by W. Johnson.
3 – http://home.cc.umanitoba.ca/~stinner/stinner/pdfs/1989-dambusters.pdf – this is another paper on the bouncing bomb providing information on the facts and figures on the bomb
4 – http://everything2.com/title/bouncing+bomb – again this is just facts about the bouncing bomb itself
5 – http://www.rafmuseum.org.uk/research/online-exhibitions/617-squadron-and-the-dams-raid/designing-the-upkeep-mine.aspx – another with facts an about the bombs dimensions and weight ect.
6 –http://simscience.org/fluid/red/DamBusters.html – this is a paper for those doing a-level so has very relevant information on it and is a reliable source
7 – http://wiki.answers.com/Q/Why_did_Barnes_Wallace_decide_to_spin_the_dambusters_bomb_backwards#slide=16&article=Why_did_Barnes_Wallace_decide_to_spin_the_dambusters_bomb_backwards – this is using Wikipedia again but it is a general statement so not needing a confirmation reference
8 – Advanced Physics (p.101/105) – this is a book used in the physics a-level it gives a great level of understanding and was a very useful book when wanting to look beyond the syllabus
9 – A.M. Kuethe and J.D. Schetzer (1959), Foundations of Aerodynamics, John Wiley & Sons, Inc., New York ISBN 0-471-50952-3. – this is a book and it explains basic aerodynamics which can also be related to the forces acting on an object in fluid which is the context used in this essay
10 –http://www.britannica.com/EBchecked/topic/357684/Magnus-effect – this briefly explains the Magnus effect which is used when the bomb is trying to sink down the dam face
11 – Pascal’s law – found on http://www.engineeringtoolbox.com/pascal-laws-d_1274.html
12 – http://www.historylearningsite.co.uk/dambusters.htm – information about the bomb.
13 – Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics (8 ed.). John Wiley & Sons – another book used briefly to explain how the bomb created a large enough force to break the dam.

[1] http://en.wikipedia.org/wiki/Bouncing_bomb
[2] Johnson, W. (1998). “Ricochet of non-spinning projectiles, mainly from water Part I: Some historical contributions”. International Journal of Impact Engineering (UK: Elsevier)
[3] http://home.cc.umanitoba.ca/~stinner/stinner/pdfs/1989-dambusters.pdf
[4] http://everything2.com/title/bouncing+bomb
[5] http://www.rafmuseum.org.uk/research/online-exhibitions/617-squadron-and-the-dams-raid/designing-the-upkeep-mine.aspx
[6] http://simscience.org/fluid/red/DamBusters.html
[7]http://wiki.answers.com/Q/Why_did_Barnes_Wallace_decide_to_spin_the_dambusters_bomb_backwards#slide=16&article=Why_did_Barnes_Wallace_decide_to_spin_the_dambusters_bomb_backwards
[8] Advanced Physics (p.101)
[9]http://wiki.answers.com/Q/Why_did_Barnes_Wallace_decide_to_spin_the_dambusters_bomb_backwards#slide=16&article=Why_did_Barnes_Wallace_decide_to_spin_the_dambusters_bomb_backwards
[10] A.M. Kuethe and J.D. Schetzer (1959), Foundations of Aerodynamics, John Wiley & Sons, Inc., New York ISBN 0-471-50952-3.
[11] http://www.britannica.com/EBchecked/topic/357684/Magnus-effect
[12] Pascal’s law
[13] http://www.historylearningsite.co.uk/dambusters.htm
[14]Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics (8 ed.). John Wiley & Sons.