Minimal-Time Ship Routing

Abstract

The effects of the state of the sea on the safety and economy of a ship's route are of considerable concern to the U. S. Navy and other organizations engaged in shipping. This article deals specifically with the problem of determining a ship's minimal-time route between two ports of call as reflecting an important aspect of desirable routing. This is a minimum value problem and the governing differential equations are derived by use of the Calculus of Variations. The basic theory is essentially the same for a ship on the sea as for an aircraft in horizontal flight, but in application the two problems differ in the manner in which the environment impedes the forward speed of the vehicle. Direct solution of the governing differential equations by numerical methods appears feasible with the aid of an electronic computer and the results of a test case are presented. For this purpose it was convenient to replace a series of empirical equations relating ship's speed to wave height and direction by a single analytical expression. Inspection of the basic differential equations and the empirical relation between ship's speed and the state of the sea indicates that in the case of moderate wave heights, the dependence on wave direction may be omitted as a good approximation. This is borne out in the test case which also shows a considerable saving in computing time because of the resulting simplification of the differential equation.