Speed/length ratio is the relationship between the length of a displacement hull and the amount of wave making resistance that hull generates. Speed/length ratio is commonly used in the form of a rule of thumb called hull speed, used to provide a quick approximation of the speed potential of a given displacement hull, such as a sailboat or rowboat.

History

---

The quantification of the speed/length ratio is generally credited to naval architect William Froude, who used a series of scale models to measure the resistance each model offered when towed at a given speed. Froude's observations led him to derive the Froude number, which allows experimental observations performed on scale models to be applied to full scale vessels.

The speed to length ratio is traditionally expressed in knots of speed and feet of waterline length:

$\backslash textrm\{Speed\; Length\; Ratio\}\; =\backslash frac\; \{V\}\{\backslash sqrt\; \backslash textrm\{LWL\}\; \}$

Physics

---

For small craft operating at low speeds, such as sailboats or rowboats, the major source of drag is the wave making resistance, or the amount of energy that goes into generating the wake of the hull. Since propagation of deep water waves is proportional to the wavelength of those waves, and since a boat's wake is based on its waterline length, there is a relationship between the wave propagation speed and the rate at which drag increases.

A simple way of considering wave making resistance is to look at the hull in relation to its wake. At speeds lower than the wave propagation speed, the wave rapidly dissipates to the sides. As the hull approaches the wave propagation speed, however, the wake at the bow begins to build up faster than it can dissipate, and so it grows in amplitude. Since the water is not able to get out of the way of the hull, the hull in essence has to climb over or push through the bow wave. This results in an exponenetial increase in resistance with increasing speed.

Hull speed

---

The speed/length ratio is strictly only useful when comparing different scalings of otherwise identical hulls. However, for many hulls, a generic speed/length ratio will provide a good general estimate of the speed potential of the hull when it is operating in displacement mode. This is commonly called the hull speed, and this term is commonly found in the boating community and among amateur builders, though it is not used by naval architects or engineers.

It does not readily apply to small, highly powered boats such as sailing dinghies and personal watercraft, which can easily plane, nor very long, narrow hulls such as those on multihulls such as catamarans and proas.

The most commonly used hull speed constant is the wave propagation speed for the hull length, and it serves well for traditional sailing hulls. Wave propagation speed is based on simple harmonic motion, and is expressed as:

$\backslash mbox\{speed\}\; =\; \backslash sqrt\; \{\backslash frac\; \{l\; \backslash times\; g\}\{2\; \backslash pi\}\}$

Plugging in the appropriate value for gravity and solving yields the equation:

$\backslash mbox\{knots\}\; \backslash approx\; 1.34\; \backslash times\; \backslash sqrt\{l\; \backslash mbox\{ft\}\}$

Or, in metric units:

$\backslash mbox\{knots\}\; \backslash approx\; 2.5\; \backslash times\; \backslash sqrt\{l\; \backslash mbox\{m\}\}$

In reality, speed/length ratios of real hulls vary from as low as 1.18 for blunt hulls such as barges to over 1.42 for long, thin hulls. Also, since hull speed takes into account only the wave making resistance, large hulls (over 200 ft or 60 m) will be more limited by other forms of drag^*.

References

---

• A simple explanation of hull speed as it relates to heavy and light displacement hulls
• Hull speed chart for use with rowed boats
• Discussion of wavemaking resistance and hull speed

Marine engineering | Rumpfgeschwindigkeit