The Sailings calculations determine the course for a vessel to follow in order to get from the point of Departure to the Destination. This is course with respect to ground which can differ from the course to steer if the vessel is deflected sideways by currents and/or winds. This leads to a class of “set and drift” problems described, for example, in the Dead Reckoning chapter in Bowditch. These problems are often solved graphically by plotting procedures. As for the equivalent numerical solution we can do the following.
The essence of these problems is the relationship between three vectors in which the ground speed is the vessel’s speed relative to the water plus the set and drift vector. The relevant geometry occurs in 2-D, so this relationship translates into two equations for the vectors’ components. Therefore, given four pieces of information on input we can solve for the remaining two.
If these two numbers both pertain to the same unknown vector (e.g. its magnitude and direction), then two Cartesian-component equations provide the solution to the problem by direct addition or subtraction of the other two (fully specified) vectors. This is the case for spreadsheets:
http://www.navigation-spreadsheets.com/dead_reckoning.html#set_and_drift
ground_speed.xls:
Calculation of the ground speed from the current’s speed and direction (i.e. set and drift) and the vessel speed relative to the water.
course_and_speed.xls:
Calculation of the required vessel speed and course from the set and drift and the desired ground speed and track.
The situation is a little bit more complicated for:
course_to_steer.xls:
Given the set and drift, the vessel’s speed and the intended direction relative to ground, this spreadsheet calculates the required vessel course and the resulting ground speed. If the vessel’s speed is too small to counteract the current, an error message is displayed in row 4.
in which the two unknowns are distributed between two partially known vectors. This problem can be solved in the following steps encoded into the spreadsheet:
1) Decompose the current’s vector into components parallel and perpendicular to the prescribed “Sailings” (ground) direction.
2) Reverse the sign of the perpendicular component; this becomes the perpendicular component of the vessel’s speed with respect to water. That way the deflecting effect of the set and drift is neutralized.
3) Use the Pythagorean theorem to determine the component of the vessel’s speed (w.r.t. water) parallel to the ground direction. The known vessel’s speed w.r.t water is the hypotenuse and the result of step 2) is one of the sides.
4) Convert the vessel’s speed’s now known two components (along with the ground direction) into course to steer.
5) Add the two parallel speeds’ components in order to obtain the ground speed. If this problem has two mathematically good solutions, this picks the “faster” one.
This problem does not always have a solution. If the vessel’s speed w.r.t. water is less than the current’s perpendicular component, the vessel is not fast enough to compensate for the deflection off course. If the current’s parallel component is negative (e.g. a strong headwind in case of an aircraft) and larger in absolute value than the parallel component of the vessel’s speed w.r.t. water, then the vessel is not fast enough to progress toward its destination. In either case, the spreadsheet zeroes out the output and displays an error message.
(first published on November 11, 2011)
Navigation Spreadsheets 