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LCDR Walsh served aboard USS NEWPORT NEWS (SSN 750) and USS MAINE (SSBN 74I)(Blue) before becoming an Engineering Duty Officer. He is now the Submarine Auxiliaries Inspector for INSUR V.

It is morning watch during the last inbound transit of the deployment. Last night the boat encountered the remnants of a late-season hurricane, and the watch team stayed below decks during the night. The rain continued at dawn, but the seas have since calmed, and it is time to shift the watch to the bridge. The relieving OOD and lookout open the hatch, and Control buzzes with activity as the off-going watch standers energetically assist with the bridge rig. As the messenger is transferring his lanyard from the cockpit to the flying bridge, the ship takes a roll, and he slips off the wet, rounded edge of the sail. Fortunately, the lookout spots him as he glances off the hull and splashes into the wake.

“Man overboard, port side!”
The OOD keys the 7MC and orders, “Left full rudder!
All stop! Man overboard, port side.”

The control room shifts efficiently to the casualty situation. The quartermaster keys the GPS for man overboard, then shifts his plot to 200 yards per inch and marks the bug. Meanwhile, the FTOW generates his own stationary contact in fire control. These three technologies are meant to help locate the man, but all three are wrong from the outset. The ship was going 14 knots at first, and has already traveled over 100 yards since the messenger fell.
The situation is changing too rapidly. The lookout keeps his eyes on his shipmate as long as he can, but loses him in the choppy seas. He searches in vain with his binoculars. The Contact Coordinator spots the sailor next. He already has his microphone in hand, and keys it without taking his eye off his shipmate. “Bridge, Coordinator, man bears 148, range 230 yards.” The quartermaster marks his plot immediately; the initial mark was way off. The FTOW adjusts his contact, too. The Contact Coordinator loses sight of the man in the heaving seas, but regains visual contact with cueing from the quartermaster’s updated plot.
“Bridge, Coordinator, man bears 135, range 260 yards.”
The lookout can’t yet see the messenger among the waves,
but the 000 knows where to steer.

“Bridge, Coordinator, man bears 095, range 120 yards.”
As the Captain arrives on the bridge, the OOD is already driving the ship for a textbook recovery. When the diver goes topside, his man is bobbing amidships just yards away. The messenger has bruises and a mild concussion, but he’ll be fine.

How did the Contact Coordinator judge the distance so precisely? He used a simple variation of the periscope ranging method taught in Submarine School. To find the range to a visual contact, submariners use the venerable masthead height equation shown in Figure 1, a diagram from the now-declassified Submarine Torpedo Fire Control Manual published in 1950.

Editor’s Note: In review it was noted that this method was previously described in a 4 December 2002 memorandum

The fonnula works through trigonometry. The masthead
height (MHH) represents the short side of a triangle, and the
number of periscope divisions corresponds to the subtended angle
8. We modify the conversion factors slightly for quick, easy use
during periscope observations (mental gym):
In low power: Range(yds) = 20 * MHH(ft) /# of divisions
In high power: Range(yds) = 80 * MHH(ft) /# of divisions

Provided we know the masthead height with some precision, this method is accurate enough for contact coordination and for computing a torpedo firing solution.

On the other hand, the formula is highly inaccurate for a small contact with uncertain masthead height:
The equation fails completely for a contact with no masthead height at all (e.g. a man overboard, a lobster pot, or a northern right whale):

This article shows how to obtain a timely, accurate periscope
range to such contacts. The method, called Height-of Eye
Ranging, works very well when certain assumptions are met.

Assumption 1: Own ship’s height of eye is known. For a submarine operating on the surface, the height of eye (HOE) is simply the height of the periscope optics above the keel (a known, fixed value) minus the keel depth (a measurable, fixed value when operating on the surface).

Assumption 2: The horizon is visible and steady. The horizon makes an excellent reference, as long as fog, land, or rain doesn’t interfere with the line of sight.

Assumption 3: The world is flat. We’ll get back to this one. Height-of-Eye Ranging works by the postulate of alternate interior angles. Consider a man floating in the ocean near a surfaced submarine. We’ll call him Oscar.

From Oscar’s point of view, the subtended angle 9 from the submarine’s waterline up to the periscope optics is a function of his range to the submarine and the periscope height of eye. From the scope operator’s perspective, the angle subtended from the horizontal plane down to Oscar’s position is the same angle 9 . Thus HOE could substitute for MHH in the masthead height equation to find the range to Oscar- if only Assumption 3 were true. The horizon would be infinitely far away and lie in the horizontal plane, and we could calculate the range to Oscar out as far as we could see him. However, the world is not flat, so Assumption 3 must be revised.

Assumption 3A: The world is 11early flat.

The actual horizon is -1/9 of a degree below the horizontal plane as seen from the periscope. For such a small angle, a linear approximation is valid over some range, then falls apart as Earth’s curvature comes into play. If Oscar is close aboard, Earth’s curvature has little effect.

To use the Height-of-Eye Ranging method, first find the height of eye of your periscope during surfaced operations. Then multiply the HOE by -17.75 to find the HOE Constant in yards. The following table gives representative values.

Periscope type HOE (ft) HOE Constant (vds)
688, #1 scope 40 710
688, #2 scope 36 640
774 38.3 680 ​
SSBN / SSGN 48.5 860​

The formula for Height-of-Eye Ranging is: In low power, range(yds) • HOE Constant 1# of divisions below
horizon In high power, range(yds) = 4 • HOE Constant 1# of divisions below horizon

In this example figure, USS NEWPORT NEWS (SSN 750) is conducting a man overboard drill during TRE workups. Oscar is visible through #2 periscope in low power. He appears 8 divisions below the horizon. His range is therefore 640 I 8 = 80 yards.

Height-of-Eye Ranging is accurate within 10 yards for objects inside a 600-yard radius from the periscope. The accuracy degrades to about 8% at 1000 yards, and rapidly falls apart beyond that. The technique easily accommodates close-aboard range tripwires for the 500-yard Naval Vessel Protection Zone and Right Whale Protective Measures. Simply plug in the tripwire range and reverse the calculation to detennine the corresponding number of divisions below the horizon.

To train your periscope operators with Height-of-Eye Ranging, start by finding the distance from the periscope to the waterline of own ship’s rudder. (The actual value can be found in the SSM or ship’s drawings.) Then practice finding the range to offshore buoys and small craft held on radar during a surface transit. A conversion table of divisions and ranges posted on each scope can be helpful.

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