﻿ Professor Robert B. Laughlin, Department of Physics, Stanford University

# Hull Speed

## Ben Shank December 12, 2007

### (Submitted as coursework for Physics 210, Stanford University, Fall 2007)

If you've spent any time near a commercial port, you have no doubt watched barges carrying cargo back and forth and seen the great mound of water that builds up at their head when they are under way. Although their massive wake helps keep the local population of JetSkiers in check, it comes at considerable expense. All of the energy transferred to those waves must be replaced by the engines to maintain a constant speed. In essence, the barge must push the water in its path aside to make headway. The inefficiency with which flat-nosed barges do this makes their wake particularly noticeable, but all ships that float by displacing water must displace different water in order to move forward. From the bow and the stern of every moving displacement hull comes a series of waves called a wake that carries away the energy of displacement. These waves travel at a speed v = (Lg/2π)1/2, where L is the wavelength and g is the local acceleration due to gravity. At sea level, where most ships travel, this works out to v = 1.34 L1/2 when L is measured in feet and v is given in knots. [2] (A knot is one nautical mile, about 6080 feet, per hour.) Because a wake arises at the bow and the stern of a ship, its wavelength can be approximated as the length of the ship at the waterline. When the ship is travelling more slowly than its wake, water is simply displaced and the associated energy travels away from the sides without further interaction. However as the ship approaches this critical speed, it will build up a barrier of water in front of it and a trough behind it because the water simply cannot get away fast enough. To go any faster the ship will have to push uphill, requiring considerably more energy. This will make the barrier of water yet higher, making the next increment in speed even more costly. Because the critical speed v depends only on the length of the hull it is referred to as the ship's "hull speed." The value 1.34 knots/ft1/2 in the equation given above is often called the speed to length ratio for a hull despite the fact that it is not strictly a speed divided by a length.

Obviously this analysis is oversimplified. Naval architects and professional shipwrights perform much more sophisticated analyses of their craft to account for interaction with ocean waves and wind, the precise shape of the hull, the modeled shape of the wake, and other factors. However the concept of a speed to length ratio is so useful and fast that many professionals work out a value for a specific hull shape and then refer to it as they scale the model up or down in size. Values range from 1.18 (in nautical units) for barges to 1.42 for very long, sleek vessels. Most amateurs use 1.34 as a good approximation for most common hull shapes.

Hull speed is sometimes treated as the highest speed a ship can attain. This is not strictly the case. It simply measures a critical speed at which the ship catches up to its own wake. Typically the energy required to speed up a displacement hull then becomes exponential in speed rather than quadratic. If the engine was already working hard to reach this speed, chances are it will not get much faster. However several options exist to beat the rising mound of water and press on to greater speed. The most common, particularly for smaller craft, is to climb up over the barrier. If the hull can be shaped in such a way as to generate lift, the boat is no longer displacing water equal to its weight and therefore experiences less displacement-related drag. This process is referred to as hydroplaning, or simply planing. Motorboats on plane have their noses raised high out of the water as they climb their own bow wake. Specialized racing boats almost seem to leap from their own wakes as they skitter across the surface of the water. Planing is useful mostly for smaller boats which almost always want to travel faster than their sluggish hull speeds and often have plenty of power to spare when they get there. Somewhat larger vessels can gain some of the benefits of planing without the inherent loss of stability by travelling in a semi-displacement mode. By receiving some lift, but not enough to balance their weight, these craft significantly reduce the hull speed barrier without removing it entirely.

Another tactic for breaking the hull speed barrier is to simply cut through one's own bow wake. [2] This method is popular with modern navies. Fast-attack warships cannot afford the instability of planing or even semi-displacement travel, but they would hardly live up to their name if they could not overtake larger vessels with greater hull speeds. These ships are given specially designed wave-piercing prows and gigantic engines. They do as much as they can to reduce wake-drag and spend most of their time patrolling well below their hull speed. However when the need arises, these ships solve the hull speed problem by throwing more power at it. As might be expected, this strategy does not result in substantial speed increases, often only allowing a two to three hundred foot destroyer to actually reach its hull speed. [3] In this limited sense, hull speed does seem to place an effective maximum on the speed of a single displacement hull.

Although almost all fighting ships today are equipped with wave-piercing bows, true fast attack is increasingly accomplished by submarines. Subs ignore the wake problem by going under it. This highlights an important aspect of wave-induced drag. It is not just the displacement of water that limits the speed of a ship so sharply. By getting away from the complex propagation of wave energy at the air-water interface, submarines have no hull speed despite the fact that they displace more water submerged than when surfaced. Because the enhanced speed of a submarine is one of its key strategic advantages, it is difficult to clearly demonstrate. All of the submarine classes identified by the US Navy [4] are listed as having a top speed of "25+ knots submerged." The smallest of these, the Cold War era Sturgeon class, has a length of 292 feet, which would give it a hull speed of 22.9 knots. The modern 350-foot Seawolf has a hull speed of 25.1 knots, but we can be certain that it, as well as the Sturgeon, are capable of far greater speeds than 25 knots when submerged.

Displacement is not the only drag force on a body moving through water. Surface tension, the tendency of water to stick to any object immersed in it, creates an effect called 'skin drag.' Long hulls which present a sufficiently small cross-section to oncoming water have their speeds dominated by skin drag. This category includes rowing shells and racing kayaks [5] as well as multi-hulled vessels such as catamarans. The wavelengths of the wakes of these vessels are not simply the length of the hull and they are often free to travel much faster as a result. Super-narrow mono-hulls show up almost exclusively in human-powered races, but multi-hulls can take more varied forms. The US Navy is experimenting with catamaran hulls as a more flexible alternative to submarines for fast response in shallow waters. The latest venture, named Swift, is 321 feet long, giving it a hull speed of 24 knots, but has a top speed twice that fast. Over the years multi-hulled designs have been used for high speed ferries, exceptionally stable fishing boats and, of course, racing.

The concept of hull speed was developed as a practical rule-of-thumb by mariners in the days of wind and steam to describe a phenomenon that arises from a basic application of the physics of waves and, for them, placed a strict limit on the speeds they could attain. But a ship's hull speed is not some unattainable velocity like the speed of light, which cannot be accessed by any imaginable application of power. Instead it is a practical limit for engineers, a place in the physics of surface vessels where the nature of the forces involved changes dramatically. Certainly fantastic power can be expended to little avail near the hull speed of a traditional hull, but creative engineers have found ways around, over or under the walls of water that bar their way.

© 2007 Benjamin Shank. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] H. Y. H. Yeh, "Series 64 Resistance Experiment on High-Speed Displacement Forms," Marine Technology 2, 248 1965.

[2] Inc. U.S. Coast Guard Auxiliary Association, Sailing Skills & Seamanship (U.S. Coast Guard Auxilary Assn., 1978).

[5] J. Winters, "Speaking Good Boat, Part II (Kayak Hull Speed and Beyond)," http://www.qcckayaks.com/resources/speakboat2.asp.