I want to do some very simplified drag calculations on a vessel. My hope was that calculating the skin-friction resistance would be enough to get a good estimate of the surge resistance.

Because wave-making resistance is very speed dependent, I assume you can neglect it when the vessel is below a certain speed. I also assume that I have to work with Froude numbers instead of velocities, to take vessel size into account.

I have seen Froude numbers below Fn = 0.1 and Fn = 0.2 mentioned in books and on the Internet, but if you calculate the velocity for a vessel with a 100 m long waterline you get:

$$V = 0.1 \cdot \sqrt{9.81\ \text{m/s}^2 \cdot 100\ \text{m}} \approx 3.13\ \text{m/s} \approx 6.08\ \text{knots}$$

$$V = 0.2 \cdot \sqrt{9.81\ \text{m/s}^2 \cdot 100\ \text{m}} \approx 6.26\ \text{m/s} \approx 12.16\ \text{knots}$$

This values seem way too high in my opinion. 12.16 knots is almost service speed for some vessels and 6 knots is also quite high.

Are Fn = 0.1 and Fn = 0.2 reasonable numbers, and if not, below what Froude numbers should I stay to be able to neglect wave-making resistance?


4 Answers 4


Your intuition is correct, those are high. However, you would need to be moving very slowly for wave-making resistance to be negligible. And since it is typically higher than skin-friction I don't think that you can realistically expect to have a significant skin friction and a negligible wave-making resistance. Perhaps a better simplified approach would be to ignore the skin friction and focus only on the wave-making resistance.


Whether these are sensible Froude numbers or not depends on the length of the vessel in question. For a 100 m ship these are probably high but for a 5 m dinghy these would be quite low numbers.

Speed-to-length ratio is the critical factor that determines the importance of skin and wave friction. Skin friction scales with $V^2$, whereas wave drag increases much more rapidly. I haven't been able to find the exact formula in a brief search, I seen to remember being told a factor of $V^6$. The wave drag places a practical limit on the speed of a displacement (non-planing) vessel of $1.34L$.

For small speed-to-length ratio skin friction will be dominant, whereas for large ratios wave friction is important. An example value I found was that skin friction is ~65% of total drag at speed/length=1.

In general large ships will have a low speed/length and skin friction will dominate. On the other hand, small displacement boats like dinghies or kayaks will have skin friction dominated by wave drag.


"Hull speed" is actually the ratio of speed to the square root of length. To make things even more confusing, length is in feet, and speed is in knots. That's how the constant 1.34 arises. (ProTip: Let's never speak of it again!)

Wave resistance ($R_w$) begins its rapid rise at a Froude number (Fr) of about 0.35. Below that Fr, $R_w$ is usually small compared to the skin-friction and other hydrodynamic drag components.

Now, for sake of example, let the wave resistance coefficient be defined as $C_w = R_w/(0.5 \rho U^2 S)$, where $\rho$ is water density, $U$ is ship speed, and $S$ is the (static) wetted surface area of the hull.

In deep water, $C_w$ increases roughly like Fr to the 6th power.

The depth-based Froude number is $F_h = U/\sqrt{g h}$, where $g$ is gravitational acceleration, and $h$ is water depth.

For finite depth water, $C_w$ can increase almost like Fh to the 10th power as $F_h \rightarrow 1$. Once through (the critical value) $F_h =1$, wave resistance begins to decrease, and it can be lower than in deep water for the same length-based Froude number (Fr).

$F_h < 1$ is usually referred to as sub-critical; $F_h > 1$ is super-critical, and (roughly) $0.9 < F_h < 1.1$ is trans-critical.

In the trans-critical regime, the hull also experiences forces and moments that significantly change its attitude with respect to the undisturbed free-surface of water. The trim and heave of a hull is known as "squat". This phenomenon is difficult to predict accurately. It can have some effects on resistance but, more importantly, in shallow water there is also a danger of the ship grounding against the sea-bed. This can cause large losses of income, and there have also been fatalities attributed to the phenomenon.

Wave patterns for finite depth are quite interesting...







As $F_h$ enters the trans-critical regime, wave patterns change dramatically. The angle of the V-shape opens out and becomes 90 degrees at $F_h = 1$.

For sub-critical speeds, transverse waves (those perpendicular to the ship's track) are apparent. In super-critical flow, transverse waves disappear. (In short, they cannot keep up with the ship).

DISCLOSURE: These patterns were made using my (free) program Flotilla.

More patterns can be found at:



You are correct that the Froude number (Fr) is very important for wave resistance.

The answer given by nivag regarding the speed-length ratio (also known as "hull speed") is not correct. That limit is often quoted, but it is a myth that it is somehow impossible for displacement hulls to surpass it. Ships can travel faster than that ratio implies, but for conventional ships the energy requirement is usually prohibitive.

Thin hulls (like rowing shells) are able to operate easily beyond the ratio of $1.34 L$. At Olympic level, rowing shells for example, operate at Fr between about 0.45 and 0.7.

Apart from skin-friction, you will also need to consider "form drag". This component can be important for stubby hulls (i.e. those with a small length to beam ratio, L/B) at low Fr. A tug boat will have greater form drag than a rowing shell at the same Fr.

If the hull has a transom (i.e. cut-off) stern there will also be a large component of resistance when the transom is not running fully dry. In that case there is a lot of eddying, and possible wave-breaking behind the stern at low Froude numbers. At higher Fr the transom is dry, and wave resistance and form drag are much lower.

If you tell us a little more about the principal proportions of the boat (e.g. displacement weight, length, beam and draft) we might be able to offer more advice.

If the hull is fairly slender, say L/B > 5, you can try some free software to estimate the total resistance (viscous + wave). See, for example, Michlet and Flotilla.

Water depth can also affect wave resistance. In this case the depth-based Froude number plays an important role, very much like the Mach number in aerodynamics. Michlet and Flotilla will both allow you to vary water depth and to see the effect on wave resistance and wave patterns.

  • $\begingroup$ Interesting answer! I don't quite understand how to use the speed-length ratio that you discuss. Speed/length has units of 1/time and the numbers that you give (i.e. 1.34 L) have units of length. What am I missing? Does the number 1.34 have units? To put it more concretely; if I have a conventional boat which is 5 meters long, what is the limit on its speed in units of m/s? $\endgroup$ Commented Jun 10, 2015 at 23:25

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