Noah’s Ark: The Problem of Violent Waves

More than 40 people died when the MV Derbyshire was lost in a typhoon in the South China Sea in 1980. The 294 m Derbyshire had been at sea for only two years. An inquiry ruled that a hatch cover had failed as huge waves buffeted the 176,369-ton bulk carrier. Further research indicated the ship failed because the waves were the exact same length as the vessel. Dr. Janet Heffernan, who analyzed the wave patterns, explained,

If the wave is smaller than the ship, then the vessel can cope with it. If the wave is much bigger, then the ship bobs on top of the wave, but if the wave is the same length, then the ship picks up the frequency of the sea…” BBC article. Technical Report

How Big and How Bad?

The size and nature of the waves during the deluge dictates the strength of the ark. The Hong et al. Korean safety study concluded that the ark was capable of riding out 30 m waves if the structure had 30 cm walls and 50 cm framing timbers. With stronger construction, the ark could survive 47.5 m waves before the water reaches the corner of the roof (heeling angle 31^o). Such waves have never been recorded in the ocean today. The highest has been measured at 26 m in the notorious North Atlantic.

Wind-Generated Waves

Generation of wind waves of such magnitude would require a constant gale over a considerable distance (fetch). This is why waves are limited in a small lake. If the wind changes direction, the wave pattern can be reduced by interference with a new wave system. Generally speaking, strong winds are of relatively short duration, making very large waves a rare event.

Thirty meters is a big wave, being a significant wave height with H_1/3 of 15 m approaching the upper limit in samples of tabulated data of North Atlantic waves. (Principles of Naval Architecture Chapter 8 [vol. III, section 2]. At such a height, weather ships in the North Atlantic (PNA VIII, Section 2, Fig 20 Roll) recorded wavelengths of around 300 m (a height/length ratio of 1:20). These observations were from regions experiencing very severe weather compared to even the North Atlantic. In a more generalized study, Hogben and Lumb compiled over one million observations spanning 10 years, where wave heights in the highest range (11 to 12 m) accounted for only 0.0013% of worldwide observations (or 0.007% of observations in the Northern North Atlantic).

For a wave with H_1/3 of 14 m to develop in the open sea, a sustained wind speed of 63 knots (117 km/h) is required (measured at 19.5 m above the water surface).1 It was also found that observed recordings slightly overestimated the actual measured wave heights in the upper ranges by around 13%. According to the Guinness Book of World Records, the highest wave ever officially recorded was observed at 34 m during a hurricane, but the record for an instrumentally measured wave is “only” 26 m.

Woodmorappe mentions the damping effect of flood debris as a factor in limiting wave development. Floating in the middle of a large mat of vegetation, the ark would be effectively shielded from wind-generated waves. Of course, one could always assume such a floating island of organic material would become waterlogged and sink to the bottom prior to burial by sediment from continental runoff in the late flood stages, which is not a bad explanation for the formation of oil reserves in the Middle East.

Data for waves up to 12 m are listed with their most probable modal wave period in Principles of Naval Architecture V3 Ch8 S2.10 Fig 26. Curve fitting to this data appears to be a second-order polynomial (parabolic) function as follows:

Wave Height = 0.0663*period² - 0.5168*period + 2.5765 (which fits the data reasonably well; R² = 0.9993.)

Then by extrapolation, we get the periods for the 30 m wave = 25 secs, 47 m wave = 30 secs. (Obviously a rather dubious extrapolation given that we just extrapolated to 300% of the actual data.)

Assuming a sinusoidal waveform, equation 16 Section 2.2 gives L_w = g * T_w² / (2 * P), which yields wavelengths of 944 m for the 30 m and 1424 m for the 47 m high wave (length-to-height ratios of approx. 30:1). Such long wavelengths were probably not employed in the Hong study since the ark would ride comfortably over a 50 m swell with a 1.5 km wavelength.

Table 1. Caption: Compiled from Ship Design for Efficiency and Economy: Table 6.17a and 6.17b after Henschke.2 A photo of each sea state can be seen at https://repository.library.noaa.gov/view/noaa/56110

Sea State	Beaufort Scale	Wind (m/s, knots)		Sea	Height (m)	Length (m)
0	0 No wind	< 0.2	< 0.4	Smooth sea	0	-
0	1 Gentle air	1.5	3	Calm sea	0.5	10
1	2 Light breeze	3.3	6.5	Rippling sea
1	3 Gentle breeze	5.4	10.5	Gentle sea	0.75	12
2/3	4 Moderate breeze	7.9	15	Light sea	1.25	22
4	5 Fresh breeze	10.7	21	Moderate sea	2.0	37
5/6	6 Strong breeze	13.8	27	Rough sea	3.5	60
6/7	7 Moderate gale	17.1	33	Very rough sea	6.0	105
7	8 Fresh gale	20.7	40	High sea	> 6.0	>105
8	9 Strong gale	24.4	47	High sea
9	10 Whole gale	28.3	55	Very high sea
9	11 Storm	32.7	64	Extremely heavy sea	20	600
	12 Hurricane	>32.7	>64	Extremely heavy sea

Table 2. Developed sea data compiled from Seaworthiness: The Forgotten Factor.3

Hong paper 1994	40.1	78	Structural limit	30	1160
Hong paper 1994	50.5	98	Stability limit	47.5	2560

Rogue Waves

There is plenty of wave data based on averages, but values such as significant wave height do not indicate the highest wave likely to be encountered. The fact that different sets of waves can be superimposed gives rise to the possibility of a freak wave appearing. These rogue waves are unusually high and unusually steep, often breaking on top of a vessel.

Rogue waves are principally wind generated, possibly a statistical freak between multiple wave sets or a perfect shape to catch the wind. There is also a theory of interplay between wind and currents, which obviously doesn’t explain how rogue waves can form in the absence of significant water current. There is considerable research underway on rogue waves, and some argue for a tightening of shipping rules—especially the increase of wave-bending moments (see appendix below) and wave-slamming loads. Very large ships normally ride several waves at once, but freak conditions such as the Derbyshire incident attest to the danger of the wavelengths equal to the length of the hull.

Tsunamis

A tsunami is a series of ocean waves generated by any rapid large-scale disturbance of the seawater. Most tsunamis are generated by earthquakes, but they may also be caused by volcanic eruptions, landslides, undersea slumps, or meteor impacts. —NOAA

In deep water, a tsunami has a gentle slope and is only a few feet high, passing under ships virtually undetected. In this case, the wave was 50 cm after several hours.4 The more dangerous wave heights only apply as the wave approaches the shore. In some instances, ships are recommended to head out to deeper water if there is enough time before a tsunami arrives.

In deep water, a tsunami develops such a long wavelength that it is very low and virtually harmless (even unnoticed) to shipping. Wavelength is related to apparent speed (celerity), so the wave formation can attain speeds of up to 500 miles per hour, crossing the Pacific in a day. It is only as it approaches the shoreline that the tsunami begins to compress and rise higher. Its momentum is known to send the water far inland. Since the height of tsunamis in deep water is not dramatic, the run-up data is often quoted. This can be very misleading since the tsunami height can be amplified 20 times as the shallow water slows it down. Cynical imagery of the ark tossed around in 500 m tsunamis (in deep water) would imply there was the potential for 10 km (six miles) of vertical run-up when it hit the shoreline—which is not likely. Such a wave would break in the comparatively “shallow” 1,000–4,000 m water, dissipating its energy and settling down to a more realistic size. In fact, the 520 m tsunami height of the 1958 landslide quoted by ark skeptics took place within a narrow Alaskan bay where there was no time for the wave to develop a standard profile. This is not a deep-sea tsunami wave height.

So a tsunami generated by earthquake, landslide, or volcanic activity could only pose a threat to Noah’s ark in shallow water. With an average depth of nearly three kilometers (two miles), which is well over the NOAA recommendation, the worldwide flood would actually protect the ark from tsunami shoreline effects. In the middle of an ocean where geological activity is concentrated on the perimeter (e.g., such as in the modern Pacific Ocean), the ark would safely ride over tsunami waves.

Waves of the Flood

Wind: After the rainfall had ceased and the fountains of the deep subsided, God sent a wind (Genesis 8:1). The ark was still afloat at this stage, so the intensity and geographical scale of these winds hold the key to wave sizes. Record-breaking waves (such as the Hong roll limit of 47.5 m) would be unacceptable at the time of the ark coming to rest on the mountains. In fact, waves of more than a few meters could pose a threat to a beached ark.

In the open sea, a stability limit of 47.5 m or a structural limit of 30 m does not tell the full story since the wavelength must also be specified.

The following screenshot shows the ark in a beam sea. The waves have the more realistic second-order Stokes profile and a very steep ratio of 1:10. For smaller waves, a limit of around 1:7 is generally as bad as it gets before the wave will break on itself. This software was begun by Tim Lovett in an effort to reproduce the roll stability calculations in the Hong paper.5

In closer detail, the ark is rolling in a double period with the Stokes waveform with the roof beginning to dip into the water. Note the center of buoyancy (B), which has not yet corrected the heel angle due to the angular momentum of the vessel generated by the passing wave crest. The wave is “traveling” from left to right in this simulation, the vessel still rolling as it descends. If waves happen to coincide with the natural roll period of the ship, then the roll is amplified. Immediate action is required, such as turning the vessel to change the frequency at which the waves meet the ship.

A more probable ratio of wave height to wavelength is around 1:30, which would look more like this:

Tsunami (based on the catastrophic plate tectonics model): Once the ark had landed, the receding water level would minimize the risk of tsunami damage, especially since the ark was in a mountain range (mountains of Ararat, with other peaks visible). Henceforth, the latter stages of the flood that involve high current velocities during continental runoff are irrelevant.

During the voyage, a tsunami could only pose a threat if the source was nearby, such as a local exploding volcano. Furthermore, the proximity to volcanic activity and its likely nature (explosive or continuous) hold the key to understanding the tsunami waves of Noah’s flood. Essentially, the deeper the water and the more distant the tsunami source, the safer the wave would be when it would’ve reached the ark.

Developing Seas

Wave Height vs. Wind Speed and Duration

It is common knowledge that wind generates waves. Stronger wind gives bigger waves.6 Initially, the waves are close together and unsorted, but with a steady wind over a long distance (fetch), the waves become well-formed and further apart. A fully developed sea is one that has reached the full height and wavelength corresponding to a particular wind speed.

The following table shows the relationship between wind speed and duration (fetch) and the effect on wave height and period. The blue figures indicate typical values in a modern sea—very high wind speeds have a fixed direction for only a relatively short time. This table is a metric conversion based on stormsurf.com/page2/papers/seatable.html.7

Wind km/h	Wind Duration (Hours)													Property
	6	12	18	25	35	45	55	70	80	90	100	120	140
41	1.74	2.38	2.74	3.05	3.35	3.66	3.66	3.66	3.66	3.66	3.66	3.66	3.66	height (m)
	6	7	8	9	10	11	11.5	12	12.5	12.5	13	13	13	period (s)
	80	185	296	463	741	1019	1296	1852	2222	2593	2871	3611	4352	fetch (km)
48	2.13	3.05	3.66	3.96	4.27	4.57	4.88	4.88	4.88	5.18	5.33	5.33	5.33	height (m)
	6.6	8	9	10	11	12	13	13.5	14	14.5	15	15	15.5	period (s)
	89	204	315	519	759	1111	1482	2037	2500	2871	3426	4167	4815	fetch (km)
56	2.29	3.66	4.27	4.88	5.49	6.1	6.1	6.71	6.71	6.71	7.01	7.01	7.01	height (m)
	7.2	9	10	11	12	13	14	15	16	16	16.5	17	17.5	period (s)
	94	232	389	556	926	1296	1667	2222	2778	3241	3704	4630	5556	fetch (km)
67	3.54	4.88	5.79	6.71	7.62	8.38	8.84	9.14	9.14	9.45	9.45	9.45	9.45	height (m)
	8	10	11.5	13	14	15	16	17.2	18	18.5	19	19.5	20	period (s)
	111	259	435	667	1000	1482	1852	2593	3148	3704	4260	5371	6297	fetch (km)
74	4.27	5.79	7.01	7.92	8.84	9.75	10.36	10.97	11.28	11.58	11.89	12.19	12.5	height (m)
	8.8	11	12.5	14	15	16.2	17	19	19.5	20	21	21	22	period (s)
	119	278	482	741	1093	1630	2222	2778	3334	4074	4630	5741	7038	fetch (km)
83	4.88	7.01	8.23	9.45	10.67	11.89	12.5	13.72	13.72	14.33	14.94	15.24	15.24	height (m)
	9.3	12	13.5	15	16	18	18.5	20	21	22	22.5	23	24	period (s)
	130	315	528	787	1167	1759	2315	2963	3704	4260	5000	6667	7593	fetch (km)
93	5.79	8.23	9.45	11.3	13.11	14.02	14.63	16.46	16.76	17.68	17.98	18.29	18.29	height (m)
	10	12.5	14.5	16	17.5	19	21	22	23	23	24	25.5	26.5	period (s)
	139	333	556	833	1296	1945	2500	3241	3889	4630	5371	7038	7871	fetch (km)
102	6.86	9.14	10.97	13.4	15.24	16.76	17.98	18.9	19.81	20.12	21.03	21.34	21.34	height (m)
	11	13	15	17	19	21	22	23	24	25	26	27	28	period (s)
	148	352	593	926	1408	2130	2685	3519	4260	4815	5741	7223	8519	fetch (km)
111	7.62	10.67	12.8	15.2	17.07	20.42	21.34	22.86	24.08	24.38	24.38	24.99	25.91	height (m)
	11.5	14	16.5	18	20	22	23.5	25	26	28	28	30	30	period (s)
	154	370	648	945	1482	2222	2778	3704	4537	5186	6019	7408	9260	fetch (km)
120	8.38	11.89	14.63	16.8	19.81	22.86	24.38	25.91	27.43	28.04	28.96	30.48	30.48	height (m)
	12	15	17	19	21	22	25	26.5	28	28.5	30	31	33	period (s)
	163	407	704	1037	1574	2315	2963	3889	4630	5463	6297	7778	9445	fetch (km)
130	9.14	13.11	16.76	18.9	21.64	24.99	27.43	29.87	30.48	31.7	33.22	35.05	36.27	height (m)
	13	16	18	20	22	25	26	29	29.5	30.5	31	32.5	35	period (s)
	169	435	732	1111	1630	2454	2963	4167	4815	5649	6667	8334	10371	fetch (km)
139	10.36	15.24	18.29	21.3	24.38	27.43	30.18	32	33.53	35.97	36.58	38.1	39.62	height (m)
	14	17	19	21	23	25.5	27	29	31	32	33	34	36	period (s)
	178	454	750	1148	1667	2593	3148	4260	5000	5834	7038	8890	11112	fetch (km)
148	11.28	16.46	19.81	22	25.91	30.48	32.61	36.27	36.88	40.54	41.45	42.67	42.67	height (m)
	14.5	17.5	20	22	23.5	26.5	28	30	32	33	34	35	36.5	period (s)
	185	472	787	1185	1806	2685	3334	4445	5278	6112	7223	9167	11297	fetch (km)
157	12.19	17.37	22.56	24.4	28.96	33.22	37.19	40.54	42.37	42.67	44.2	47.24	48.77	height (m)
	15	18	21	22	25	27.5	30	32	33.5	35	35.5	37.5	39.5	period (s)
	191	482	824	1259	1852	2778	3519	4630	5556	6482	7501	9353	12038	fetch (km)
167	13.72	19	24.38	28	32.61	36.58	39.62	42.67	44.81	47.24	50.29	51.82	57.91	height (m)
	16	19	22	24	26.5	29	31.5	33	34.5	36.5	37	40	44	period (s)
	204	500	852	1296	2037	2871	3704	4815	5741	6945	7871	9630	12594	fetch (km)

Converting to wavelength: For harmonic waves in deep water, as shown previously, the wave period (T_w in seconds) is related to wavelength L_w.

L_w = g * T_w²/2 π8

So, after 120 hours, a steady 157 km/h wind generates waves approaching the 47.5 m mark with a period of 37.5 seconds, giving a wavelength of L_w = 9.8 * 37.5²/(2*pi) = 2193 m (2.2 km). This requires a fetch of over 9,000 km, over which the wind has been blowing steadily.

If you were to run those numbers on the Roll Simulator, you will get a rather tame motion, especially at 1x speed. Even in a beam sea, the roll is only six degrees, so the main sensation would be a gentle vertical acceleration—something like being in an elevator. (Stokes waveform gives max acceleration on wave crest of -0.7 m/s², or 0.07 g.) Compare this to the passenger ship acceleration limit of 0.34 g (at forward perpendicular), which is nearly five times higher. Modern passenger ships are, of course, designed to be very comfortable (low accelerations), so this big (and very long) wave is not presenting a problem.

All of this assumes an ideal well-developed sea without interference from other waves. A more realistic situation would be a dominant well-developed waveform with smaller superimposed waves causing some randomness. In other words, somewhere between ideal wind-driven waves and a totally random sea, but with the globality of the wind (Genesis 8:1) causing a bias toward a regular and fully developed sea.

There is more to the story, however. The vessel will experience wind loads that will cause the vessel to heel (roll). A lightly loaded ark will be worse here because of the larger area for the wind to push against. Collins calculated that a wind of 210 knots (388 km/h) would be required to hypothetically capsize the ark in flat water, so the 157 km/h wind (16% of the energy of 388 km/h) looks feasible even when side-on (broaching).

Of course, the balance to all this is that Noah’s ark should not be side-on to the wind for very long anyway. The ship should align itself with the wind and experience a relatively consistent head sea.

Conclusion

Wind-generated waves under global wind conditions could reach abnormal wave heights but are likely to be very regular and fully developed with long wavelengths. Noah’s ark (or any other decent ship) could comfortably ride huge waves in a fully developed sea because the waves were not steep.9 If the global wind was not so uniform, interference of wave sets could produce a more random sea, increasing the likelihood of steep waves and rogue formation.

The biggest threat to the ark would be a localized storm, which is not likely to form when there is a global wind blowing.10

Appendix: Standard estimate of loads applied by waves

Long Hull Syndrome

The proportions of Noah’s ark are explicitly stated in Genesis 6:15—300 x 50 x 30 cubits. The vessel is 10 times as long as it is high, which means that the bending loads applied by waves will be significant. The ark has similar proportions to a modern ship, and ships are not supposed to break in half. To avoid making a ship that is too weak in the middle, there are rules.

Wave bending moment (M_w) is about waves trying to bend the hull. If the hull can be built to withstand this amount of bending, then it should be strong enough in the worst modern sea. What about the waves of the flood?

The Calculator. Select a cubit and a general shape. The numbers in red indicate how strong the hull needs to be in the middle. Of course, that still leaves all the engineering associated with hull construction yet to be done.

CALCULATOR: Wave Bending Moment andShear Force

Select Cubit size	Short Hebrew Short Egyptian Common Babylonian Royal Long Hebrew Royal Egyptian Long Babylonian
Ark Dimensions
Select Block Coeff	Double ended Ferry Naval Frigate Passenger Liner Barge Carrier Cargo/Container Ship Crude Oil Carrier James King Ark Gt Lakes Ore Carrier Rod Walsh Ark Pure Block
ABS (Hogging) Mw
ABS (Sagging) Mw
Lloyd's Mw
B.Veritas Mw
ABS Wave Shear Fw

Note:

• Block Coeff is a measure of how closely the hull approximates a rectangular block.
• The American Bureau of Shipping (ABS) has the highest M_w values, which means they are the most conservative.
• If Noah’s ark was built to handle the ABS bending loads, then it should be strong enough for most sea conditions.
• Shear force is related to the tendency for planks to slide against each other.

Calculation of Wave Bending Moment

How strong did the ark have to be?

To endure several months in the open sea, the wooden hull of Noah’s ark must have a certain minimum strength. Factors such as uneven cargo distribution, increased length, or a more “block-shaped” hull (block coefficient) accentuate the need for a strong hull. Another factor is the severity of the sea state.

For a first approximation, we will consider the worst seas of today. In ship design, one of the first things to check is the bending strength of the hull. A ship riding over large waves experiences bending forces (causing a moment or torque) that flex the hull up and down along its length (hogging and sagging). Without adequate strength and rigidity, the ark could leak or break when it meets high seas. The following calculations are based on standard procedures for ships operating in the open sea.

Preamble

Applicability: The following calculations apply to ships longer than 90 m. Cargo ships with homogenous loading or less than 250 m long only require a still water bending moment to be calculated amidships.

Wood in place of steel: The applied wave loads are related to hull geometry and are independent of hull material. The data will be suitable for the timber ark up to the point where material properties such as stress and stiffness are investigated. In other words, the wave bending moment is externally applied so is independent of hull material (assuming adequate stiffness as dictated by waterproofing requirements).

These approximations are based on the vessel’s length, width, and shape, using a worst-case sea state. Several standards are compared, with the most conservative estimate recommended (ABS Rules).

Estimation of Longitudinal Wave Bending Moment

Bending moment is the amount of “bending” the hull experiences. It is highest in the middle (amidships) and occurs when the hull is bridging two waves (sagging or positive bending). Another situation is when a wave is supporting the hull amidships as if the ship were riding a wave (hogging or negative bending). Both need investigation since either case might be the failure mode at sea, and they represent the maximum amplitudes of fatigue loading.

Firstly, we must define the hull size by choosing the most suitable cubit.

Next, the hull shape: The hull of a ship has certain coefficients of form. One of the most fundamental is the block coefficient, C_b, which describes how well the hull approximates a rectangular prism. It is calculated by comparing the design displaced volume with the enclosed volume of its maximum wetted dimensions.

C_b = (Displaced Volume) / ( L * B * T )

Where L is length, B is beam or breadth, and T is the draft. The density of seawater (1.025) may need to be accounted for in determining the draft.

The sculptured hulls of a small ship such as a harbor ferry might have a C_b of only 0.4, whereas the rectangular cross section of large crude oil carriers can have a C_b of almost 0.9. Ark depictions shown as almost a pure rectangular prism could have a C_b as high as 0.98.

Using the popular choice of 18” for the cubit,12 the ABS wave bending moment rules to calculate the hogging and sagging moments, with a block coefficient of 0.9, gives the following values: Hogging (tf-m): 65071.4912, Sagging (tf-m): -67008.743. Use the calculator above to test the effect of hull changes.

Wave BM Calculated According to Shipping Rules

Definition of symbols;
L = length (m)
B = beam or breadth (m)
C_b = Block Coefficient as defined above.

Method 1: ABS Rules

Steel Vessel Rules 2004, Part 3, Hull Construction and Equipment

https://ww2.eagle.org/en/rules-and-resources/rules-and-guides.html

Bending Moment

ABS Rules for Building and Classing Steel Vessels 2004. Part 3, Chapter 2, Section 1, Subsection 3.5.1 “Wave Bending Moment Amidships”

(Note that the ABS rules make a distinction between maximum hogging and sagging conditions. Also note that the ABS rules are the most conservative of these classification societies.)

M_w = -k1 * C₁ * L² * B * (C_b + 0.7) / 1000 (sagging)
M_w = k2 * C₁ * L² * B * C_b / 1000 (hogging)

Where:

C₁ = 10.75 - (300-L) / 100) ^3/2 (for 90<= Length <= 300m)
Sagging k1 = 11.22 (for units in tf.m)
Hogging k2 = 19.37 (for units in tf.m)

Method 2: Lloyd’s Register

Rules for Steel Ships

Bending Moment

Lloyd’s Chapter D, part 315 gives a formula for estimating the bending moment midway along the hull (amidships) that could be applied by typical sea waves. The factor C₁ is tabulated and based on shipping data.

Linear interpolation of tabulated values near the ark length gives C₁ = 6.2527 + 0.0178*L

Lloyd’s specifies a material stress of 98.1 MPa (assumes a steel hull). All other variables are as previously defined.

Method 3: Bureau Veritas

Rules and Regulations for the Construction and Classification of Steel Vessels. 1977. International Register for the classification of ships and aircraft. (Henri-Rochefort 75821 PARIS CEDEX 17)

Bending Moment

BV 5-23-31 The maximum rule value of wave bending moment in kN.m is given by the formula:
M_H = H L² B ( C_b + 0.7 ) 10^-3
Where
H = 703 - 65 ( (300 - L) / 100)^3/2 if L < 300

Wave Shear Force (ABS)

American Bureau of Shipping. 2014. ABS Rules for Building and Classing Steel Vessels 2004. Part 3, Chapter 2, Section 1, Subsection 3.5.3 “Wave Shear Force.”

The maximum shearing force induced by wave, in kN (tf, Ltf)

Fwp = + k1 * F1 * C₁ * L * B * (C_b + 0.7) / 100 (pos shear)
Fwn = - k1 * F2 * C₁ * L * B * (C_b + 0.7) / 100 (neg shear)

Where:

k1 = 30 (3.059, 0.2797) kN (tf,Ltf)
F1 = distribution factor, as shown in 3-2-1/Figure 3
F2 = distribution factor, as shown in 3-2-1/Figure 4

This factor is 0.7 amidships rising to 1.0 at approx. 25% from bow/stern. We assume the highest value 1.0.

C₁ = C₁ = 10.75 - ((300-L) / 100) ^ 3/2 (for 90<= Length <= 300 m) The same as the bending moment.
L = length of vessel in m (ft.)
B = breadth of vessel in m (ft.)
C_b = block coefficient, but not to be taken less than 0.6

Comments

The calculations follow the same general form, where the wave bending moment is proportional to L^3.5 and also to B¹. For ships up to 300 m long, the intensity of bending in the hull is very sensitive to the ship’s length (L). So, a long, wider hull is more likely to break in half (or leak badly) due to the wave-induced bending moment.

Supporting these relatively simple calculations is an extensive record of monitoring ship stresses during service. These formulae have been determined on the basis of sea states throughout the world and take account of storms and worst-case uncertainties (just the wrong combination of wave height, length, etc.). While direct stress analysis using the finite element methods is possible today (barely!), these simple formulae are the trusted solution. Highly detailed analysis is required if the designer intends to use a number lower than these.