What is the Theoretical Maximum Hull Speed?

The Theoretical Maximum Hull Speed is defined as the speed of a water wave with the same wave length as the length of the water line of the boat.
When reading about Maximum Hull Speed, the number 1.34 pops up again and again. This page explains this number and its limitations, *i.e.* is 1.34 always 1.34?

How fast is a water wave?

The speed of a water wave is dependent on many factors as water depth, salt density, temperature, and many more.

When scientists makes theories they usually creates a simplified model of the real world because the real world is far too complex. This is also the case for the theories for propagation of water waves. We will not dig into these models on this page before this is out of scope. We just mention this because many people seems to believe that the number 1.34 is not to discussion. In the following, we will skip the initial steps and go straight to the resulting formula for a wave propagation, and we will rely on that the scientists have made reasonable assumptions and conditions. If the reader is interested in this model's many preconditions and assumptions we suggest to google for terms like "Navier-Stokes equation" and "dispersion relation of Lamb".

This formula is called dispersion relation of Lamb

Formula for wave speed

Formula for wave number

Combining the three formulas above, the wave speed C can be derived as:

This is the wave speed (C) as function of the water depth (h) and the wave length (λ).

In the following, we will distinguish between shallow-water waves Cs and deep-water waves Cd. This distinction between shallow-water and deep-water waves has nothing to do with absolute water depth. It is determined by the ratio of the water's depth to the wavelength of the wave (h/λ). If h/λ is large than we have deep-water waves, if h/λ is small then we have shallow-water waves.

*Shallow-water waves:*

First, let's look at the speed for a shallow-water wave Cs. For a shallow-water wave h/λ is small. This allow us to simplify the Lamb equation. Mathematically this is expressed as:

This means that if h/λ is very small then the tanh part of the Lamb equation can be substituted by 2πh/λ.
This results in this simple equation for wave speed Cs:

As you can see wave speed Cs is only dependent of the depth (h) of the water (because g is a constant).

An example of a shallow-water wave in the ocean is a tsunami wave. A typical wavelength of a tsunami wave is 200 km (120 miles). The dept of the ocean is about 4 km (2.5 miles). h/λ = 4km/200km = 0.02. 0.02 is small so we must use the formula for shallow water waves Cs:
Cs = 198m/s = 713km/h = 581mph = 385kt

So you will most likely have to upgrade your motor, if you want to outrun a tsunami wave.

*Deep-water waves:*

Let's look at the speed for a deep-water wave Cd. For a deep-water wave h/λ is large. Mathematically this is expressed as:

This shows us that if h/λ is very large then the tanh part can be substituted by 1.
This allow us to simplify the Lamb equation to this:

Speed of deep-water wave

And if we isolate the constant part, we have the following expression:

Which is the wave speed formula for a deep water wave. This shows that the wave speed Cd is only dependent of the wavelength (because g/2π is constant). Waves with longer wavelength move faster than waves with shorter wavelength.

An example of a deep-water wave in the ocean is a 10 meter long wave. The dept of the ocean is about 4 km (2.5 mile). h/λ = 4000m/10m = 400. 400 is a large number so we must use the formula for deep-water waves Cd.

So, for a surface gravity wave of length 10m, the packet would travel at approximately 1.97 m/s.

When scientists makes theories they usually creates a simplified model of the real world because the real world is far too complex. This is also the case for the theories for propagation of water waves. We will not dig into these models on this page before this is out of scope. We just mention this because many people seems to believe that the number 1.34 is not to discussion. In the following, we will skip the initial steps and go straight to the resulting formula for a wave propagation, and we will rely on that the scientists have made reasonable assumptions and conditions. If the reader is interested in this model's many preconditions and assumptions we suggest to google for terms like "Navier-Stokes equation" and "dispersion relation of Lamb".

This formula is called dispersion relation of Lamb

Formula for wave speed

Formula for wave number

Abrev. | Unit | Description |
---|---|---|

ω | s^{-1} | Angular frequency |

g | m/ s^{2} | Gravitational constant g = 9.81 m/s ^{2} |

κ | m^{-1} | Wave number, also called frequency |

h | m | Height of the water surface above the ground. (traditionally the letter h for height is used instead of the letter d for depth) |

λ | m | Length of the wave (length between wave crests) |

C | m/s | Wave speed (distance a wave crest travels per unit time) |

Cs | m/s | Shallow water wave speed |

Cd | m/s | Deep water wave speed |

Combining the three formulas above, the wave speed C can be derived as:

This is the wave speed (C) as function of the water depth (h) and the wave length (λ).

In the following, we will distinguish between shallow-water waves Cs and deep-water waves Cd. This distinction between shallow-water and deep-water waves has nothing to do with absolute water depth. It is determined by the ratio of the water's depth to the wavelength of the wave (h/λ). If h/λ is large than we have deep-water waves, if h/λ is small then we have shallow-water waves.

First, let's look at the speed for a shallow-water wave Cs. For a shallow-water wave h/λ is small. This allow us to simplify the Lamb equation. Mathematically this is expressed as:

As you can see wave speed Cs is only dependent of the depth (h) of the water (because g is a constant).

An example of a shallow-water wave in the ocean is a tsunami wave. A typical wavelength of a tsunami wave is 200 km (120 miles). The dept of the ocean is about 4 km (2.5 miles). h/λ = 4km/200km = 0.02. 0.02 is small so we must use the formula for shallow water waves Cs:

So you will most likely have to upgrade your motor, if you want to outrun a tsunami wave.

Let's look at the speed for a deep-water wave Cd. For a deep-water wave h/λ is large. Mathematically this is expressed as:

Speed of deep-water wave

And if we isolate the constant part, we have the following expression:

Which is the wave speed formula for a deep water wave. This shows that the wave speed Cd is only dependent of the wavelength (because g/2π is constant). Waves with longer wavelength move faster than waves with shorter wavelength.

An example of a deep-water wave in the ocean is a 10 meter long wave. The dept of the ocean is about 4 km (2.5 mile). h/λ = 4000m/10m = 400. 400 is a large number so we must use the formula for deep-water waves Cd.

So, for a surface gravity wave of length 10m, the packet would travel at approximately 1.97 m/s.

How to calculate the Theoretical Maximum Hull Speed?

The Theoretical Maximum Hull Speed is defined as the speed of a water wave with the same wave length as the length of the water line of the boat.
It is important to note that Theoretical Maximum Hull Speed is a definition.
It is not a physical law.

The background for this definition is experiments in the 1950's^{[1]} which has shown that the speed at which wave resistance is accumulating most rapidly, is the speed of an ocean wave the length of which, from crest to crest, is about that of the ship from end to end.
It is these experiments which resulted in the definition.
Given that definition, we can substitute the wave length of the wave (λ) with the water line length of the boat (lwl).

Using g = 9.81 m/s^{2}, and π = 3.1415, we have the following simple formula for Cd:

In the scientific world, the fundamental units are meter, kilogram, second, and ampere. In the nautic world other units are convenient. The conversion rules are as follows:

Using the conversions above, we get the equivalent formula in imperial units.

The background for this definition is experiments in the 1950's

Using g = 9.81 m/s

In the scientific world, the fundamental units are meter, kilogram, second, and ampere. In the nautic world other units are convenient. The conversion rules are as follows:

Unit | Conversion |
---|---|

Length | 1 nautical mile = 1852 meter 1 meter = 3.2808399 feet |

Time | 1 hour = 3600 seconds |

Speed | 1 m/s = 3.6 km/h = 1.9438 knots 1 knot = 1 nautical mile / hour = 1852 m/h = 1852 * 3.2808399 ft / 3600 s = 1.68780985999 ft/s |

Other | g = 9.81 m/s^{2} ≈ 32.2 ft/s^{2} |

Using the conversions above, we get the equivalent formula in imperial units.

When is 1.34 not 1.34?

In the calculations above, we have three assumptions:

*h/λ is large:*

For a deep water wave the condition was that h/λ is large, and as a result the tanh expression could be substituted by 1

Let's investigate the limitation of this tanh expression.

Example 1: The boat's waterline is 30 ft and the water is 100 ft deep. h/λ = 100/30 ≈ 3.3

Then

Example 2: The boat's waterline is 30 ft and the water is 30 ft deep. h/λ = 30/30 = 1

Then

Then

Conclusion: The value 1.34 can be used if the water depth is larger than the length of the boat,*i.e.* if h/λ > 1. But if the water is very shallow, then 1.34 is not valid.

*g = 9.81 m/s*^{2}:

Now, let's investigate variations of the gravitational value.

The nominal "average" value at Earth's surface, known as standard gravity is - by definition - 9.80665 m/s^{2} (32.1740 ft/s^{2}) at a latitude of 45° at sea level.

This average value was defined at a meeting in CGPM (Conférence Générale des Poids et Mesures) in Paris in 1901. But the earth is not a perfect sphere, the earth is rotating, and pulled by the moon and the sun. Therefore the actual value of g varies from the equator to the poles and even from day to day.

At the poles g = 9.832 m/s^{2} (32.2572 ft/s^{2}), the constant is: 1.34243

At 45° latitude g = 9.80665 m/s^{2} (32.1740 ft/s^{2}), the constant is: 1.34069

At equator g = 9.78033 m/s^{2} (32.0877 ft/s^{2}), the constant is: 1.33889

Conclusion: From a gravitational perspective the value 1.34 is valid from the poles to equator.

*The boat is a displacement boat:*

Do not forget that the background for the definition for Theoretical Maximum Hull Speed is the experiments with displacement boats. If your boat is not a displacement boat, then the value 1.34 cannot be used.

In fact, some use a derived definition: if a boat can sail between 1 - 2.5 times the Theoretical Maximum Hull Speed, the boat is considered a semi-planning boat. Likewise, if a boat can sail above 2.5 times the Theoretical Maximum Hull Speed, the boat is considered a planning boat.

Also note that modern racing sailboats has a transom which extends into the water. This design trick add a kind of virtual length to the boat with the same result as had the boat been longer. For these boats you should substitute the LWL with this, let's call it virtual LWL (vLWL). As the vLWL is always greater than LWL it is equivalent to increase the 1.34 value. This is the actual reason why you will see that some people claim that the 1.34 is higher for these boat designs. In stead of claiming that the 1.34 should be higher, it would be more correct to keep the value 1.34 and enter a larger value for LWL because the boat's water line - by the design - is extended behind the boat. This will also make it easier to understand what is actual happening.

Conclusion: For the mainstream displacement hull design with the average motor/sail power, the value 1.34 is just fine. But the value 1.34 (or more precise the LWL) is not valid for certain hull designs and for boats with huge power.

- h/λ is large
- g = 9.81 m/s
^{2} - The boat is a displacement boat

For a deep water wave the condition was that h/λ is large, and as a result the tanh expression could be substituted by 1

Example 1: The boat's waterline is 30 ft and the water is 100 ft deep. h/λ = 100/30 ≈ 3.3

Then

1.34 * = 1.34 * 0.99999999999999999935 ≈ 1.34

Example 2: The boat's waterline is 30 ft and the water is 30 ft deep. h/λ = 30/30 = 1

Then

1.34 * = 1.34 * 0.9999965 ≈ 1.34

Example 3:
The boat's waterline is 30 ft and the water is 10 ft deep. h/λ = 10/30 ≈ 0.33Then

1.34 * = 1.34 * 0.985 ≈ 1.32

Conclusion: The value 1.34 can be used if the water depth is larger than the length of the boat,

Now, let's investigate variations of the gravitational value.

The nominal "average" value at Earth's surface, known as standard gravity is - by definition - 9.80665 m/s

This average value was defined at a meeting in CGPM (Conférence Générale des Poids et Mesures) in Paris in 1901. But the earth is not a perfect sphere, the earth is rotating, and pulled by the moon and the sun. Therefore the actual value of g varies from the equator to the poles and even from day to day.

At the poles g = 9.832 m/s

At 45° latitude g = 9.80665 m/s

At equator g = 9.78033 m/s

Conclusion: From a gravitational perspective the value 1.34 is valid from the poles to equator.

Do not forget that the background for the definition for Theoretical Maximum Hull Speed is the experiments with displacement boats. If your boat is not a displacement boat, then the value 1.34 cannot be used.

In fact, some use a derived definition: if a boat can sail between 1 - 2.5 times the Theoretical Maximum Hull Speed, the boat is considered a semi-planning boat. Likewise, if a boat can sail above 2.5 times the Theoretical Maximum Hull Speed, the boat is considered a planning boat.

Also note that modern racing sailboats has a transom which extends into the water. This design trick add a kind of virtual length to the boat with the same result as had the boat been longer. For these boats you should substitute the LWL with this, let's call it virtual LWL (vLWL). As the vLWL is always greater than LWL it is equivalent to increase the 1.34 value. This is the actual reason why you will see that some people claim that the 1.34 is higher for these boat designs. In stead of claiming that the 1.34 should be higher, it would be more correct to keep the value 1.34 and enter a larger value for LWL because the boat's water line - by the design - is extended behind the boat. This will also make it easier to understand what is actual happening.

Conclusion: For the mainstream displacement hull design with the average motor/sail power, the value 1.34 is just fine. But the value 1.34 (or more precise the LWL) is not valid for certain hull designs and for boats with huge power.

Final thoughts

Again, Theoretical Maximum Hull Speed is just a definition, not a physical law.
If you install an infinite powerful motor the boat can sail infinite fast, even as a displacement boat.^{[note 1]}

This said, as rule of thumb, if you want to sail faster than this theoretical speed, you will have to will have to provide proportional extra power. So, if you have a displacement motorboat, you might want to sail below this speed for economical and environmental pollution reasons.

This said, as rule of thumb, if you want to sail faster than this theoretical speed, you will have to will have to provide proportional extra power. So, if you have a displacement motorboat, you might want to sail below this speed for economical and environmental pollution reasons.

Notes

[Note 1]: According to the Einstein's general theory of relativity^{[2]} there is a maximum speed in every media.
And this **is** a physical law.
Unfortunately, the relativistic maximum speed of displacement boats in water is not mentioned anywhere in his theory :-)

References

[Ref 1]: Froude, W. 1955, "The papers of William Froude", edited by A.D. Duckworth, London: Institution of Naval Architects, page 280

[Ref 2]: Prinzipielles zur allgemeinen Relativitätstheorie, Annalen der Physik, Vol. 55 (1918), pp. 241-4. Leipzig

[Ref 2]: Prinzipielles zur allgemeinen Relativitätstheorie, Annalen der Physik, Vol. 55 (1918), pp. 241-4. Leipzig