‘What Course Should You Steer?’

Extract

Van de Ree has provided a solution to what must be a fairly common problem in navigation. The problem may be stated as follows.

A ship on a rhumb-line course encounters head-winds and heavy seas requiring her to reduce speed. If she alters course, but does not change the rate of fuel consumption, her speed will be given by (a + b sin A), where A is the course change, a is the speed into the head sea and a + b is the speed in a beam sea. How far should the navigator alter course so that, after a given time, the ship will be nearer the destination than on any other course and, in addition, the labouring of the vessel will be reduced?

Van de Ree's trial-and-error solution assumed the destination to be initially 63 miles away, a = 12, b = 5, and the optimization period 1 hour. His solution was to alter course by 20(± 2·5)°. It is possible to obtain a general algebraic solution to this problem, but it requires differentiating complicated trigonometric functions and solving a complicated trigonometric equation, and so the algebraic solution is not practically useful. It is also possible to solve the problem graphically. This solution is by far the simplest one and would be recommended in place of either Van de Ree's laborious numerical solution, or the equally difficult algebraic solution.

However, one could question whether the chosen tactic is the appropriate one.