IIRC (as in, reciting from memory) the range to the target was measured with a trigonometric equation involving a right-angle triangle. starting with the periscope head at the bottom end, the known height of the ship as the vertical line, one must find the length of the hypotenuse to determine range (the horozontal line in a right-angle triangle)only knowing how tall the ship is.
I do not know the math involved, but thats the gist of it.

sine alpha = opposite/hypotenuse

alpha being angle subtended on periscope

opposite = mast height
hypotenuse = range

so range = mastheight / sine angle subtended

but the earlier scopes with the split prism had this calculation automated, as the american scopes do in sh4

Hitman fairly recently demonstrated that the telemetry markings on the periscope reticule were in radians rather than degrees, which apparently makes the range calculation easier

Don't we use the two catheti and tangens of the angle to calculate the range?

range = height / tangens (angle)

Anyway, for such a small angle the difference between the hypotenuse and longer cathethus is minimal.

Maybe so.
However, it is correct physics.
A torpedo travels about 1000m/min
A 300 meter ship at ten knots travels about 300m/min
If a sub at 1000m, perpendicular to the target fires a torpedo at the middle of the ship, the middle of the ship will have traveled 300 meters by the time the torpedo arrives at the spot at which it was aimed, missing the ship by 150 meters.
If the sub is at 2000 meters, and it fires a torpedo at the center of the ship, it will miss the ship by 450 meters.
Etc.
It appears prudent to lead the ship.

The AOB calculation corrects the equation for the sub not being perpendicular, by adjusting the amount that you lead the ship with your torpedo.

This is done by the torpedo computer when you feed the data into it.
If the torpedo doors are not open when the fire order is given, there will be a delay in the firing of the weapon by several seconds, making the torpedo arrive late and perhaps missing the ship.

The only time range is not important is if both the sub and the target are stopped, and the ship is at least 350 meters away, and no further than the range of the torpedo (?5Km).

this still not correct physics, either;)

innit, and the difference is in fact vanishingly small except at very close ranges. with sine you are measuring range to top of mast, with tangent you measure range to waterline.

e.g 25m mast subtends 1 degree
range to mast top (using sin) = 1432.47m
range to waterline (using tan) = 1432.25 m

a difference of 22 cm

e.g. 25 m mast subtends 10 degrees
range to masttop (using sin) = 143.97m
range to waterline (using tan) = 141.78m

a difference of just over 2 m, and in either case within minimum torpedo range.

Using sine is more practical for the simple reason that the same circular sliderule can be used to solve this calculation as for all the other sine-based calculations you may need to solve!

oh my...math gibberish...:dead:

basically the firing solution is a triangle, which is why trig is so useful. One side of the triangle is made up of the target speed, and another the torpedo speed. All that is then needed is the lead angle. Range is not important. The angle deals with it.

test it yourself. Set the TDC up to show an AOB of say port 80 and a bearing of say 10 and a speed of between 3 and 15 and see what happens to the calculated gyro with different ranges. Try with different speeds.

But it is only really a triangle with straightfire - say gyros between +-15

more than that, it becomes curved fire and to make the torpedo meet the aiming point requires factoring in another triangle, which is the difference between the line of sight between the scope and the target, and the fact that the torpedo exits from the front of the submarine, goes straight until it has cleared the boat, then begins a turning circle to the new gyro. This requires range to provide the necessary information.

test this one too, put in an AOB of port 80 and a bearing of 280 and a speed of 10. See what happens to the gyro when you put in different ranges.

You'll see!

Good point. Thanks for explaining!

Oooooh my aching head! :88)
It is waaay too early in the morning for this.
:D

And one facet of the original question - "how did real u-boat sailors determine range" - is the hardest one of all. Submerged, the split-prism rangedfinder made it easy even when the scope was bobbing and weaving. Surfaced, the had one thing we can never have: experience. On the surface an experienced sailor can look at a ship through binoculars and decide what the type is. Once he knows that he can - if he's any good - estimate the range quite closely just by eyeballing it.

Too bad it's pretty much impossible to do that with pixels on a screen, which is another reason why I still use the Weapons Officer (the other is that I'm lazy).