معادلات الملاحة

Lighthouse Formula

D = 2.08 × (√H + √h)

• D= Geographic range — distance at which the light rises or dips (nautical miles)
• H= Height of the light above sea level (metres, from the chart)
• h= Height of observer's eye above sea level (metres)

The instant a light 'rises' (first seen) or 'dips' (disappears) gives an instant position line — clock the bearing at the same moment.

Distance Off by Vertical Sextant Angle

D = 1.854 × H ÷ θ

• D= Distance off the object (nautical miles)
• H= Height of the object above sea level (metres)
• θ= Corrected vertical sextant angle in minutes of arc (1° = 60′)

Measure the height from the object's charted elevation. If its base is at sea level, correct for the height of tide.

Vertical Danger Angle (limiting line)

VDA = 1.854 × H ÷ d (minutes of arc)

• VDA= Maximum allowable vertical sextant angle (arc minutes)
• H= Height of the object (metres)
• d= Minimum safe passing distance from the danger (nautical miles)

If your measured vertical sextant angle grows larger than the VDA, you have crossed inside the limiting danger line — alter away.

Doubling the Angle on the Bow

Distance off = Distance run (when bow angle doubles)

• A (first angle)= Bearing of the object measured from dead ahead at the first observation (degrees)
• 2A (second angle)= When the bow angle doubles exactly — note the log and take the bearing
• Distance run= Distance steamed between the two observations (nautical miles)
• Distance off= Distance from the object at the moment of the second bearing — equals the distance run

Works for any starting angle A provided the second bearing equals exactly 2A. Common exam pairs: 30°/60°, 22½°/45°, 26½°/45°. The object is still forward of the beam when the second bearing is taken, giving time to act. See also: Four-Point Bearing (45°/90°) — the most frequently examined special case.

Four-Point Bearing (45° Rule)

Distance off (abeam) = Distance run (from 45° to beam)

• 45° bearing= Object bears 45° on the bow (four compass points from ahead) — start the log
• Beam bearing= Object comes abeam (90° on the bow) — read the log
• Distance run= Distance steamed between the 45° bearing and the beam (nautical miles)
• Distance off (abeam)= Closest point of approach as you pass the object — equals the distance run

'Four points' is the old compass term for 4 × 11.25° = 45°. This is the most-examined form of the doubling-angle rule. Useful at night to gauge clearance off headlands without a sextant — note the bearing when 45° on the bow, read the log when the object comes abeam.

Safety (Anchor) Swing Circle

R = LOA + cable veered + safety margin

• R= Radius of the swing circle drawn around the anchor
• LOA= Length overall of the vessel (metres)
• Cable veered= Maximum length of cable paid out (1 shackle = 27.5 m)
• Safety margin= Extra clearance you add for safety (metres)

Plot the circle centred on the anchor position. If any danger falls inside it, choose a different berth, less scope, or accept a larger margin.

Bridge Swing Circle

R = (stem → standard) + cable paid out

• R= Radius of the circle the bridge position should stay within
• stem → standard= Distance from the stem to the standard compass / bridge position (metres)
• Cable paid out= Length of cable veered (1 shackle = 27.5 m)

Because the GPS antenna is by the bridge, this circle lets you confirm at a glance that the vessel has not dragged.

Stern Swing Circle

R = waterline length + cable paid out

• R= Radius of the circle swept by the stern as the ship swings
• Waterline length= Length of the vessel on the waterline, LWL (metres)
• Cable paid out= Length of cable veered (1 shackle = 27.5 m)

Use this when berths are tight or you anchor near other vessels — the stern sweeps the largest arc as the ship swings.

LDL Anchor (no squat)

LDL = Draught + Safety margin − Height of tide

• LDL= Least charted depth (sounding) needed at the anchor berth (metres)
• Draught= Static deepest draught of the vessel (metres)
• Safety margin= Required under-keel clearance (metres)
• Height of tide= Height of tide above chart datum at the time (metres)

No squat term — the ship is stopped at anchor. Check the depth against the LOWEST height of tide expected during the stay.

Speed · Time · Distance

D = S × T (6-min rule: D = S ÷ 10)

• D= Distance (nautical miles)
• S= Speed (knots)
• T= Time (hours)

In 6 minutes (1/10 of an hour) a vessel runs one-tenth of its speed in nautical miles — handy for quick pilotage checks.

1-in-60 Rule (track error)

Error (°) = (off-track distance ÷ distance run) × 60

• Error= Course correction needed (degrees)
• off-track distance= Cross-track distance from the intended track (nm)
• distance run= Distance travelled along the track (nm)

A 1 nm error over 60 nm equals 1°. The rule scales for any leg length and is the basis of quick visual position-keeping.

Rate of Turn for a Radius

ROT (°/min) = 0.955 × S ÷ R

• ROT= Rate of turn (degrees per minute)
• S= Speed over ground (knots)
• R= Radius of the turn (nautical miles)

Set this on the rate-of-turn indicator to hold a planned curved track through a channel bend.

Squat — Quick Estimates

Squat ≈ V² ÷ 100 • 0.3 m per 5 kn • 10% of draught

• V= Ship's speed through the water (knots)
• V² ÷ 100= Quick open-water estimate of maximum squat (metres)
• 0.3 m / 5 kn= Allow 0.3 m of squat for every 5 knots of speed
• 10% of draught= Rule-of-thumb maximum squat in shallow water

These are quick rule-of-thumb checks. Squat rises with the SQUARE of speed, so slowing down is the most effective way to cut it. Barrass formula: Squat ≈ Cb × V² ÷ 100 (use ÷ 50 in confined / shallow water).

Cable to Veer — Forged Steel & Aluminium Bronze

Forged steel: 1.5√d (m) or 2√d (fm) Aluminium bronze: √d (m) or 1.3√d (fm)

• Shackles= Amount of cable to veer (in shackles; 1 shackle = 27.5 m)
• d= Depth of water at the berth
• Forged steel= 1.5 × √d if depth is in metres, or 2 × √d if depth is in fathoms
• Aluminium bronze= √d if depth is in metres, or 1.3 × √d if depth is in fathoms

Use the factor that matches your cable's material and your depth units (metres or fathoms). Increase scope for strong wind, tide or poor holding ground.

Echo Sounder — Acoustic Depth Formula

D = (V × T) ÷ 2 · Shortcut: D = 750 × T

• D= Depth of water below the transducer (metres)
• V= Speed of sound in seawater — standard value 1,500 m/s (use any stated value in the question)
• T= Total two-way travel time: from transmission to receipt of the echo (seconds)
• ÷ 2= The measured time is for the downward AND return journey — divide by 2 for one-way depth
• 750 × T= Quick shortcut: because 1,500 ÷ 2 = 750, depth (m) = 750 × time (s) when V = 1,500 m/s

The standard exam speed is 1,500 m/s. If the question gives a different value (e.g. 1,520 m/s in warm, saline water), substitute it and still divide by 2. The shortcut 750 × T only works when V = 1,500 m/s. Add the transducer depth to get the true water depth from the surface (see the 'Echo Sounder — True Depth & UKC' formula).

Echo Sounder — True Depth & UKC

True depth = ES reading + transducer depth UKC = ES reading − (Draught − transducer depth)

• True depth= Actual depth of water below the sea surface (metres)
• ES reading= Depth displayed by the echo sounder — measured below the transducer (metres)
• Transducer depth= Depth of the transducer below the waterline (metres) — given in the echo sounder's specification
• Draught= Ship's deepest static draught (metres)
• UKC= Under-keel clearance — water between the keel and the seabed (metres)

The echo sounder measures from the transducer downwards, not from the sea surface. Always add the transducer depth to convert the reading to true depth. Subtract the ship's draught from true depth to get UKC — or use the shortcut UKC = ES reading − (Draught − transducer depth). In pilotage, add squat to the draught before computing UKC.

LDL Pilotage (with squat)

LDL = Draught + Squat + Safety margin − Height of tide

• LDL= Least charted depth (sounding) you can safely pass over (metres)
• Draught= Static deepest draught of the vessel (metres)
• Squat= Dynamic bodily sinkage at speed (metres)
• Safety margin= Required under-keel clearance (metres)
• Height of tide= Height of tide above chart datum at the time (metres)

Underway you carry squat, so it is included (unlike at anchor). The height of tide is subtracted because it adds water above the charted sounding.

Rule of Twelfths — Tidal Height

Hours 1–6: ¹⁄₁₂ · ²⁄₁₂ · ³⁄₁₂ · ³⁄₁₂ · ²⁄₁₂ · ¹⁄₁₂ of the range

• Range= Tidal range = HW height − LW height at the port (metres)
• 1/12= Hours 1 and 6: tide rises or falls 1 twelfth of the range (≈ 8%)
• 2/12= Hours 2 and 5: tide rises or falls 2 twelfths of the range (≈ 17%)
• 3/12= Hours 3 and 4: tide rises or falls 3 twelfths of the range (≈ 25%)

Quick check: after 3 hours (midway) the tide has covered exactly half its range (1+2+3 = 6 twelfths). The rule assumes a smooth sinusoidal curve — it is not reliable for ports with double HWs, tidal bores, or strong shallow-water effects. For accurate work, always use the published tidal curve.

Compass Error — CADET Mnemonic

C + Dev(E) = M · M + Var(E) = T Error = True − Compass (E is +, W is −)

• C= Compass bearing — the raw reading from the compass
• Dev= Deviation — error caused by the ship's own magnetism (E = add, W = subtract)
• M= Magnetic bearing — compass corrected for deviation
• Var= Variation — angle between magnetic and true north from the chart (E = add, W = subtract)
• T= True bearing — corrected for both deviation and variation
• CADET= Compass → Add Deviation East → Magnetic → Add Variation East → True

CADET going Compass → True: add East values, subtract West. To reverse True → Compass (TVMDC): subtract East values, add West. Total compass error = True − Compass; named East if True is greater, West if True is less.

True to Compass — TVMDC / CDMVT

T − Var(E) = M · M − Dev(E) = C (CDMVT read right-to-left; E is −, W is +)

• T= True bearing — the course or bearing laid off on the chart
• Var= Variation — from chart or compass rose (E = subtract going toward compass, W = add)
• M= Magnetic bearing — true corrected for variation
• Dev= Deviation — from the ship's deviation card (E = subtract, W = add)
• C= Compass bearing — the course you order the helmsman to steer
• TVMDC= True → Subtract Variation East → Magnetic → Subtract Deviation East → Compass
• CDMVT= The sequence Compass-Deviation-Magnetic-Variation-True, read right-to-left gives True → Compass

Reverse of CADET: to go True → Compass, subtract East values and add West. TVMDC = 'True Virgins Make Dull Companions'. CDMVT is the same sequence read right-to-left. Cross-check with CADET: 085°C + 3°E Dev + (−5°W Var) → 083°T ✓.

Course Made Good (CMG)

CMG = the bearing of the straight line drawn from FIX to EP

• FIX= Last confirmed position — the starting point of your plot
• DR= Dead Reckoning position — where you'd be with no current (course steered × speed × time)
• EP= Estimated Position — start from DR, then draw the current vector to arrive at your best estimated position
• Set= Direction the current flows toward (degrees true)
• Drift= Speed of the current (knots)
• CMG= Course Made Good — the actual track over the ground from FIX to EP, in degrees true

What is arctan / atan2? It is the calculator function that works backwards from a right-angle triangle — if the east leg is 6 nm and the south leg is 3.5 nm, arctan(3.5 ÷ 6) gives the angle between them (≈ 30°). On the chart you never need to calculate it: just draw the FIX→EP line and measure the bearing with parallel rulers or a plotter. The diagram above shows the three-vector triangle: Course Steered (FIX→DR) + Current (DR→EP) = Course Made Good (FIX→EP).

Speed Made Good (SMG)

SMG = distance FIX→EP ÷ elapsed time

• SMG= Speed Made Good over the ground (knots)
• D (FIX→EP)= Straight-line distance from FIX to EP in nautical miles — measure it with dividers on the chart
• T= Elapsed time from FIX to EP in hours

On the chart: measure the FIX→EP line with dividers against the latitude scale — no maths needed. Use SMG (not engine speed) for ETAs, and CMG (not course steered) to check you're on track.