Lanchester's laws are mathematical formulae for calculating the relative strengths of a predator/prey pair. This article is concerned with 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.[1][2]
In 1916, during the height of World War I, Frederick Lanchester 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).
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In ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men 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 firing. If two forces, occupying the same land area and using the same weapons, fire 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.
With firearms engaging each other directly with aimed fire from a distance, they can attack multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons firing. Lanchester determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law.
More precisely, the law specifies the casualties a firing force will inflict over a period of time, relative to those inflicted by the opposing force. In its basic form, the law is only useful to predict outcomes and casualties by attrition. It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time. It only works where each man (or ship, unit or whatever) can kill only one equivalent enemy at a time (so it does not apply to machine guns, artillery or—an extreme case—nuclear weapons). The law requires an assumption that casualties build up over time: it does not work in situations in which opposing troops kill each other instantly, either by firing simultaneously or by one side getting off the first shot and inflicting multiple casualties.
Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.
Suppose that two armies, Red and Blue, are engaging each other in combat. Red is firing a continuous stream of bullets at Blue. Meanwhile, Blue is firing a continuous stream of bullets at Red.
Let symbol A represent the number of soldiers in the Red force at the beginning of the battle. Each one has offensive firepower α, which is the number of enemy soldiers it can knock out of battle (e.g., kill or incapacitate) per unit time. Likewise, Blue has B soldiers, each with offensive firepower β.
Lanchester’s square law calculates the number of soldiers lost on each side using the following pair of equations [3]. Here, dA/dt represents the rate at which the number of Red soldiers is changing at a particular instant in time. A negative value indicates the loss of soldiers. Similarly, dB/dt represents the rate of change in the number of Blue soldiers.
Lanchester’s equations are related to the more recent Salvo combat model equations, with two main differences.
First, Lanchester's original equations form a continuous time model, whereas the basic salvo equations form a discrete time model. In a gun battle, bullets or shells are typically fired in large quantities. Each round has a relatively low chance of hitting its target, and does a relatively small amount of damage. Therefore Lanchester’s equations model gunfire as a stream of firepower that continuously weakens the enemy force over time.
By comparison, cruise missiles typically are fired in relatively small quantities. Each one has a high probability of hitting its target, and carries a relatively powerful warhead. Therefore it makes more sense to model them as a discrete pulse (or salvo) of firepower in a discrete time model.
Second, Lanchester's equations include only offensive firepower, whereas the salvo equations also include defensive firepower. Given their small size and large number, it is not practical to intercept bullets and shells in a gun battle. By comparison, cruise missiles can be intercepted (shot down) by surface-to-air missiles and anti-aircraft guns. So it is important to include such active defenses in a missile combat model.
In modern warfare, to take into account that to some extent both linear and the square apply often an exponent of 1.5 is used.[4][5][6]