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Concept
Lanchester's laws
Summary
Lanchester's laws are mathematical formulae for calculating the relative strengths of military forces. The Lanchester equations are differential equations describing the time dependence of two armies' strengths A and B as a function of time, with the function depending only on A and B.
In 1915 and 1916 during World War I, M. Osipov and Frederick Lanchester independently devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's linear law (for ancient combat) and Lanchester's square law (for modern combat with long-range weapons such as firearms).
As of 2017 modified variations of the Lanchester equations continue to form the basis of analysis in many of the US Army’s combat simulations, and in 2016 a RAND Corporation report examined by these laws the probable outcome in the event of a Russian invasion into the Baltic nations of Estonia, Latvia, and Lithuania.
For ancient combat, between phalanxes of soldiers with spears for example, one soldier could only ever fight exactly one other soldier at a time. If each soldier kills, and is killed by, exactly one other, then the number of soldiers remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons.
The linear law also applies to unaimed fire into an enemy-occupied area. The rate of attrition depends on the density of the available targets in the target area as well as the number of weapons shooting. If two forces, occupying the same land area and using the same weapons, shoot randomly into the same target area, they will both suffer the same rate and number of casualties, until the smaller force is eventually eliminated: the greater probability of any one shot hitting the larger force is balanced by the greater number of shots directed at the smaller force.
Lanchester's square law is also known as the N-square law.
With firearms engaging each other directly with aimed shooting from a distance, they can attack multiple targets and can receive fire from multiple directions.
Official source
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