A while back, I was reading the chapter in Bowditch dealing with dead reckoning. I was especially interested in the section dealing with course and speed made good. Well, last night on my lunch break (I work nights), I decided to solve a quick problem I made up:
"If I were on a ship which was 10nm due East of port, with a set of 360° and drift of 3 kts., and I needed to arrive in port in exactly one hour, what course and speed should I use?"
I figured the answer to be 253° at 10.5 kts. But then I started to wonder: That's all well and good for a vessel under engine power, but what about a sailboat? I know the way to figure course to steer at a given speed, but wouldn't turning a sailboat into the current (as in my example) actually slow the speed through the water enough to change the necessary course to steer? Or is this error so small as to be negligible? Or is there some other way to factor that in?
Determining the course to steer when underway in a sailing yacht is an iterative process. It can be done "on the water" based upon ones actual speeds achieved or--with somewhat lesser accuracy--with one's Polars, although Polar plots don't account for variations in sea state which can effect ones speeds regardless of what the Polars indicate (hence they are actually relative speed diagrams rather than actual).
The solution can be done mathematically although a discourse on the trig would go over a lot of heads. A graphical method, tho' is pretty easy. One begins with ones point of departure, the rhumb line to one's destination, one's assumed speed, and the average set and drift of the currents across the rhumb line. One first sets ones dividers to the distance of one's assumed speed along the rhumb line. (For example, if the scale of the chart is such that an inch = 2 miles, and ones assumed speed is 6 knots, one would set the points of one's divider 3" apart). One then strikes a line through the point of departure at the angle of the average "set" of the current across the rhumb line and marks a point that is the distance along the set line equal to the "drift". (For example, the average "set" might be 45º, with an average "drift" of 2.5 knots, hence one would place a mark along the set line that is 1-1/4 inch from the point of departure.) One then swings the divider in an arc (a "speed arc") from the "set point", to the rhumb line, marking the point of intersection. A line from the "set point" to the intersection of the speed arc with the rhumb line is the course to steer. (This is where Polars come in handy.) If the course to steer is closer to the wind, one might find that ones speed is somewhat less than the speed one assumed in making one's "speed arc", hence the radius of the speed arc needs be reduced. Likewise, if the course to steer is further off the wind, such that one's speed is greater than the speed assumed in making the speed arc, one would increase the radius of the speed arc. In either case, one then marks the intersection of the "adjusted" speed arc--around the set point--with the rhumb line and defines an adjusted course to steer. How many iterations of the foregoing are necessary depends upon the shape of the Polars of one's own yacht (in the sense of how much one's yacht's speed with vary relative to changes in the apparent wind). The foregoing process can be done without Polars, simply by observing ones actual speed through the water, which is the reference frame one is relying upon (even though the reference frame is moving itself, at the set and drift of the current) and making periodic adjustments from new points of origin as the voyage progresses.
The foregoing process is somewhat more difficult to describe than to actually perform and, unfortunately, in these daze of GPS navigation, not often performed even though it can shorten the length and time of a passage (and more significantly as the distance between point of departure and destination increases). The passage between say Miami and Bimini, across the Gulf Stream, is a good example of where the process can be used to advantage.
In terms of your question seeking a specific time of arrival, that can be achieved to the extent that one can slow ones progress, perhaps by "scandalizing ones rig", as one might do to arrive at a specific point at a specific time for, perhaps, a Bar passage or to save ones tide, but, of course, one cannot sail faster than ones maximum potential speed at a given apparent wind angle. In such cases ones arrival point may be unknown, but one can define a "Known Unknown" rather than allowing oneself to confront an "Unknown Unknown". For example, on passages between the Channel Islands and the Alamitos Bay Channel entrance in Long Beach, we would always steer to favor an arrival point northwest of the Channel entrance so that, if we didn't hit the Channel bang one, we knew that once the coast came into view we could merely head east along a bottom contour line to find our entrance. (I.e. we didn't know where the channel was, specifically, but we knew it was to the east of us; and, we knew if we stayed in 10 fathoms, we weren't going to bang into anything while we sailed down to it--hence a known unknown.)