Recently my ability to understand geometry and mathematics has been impugned somewhat, and although I am impervious to such opinions, I thought I would try and explain my current understanding about something that has puzzled me for a while. It is a long winded explanation and I would welcome comments from those who will bear with me.
Reading Patrick Beesley's book Very Special Intelligence, about the work of Bletchley Park and the radio interception Y stations during WW II, I was considering again the apparent failure to plot accurately the bearings for Lutjens long winded transmission which should have allowed earlier interception by Tovey. I also noticed that Wellings' claims that the same same bearings were successfully plotted in Rodney. How could this be?
Even Rhys-Jones in The Loss of the Bismarck writes a whole chapter and yet does not really pin things down. Tovey's navigator, who has been accused of making a tyro's error, apparently suffered no censure and had a successful career through the rest of the war. Yet the error resulted in KG V steaming in the wrong direction for hours and nearly let Bismarck get away.
The standard marine chart has a lat/long grid at right angles to one another, the meridians all pointing to true north. This is the Mercator chart. However since on the globe meridians all converge at the pole, this is not represented and the effect is more pronounced at higher latitudes. In fact the grid pattern remains and has the effect of distorting distance between points away from the equator.
Ships (and aircraft) travelling long distances choose the Great Circle route which represents the shortest distance ie a straight line over the curved surface of the globe, but this crosses meridians at different angles along its length and so the vehicle is actually steering a continuously varying heading. In practical terms the course to steer consists of a number of straight elements on a series of headings approximating to the curved shape as represented on a Mercator projection chart.
For long ocean passages ships will carry a Gnomonic projection chart, which has a single straight meridian, located around the middle of likely routes, but all other meridians end up with curvature, the effect becoming more extreme at the east and west extremities. Having the straight meridian in mid Atlantic, when the straight line is drawn from say Land's End in Cornwall to New York harbour, course to steer can be derived, with some difficulty, given the curved meridians at the extremities, but as the meridians in mid Atlantic become more straight, the course is easier to derive. The navigator would likely transfer these meridian crossing angles to a Mercator chart, and derive the local courses for his series of straight elements.
Another thing travelling the great Circle route are radio waves as transmitted by Bismarck and so the bearings picked up by interception stations on the UK coast, mostly located for WW I purposes at this time, and thus the East coast, were measured and sent to the Operations Information Centre at the Admiralty London. Beesly implies, but does not actually say, that special Gnomonic charts created for the longitudes (meridians) of the interception stations were then used to plot the bearings out into the Atlantic. Even then the intersection was of a number of lines very nearly parallel to one another with a ellipse of error very wide in longitude and somewhat narrower in latitude. Stations in Iceland and Gibraltar did not pick up the signal so could not improve the geometry. The experienced team at OIC concluded a position and compared with Bismarck's previous location, before Wake-Walker had lost contact, showed she was heading for France.
Having spent some time on deriving this elongated position, the bearings alone, apparently at Admiral Tovey's insistence, were sent to him but also intercepted and at least in Rodney's case independently plotted. What could Captain Frank Lloyd Tovey's Master of the Fleet ie Chief Navigator do with these bearings?
It is clear he did not have gnomonic charts with a central straight meridian for the interception stations. These would be very specialist items. He very likely would have an ocean passage gnomonic chart with a central meridian located in the middle of the Atlantic. However on such a chart the distortion over on the East Coast of the UK would have meant there were curved meridians and somewhere between them were the intercept stations whose local north would be difficult to derive accurately. An error of less than a degree would swing the line many miles either north or south out in the Atlantic.
However radio navigation by bearings of shore transmitters was a well understood exercise, used for many, many years, albeit for usually much shorter ranges and a description of the half-convergency correction technique is included in the Basic RN navigation manual. The allows the navigator to compute an angular correction to the bearing so he can plot a corrected bearing as a straight line on a conventional Mercator chart. With several of these he will get an intersection of varying quality which will have his location within it. It is likely Lloyd used this technique and yet came up with an answer which sent Tovey off in the wrong direction for hours. He has been castigated by some writers and yet how could the technique fail if he carried it out competently?
Well, the Admiralty Manual of navigation says it can be used up to a hundred miles from the transmitter and this example was much more than this, so this is one factor, but also the technique assumes you have a reasonable estimate of position (EP) to input in the first place. In the normal course of navigation one has this as an estimated position from dead reckoning sun sights etc. In the case of the Bismarck where Lloyd was attempting to use the technique in reverse, he didn't have any idea where Bismarck was. The EP is an important parameter in the equation because effectively it draws the curvature of the Great Circle as if on a Mercator chart and then computes the angle of the chord joining the two ends. One end is the known location of the receiving station ashore, but the other end is the location out in the ocean. If this outer end is not constrained in longitude/latitude the bearing of the derived chord could vary considerably.
The OIC had the Gnomonic charts, with the correct central meridian to plot accurately. Lloyd was making do with Mercator charts and had to guess a reasonably accurate position for Bismarck before he could get bearing corrections to apply at all. Maybe he assumed Bismarck was heading for Norway and put an EP into his calculations based on that. Maybe Rodney's navigator, Gatacre input a more realistic EP for Bismarck into his calculations and came up with corrections to the transmitted bearings which allowed a position to be plotted on their Mercator that was closer to OIC's estimate.
If only the Admiralty had overridden Tovey's ridiculous stipulation and sent the derived position as well as the bearings, there would have been error trapping and Lloyd would have suspected there was something wrong. If only the men at sea had trusted OIC's expertise and just trusted a location sent to them. If only it had been envisaged that this might happen and the necessary gnomonic charts with east coast straight meridian had been sent to Lloyd in case they were required.
Thus Lloyd, although sometimes pilloried by those who don't understand the problem, had done nothing wrong. He had done the best he could with the information he had. If Tovey had truly insisted that he wanted no worked out position sent, then he was foolish. However far worse were the decisions of the Admiralty not to send a worked out solution anyway and worse still not to comment for hours when Tovey's transmitted solution differed so much from their own. A hugely valuable intelligence advantage was thrown away for the lack of a little preparation and thought.
It parallels the situation before Jutland when Room 40 told Jellicoe the High Seas Fleet was coming out, so sending the Grand Fleet out to do battle, but subsequently the naval liaison officer assured the Admiral that Scheer's flagship was still in the Jade, because he was unaware of an important fact and never asked the right question. When Scheer went to sea he exchanged call-signs with the Shore Base, so Jellicoe didn't realise his opponent had actually sailed and didn't concentrate with Beatty as early as he might.
Thanks for bearing (joke) with me on this navigational conundrum, if you have.
All the best