Calculus at the Battle of Trafalgar The summer of 2005 marked the 200^{th} anniversary of the British naval victory over a combined French and Spanish fleet in the waters off Cape Trafalgar. During the Napoleonic wars, naval warfare followed certain rules that seem rather formal to us today. The ships in each fleet lined up in a row sailing parallel to its opponent and fired as they sailed past each other (see Figure 1). This maneuver was repeated until one fleet was disabled or sunk. This is known as the directed fire model or conventional combat model.

Figure 1: The White Fleet takes a beating

In such an engagement, the fleet with superior firepower will inevitably win. To model this battle, we begin with a system of differential equations that models the interaction of two fleets in combat. Suppose we have two opposing forces, fleet A with and fleet B with ships initially, and and ships t units of time after the battle is engaged. Given the style of combat at the time of Trafalgar, the losses for each fleet will be proportional to the effective firepower of the opposing fleet. That is,
and ,
where a and b are positive constants that measure the effectiveness of the ship’s cannonry and personnel and A and B are both functions of time. These equations indicate that the rate at which one navy lost ships depended only on two things: the number of ships in the opposing fleet and the effectiveness of the opposition fire. It is assumed that the effectiveness does not change throughout the battle, so the rate at which a navy lost ships was proportional to the number of ships in the opposing fleet.

Part I: Numerical Exploration

Suppose a force of 30 ships engages a force of 40 ships in conventional battle and each has a proportionality constant of 0.04. The battle continues until one force is completely deleted, that is, when it has zero ships left fighting. Use Euler’s Method to determine how many ships remain on the winning side when the battle is over.

Suppose a force of 35 engages a force of 40 and both fleets have the same proportionality constant. How many will remain on the winning side? What is the outcome if 20 ships fight against 40 with the same proportionality constants?

Suppose A has 30 ships and B has 40. If the ships in A are “twice as good” as those in B and the engagement continues until one fleet reaches zero, does A win? That is, does having more effective ships compensate for not having as many ships?

Part II: A Simplified Model It is helpful to begin investigating complex phenomena analytically by simplifying our assumptions. In this case, we will first consider two fleets that are equal in battle. That is, we will assume .

Using the above differential equations, set up an equation for . Solve this differential equation to show that the total number of ships still fighting is decreasing exponentially. (Hint: Let _{} and write the differential equation in terms of _{} and _{}.)

Set up an equation for . Assume that . Solve the differential equation to show that the positive difference in the size of the two fleets is increasing exponentially. (Hint: Let _{} and proceed as in the hint to question 1.)

Use the results in questions 4 and 5 to solve for A as a function of time and for B as a function of time.

Part III: A More Inclusive Model If, the techniques used in part I cannot be used to determine the number of ships in each fleet as a function of time.

Use and to write in terms of A, a, and b. Solve for. There will be two solutions to this 2^{nd} order differential equation.

Verify that the sum of the two solutions from questions 7 is also a solution to the 2^{nd} order differential equation. We call this the general solution.

Use the general solution obtained in question 8 to find. Using the initial conditions _{} and _{}, find values for the constants in your equations for _{} and _{}.

In your equations for and replace a and b with k. Simplify and confirm that the simplified solutions obtained in question 6 are special cases of the solutions from questions 8 and 9.

If and, find and solve for A in terms of B. This equation will give you the expected number of ships remaining in fleet A when.

Write an equation that gives the expected number of ships remaining in fleet B when A = 0.

Part IV: A Closer look at Nelson’s Strategy The commander of the British fleet was Admiral Nelson. In the now famous Battle of Trafalgar, he exhibited cunning military strategy. In one account of the battle, Nelson expected to have 27 ships in the British fleet (B) and predicted that the French/Spanish Armada (A) would have 34 ships. In planning his strategy, Nelson believed that the British fleet was better prepared (and better led) than the French/Spanish Armada. Suppose that.

If Nelson’s 27 ships fought a conventional battle against the 34 ships in the French/Spanish Armada with, would he win? How many ships would remain in the winning fleet?

Instead of sailing parallel to the French/Spanish Armada, Nelson planned to sail through the middle of the fleet, cutting it in half and fighting two separate conventional battles. In one battle, he would have numerical superiority and consequently win that battle. In the other, he would have fewer ships and lose. It was Nelson’s hope that, in a third and decisive battle, the British fleet would be victorious. It should be noted that Nelson assigned himself the task of leading the portion of his fleet that was expected to lose its battle.

Determine if Nelson could have arranged his 27 ships to defeat a larger fleet of 34 ships using the 3-battle plan as described above with by using the results from questions 8 and 9 and experimenting with different configurations of the British fleet. According to our model, how many ships would be expected to survive the final battle?