What is wave cancellation, how does it work, and how we hope to use the Deltaform configuration to further research and advance the understanding of wave cancellation as it applies to multihull vessels?


 Wave-making resistance

Wave resistance in a marine vessel is a measure of how much propulsive energy is expended to push the water out of the way of the hull. This energy goes into creating the wave.

The waves that are generated by the hulls moving through the water are determined by the vessel's displacement, how much of the submerged body is close to the surface, the vessel’s speed and to a lesser extent by it's' hull form.


The waves (or the wake) created by the vessel consists of divergent and transverse waves.

(i) Transverse Waves

Transverse waves are those waves traveling roughly perpendicular to the ship’s track. They can be seen extending across the otherwise relatively calm area between the sides of the wake. The transverse displacement waves are always contained within the Kelvin Wake (explained later) and always reach its' outer boundary.

(ii) Diverging Waves

Diverging waves are those waves traveling diagonally outwards.

The divergent waves are observed as the wake of a ship with a series of diagonal or oblique crests moving at an angle to the ship’s track.


Minimising Resistance Waves

The wave-making can be minimised by using fine hulls (often referred to as semi displacement hulls), hydro-foiling, planing, or in the case of ocean going cargo ships employing a bulb to modify the bow wave. And of course resistance can also be reduced by minimising displacement.

The wake pattern strongly depends on the Froude number.

As the Froude number is increased the relative amplitude of the transverse waves decreases until they are barely noticeable at a Froude Number of 3, a regime where many high speed planing boats operate.



Cancelling the Waves

While large cargo vessels make use of bulbs to modify the bow wave and reduce wave making resistance (the first known use of a bulb bow was on the USS Delaware in 1907), the Deltaform configuration naturally, and somewhat uniquely, lends itself to wave cancellation technology.

By strategically placing the outrigger hulls in the fore and aft and athwarthships plane we have the opportunity to modify the transverse displacement waves generated by the centre hull, and coincidentally the transverse waves from the inboard side of the outrigger hull.

Wave cancellation is achieved when the crests from one set of waves are in phase with the troughs of another set of waves. The maximum wave height, (and consequently drag) is reduced or potentially even totally cancelled in that region where the crests and troughs are in phase. This is known as destructive interference.


If the crests of each set of waves were in phase with each other we would have constructive interference and resistance would be increased.


Transverse wave cancellation is a complex exercise and it helps to have at least a basic understanding of wave theory, especially where there is interference from waves being refracted and reflected.

One good source of information on the topic, especially in relation to multihulls, is a research paper by Ernie Tuck and Leo Lazauskas titled "Optimum Spacing of a Family of Multihulls" which looks at various multihull configurations from two hulls up to an infinite number of hulls and provides a means of calculating the optimum hull spacing to maximise wave cancellation. Careful choice of hull placement and separation in a given speed range is critical .



Of all the configurations studied in the Tuck and Lazauskas paper, if you disregard the more impractical arrangements, the CAT, the TRI and the ARR offer the best opportunity to cancel waves, and the ARR is the best of these for low speed cancellation. We are not concerned with high speed cancellation because the hulls be foil supported and will be flying in any case.


Froude Number

The Froude number can be used to determine the resistance of vessels and other partially submerged objects moving through water, and permits the comparison of vessels of different sizes with a non dimensional number.

The Froude number is thus analogous to the Mach Number. The greater the Froude number, the greater the resistance.

The Froude number is fundamantal to studying the resistance and wake formation in vessels.


For metric units Fn=V/SQRTL x .1642


v = speed in knots

LWL = square root of the WL length in metres.


The Froude number is not a measurement of a vessel’s efficiency. A small vessel can be very light and easily driven, but if only a small amount of energy is applied and it doesn’t go very fast, it will have a low Froude number.

To measure a vessel’s efficiency the displacement to length ratio is used.


Some examples of typical Froude numbers for a range of vessels is provided at the end of the document and we give the example of a 30’ sailing boat with three different Froude numbers depending on wind speed (or sail area) to illustrate how the Froude number applies.


The Kelvin Angle

The angle the wake makes to the direction of travel was calculated by Lord Kelvin in the 1880's to be a constant 19.47˚ either side of the ship's centreline, and remains constant independent of the vessel's speed or displacement.



Kelvin reasoned that no matter the speed of the boat the wake angle will remain constant and he based this reasoning on two fundamental arguments.

First, the speed or "phase velocity" of water waves varies with their wavelength, with longer wavelengths traveling faster than shorter ones do. As the boat moves, it creates waves of all speeds slower than the boat itself. And the longer waves generally spread out behind it faster than the shorter ones.

To make a stable wake, however, the waves also have to overlap and "interfere" in the right way. For waves moving almost as fast as the boat, that interference occurs only right behind the boat. So the fastest waves also produce a narrow wake.

Putting these two factors together, Kelvin demonstrated that the width of the wake is determined by waves traveling at a fixed fraction of the boat's speed: 81.6%. Because of that proportionality, the wake angle is always the same, as the faster the boat goes ,the faster the wake spreads.


Over time the Kelvin angle has been called to question with some observers noting that some ships, particularly faster ones; appear to have Kelvin angles noticeably less than 19.47˚. This argument, and doubts about the validity of the Kelvin Angle has likely been invigorated by developments in fast ferries and the recent additions of the USS Liberty and USS Independence Littoral Ships to the US Navy’s fleet.

Both ships employ advanced hull design and high powered gas turbine engines, and both, but especially USS Independence exhibit very flat and very straight wakes that are almost parallel to the track of the vessel when operating at high speed.



The Relationship between the Froude Number and the Kelvin Angle

Until fairly recently the discrepancy between the theoretical Kelvin Angle and what is being observed has been explained away by such possible phenomena as the effects of finite depth, waves already present on the surface, or complicated, "non-linear" interactions of the waves,. It has also been blamed on inaccurate observations.

However two recent studies have shown that in fact there is more going on here than just the Kelvin angle, and that while the Kelvin angle has been confirmed and acknowledged as being correct by both studies, the Kelvin angle is not necessarily the angle of the most predominant wave crests, and at High Froude numbers the Kelvin wake as defined might be virtually imperceptible.

USS Littoral Ship Independence. A straight wake if ever I saw one.
USS Littoral Ship Independence. A straight wake if ever I saw one.

Recent Studies of Kelvin Angle

(i) The Rabaud Moisy Study titled “Ship Wakes: Kelvin or Mach Angle?”

First Published in May 2013

The study by Rabaud and Moisy supported by the observation of airborne images of ships wakes from the Google Earth Database, showed that the wake angle seems to decrease as the Froude number Fr increases, scaling as 1/Fr for large Froude numbers.

For the most part the study showed that at low Froude numbers the wake angle did indeed appear to very close to Froude’s 19.47˚, (they measured a plateau 18.6˚ despite a certain amount of scatter in the data) but examples were found at 10˚ and even as low as 7˚ for the fasted ships in the data set.


To explain these observations Rabaud and Moisy hypothesise that an object of size b cannot generate wavelengths larger than b, thus leading to unrealistic pressure fields to model the object.

If a boat is going faster than a certain speed determined by its hull length, they argue, it cannot create waves with wavelengths longer than the hull. That wavelength "cutoff" then limits the speed of the waves and therefore how fast the wake can spread. So as the boat picks up even more speed, its wake stretches and narrows.

From their observations Rabaud and Moisy postulate that “ship wakes undergo a transition from the classical Kelvin Regime at low speeds to a previously unnoticed high speed regime that resembles the Mach cone prediction for supersonic aircraft”. (quoted from Physical Review Letters 24th May 2013).


Rabaud and Moisy go on to propose a model that takes into account the finite length of the ship and which successfully predicts that the point where the wave propagation transitions from a Kelvin regime to a Mach regime at a critical Froude number of Fr=0.5



(ii) More recently a study published by Darmon, Benzaquen and Raphael titled “A Solution to the Kelvin Wake Angle Controversy” does not contradict Rabaud and Moisy’s observations but offers an alternate and simpler explanation for the behavior of the wake at higher Froude numbers.

Rabaud and Moisy hypothesise that an object of size b cannot generate wavelengths greater than b. Darmon, Benzaquen and Raphael propose an analytical explanation of Rabaud and Moisy's results that does not require the use of the maximum wavelength argument.


The graphics above show a series of five relief plots of the surface displacement waves generated by a single hull vessel at five different Froude Numbers. The red line is the maximum envelope for the surface displacement using Darmon, Benzaquen and Raphael's analytic methods which are described in full in their paper arXiv:1309.6751v1


Two sets of waves can be distinguished in the wake, transverse and diverging.

Their amplitude directly depend on the Froude Number. As the Froude Number increases the amplitude of the diverging waves increases also, but the transverse waves decrease rapidly and seem to vanish for sufficiently high Froude Numbers.


What Darmon, Benzaquen and Raphael's analysis shows is that while the Kelvin Angle remains constant for all Froude Numbers, there is a different angle corresponding to the maximum amplitude of the waves that can be identified and that at high Froude Numbers this angle behaves as a Mach angle as highlighted by Rabaud and Moisy.


To quote from “Rabaud and Moisy may not be measuring the right angle, says Robert Beck, a hydrodynamicist in the Department of Naval Architecture and Marine Engineering at the University of Michigan, Ann Arbor. They measure the angle between the most intense waves on either side of the boat. But in mathematical terms, Beck notes, the wake angle is defined as a particular crease in the wave pattern—and that crease could be fainter for faster boats. "If you want to define the wake angle as what's most visible, then that's okay, they're doing it right," Beck says.



Canceling Waves with the Deltaform Configuration

One particular aspect of wave cancellation is that it will only work at optimum efficiency for a specific wavelength, and this dictates a specific operating speed. This is not necessarily a problem for the Deltaform boats, especially if they employ foils, because the optimum speed can be designated as the maximum cruise speed just below the speed where the foils will lift the hulls clear.

So planning the ideal hull length, lateral spacing, and fore and aft placement has to be integrated with power requirements and the preferred operating speed both in displacement mode and the intended lift off speed for the foils.


At the time of writing this article we are in the process of studying the wake patterns under various Froude numbers to determine whether we can make substantial performance gains for the Deltaform configuration using cancellation technology.



Graphic-01 Divergent waves only
Graphic-01 Divergent waves only
Graphic-02 Transverse waves only
Graphic-02 Transverse waves only
Graphic-03 Divergent and transverse waves superimposed
Graphic-03 Divergent and transverse waves superimposed



The diagrams above are wakemapes of a Deltaform (or ARR) configuration that help to visualise the wave pattern of transverse and divergent waves. In particular it shows where there will and will not be zones of interference and where possible cancelling can occur for waves of each type.





We are very fortunate to have access to excellent papers and discussions on web forums by a number of highly qualified people who are very generous with their time and the extent of knowledge they make available.

In particular I would like to thank Tom Speer, Leo Lazauskas, and posthumously Ernie Tuck for their contribution to numerous technical papers and online advice.



On interpreting the Science ( a designer’s perspective).

I was amused by T.S’s observation in regard to the two recent papers debating the Kelvin angle. The first by Rabaud and Moisy and the second by Darmon, Benzaquen and Raphael as discussed above. Tom’s observation was; “It’s a bit like the blind man and the elephant story, they’re both right”.

And I guess we could add; “So was Kelvin”


And later I came across a forum post by Leo who recounted the problem of the incorrect diagram of the wake formation that keeps cropping up in text books.

“The incorrect graphs seem impossible to eradicate. It's like the myth of flowing glass windowpanes”

If you’re not familiar with the story (of the flowing glass windowpanes) it is based on the long held (and totally ridiculous) theory that 50% of old window panes can be observed to be thicker at the bottom than at the top; therefore glass must be liquid even after it’s cooled!




One paper I came across in my research that I highly recommend to anyone interested in a further understanding of hull resistance and wakes is produced by the Tasmanian Department of Primary Industries, Parks, Water and the Environment. It is titled “Knowing your boat means knowing its wake.” and you can find it here;$FILE/Understanding%20Wave%20Wake.pdf


The paper includes plots from the Michlet computer program developed by Australian Mathematician and avid hydrodynamicist Leo Lazauskas (unfortunately spelled incorrectly in the attribute), which in turn is built upon the work of acclaimed Australian mathematician John Henry Michell (1863-1940).

Michell excelled in his knowledge of hydrodynamics and elasticity and his paper “The wave resistance of a ship” which was published in 1898 was taken up in earnest by German and English hydrodynamicists and ship designers some 30 years later.


And I can highly recommend just about anything on Vimeo or YouTube featuring interviews with, or lectures by Richard Feynman. Not sure if he was into boats but he sure as hell was passionate about understanding waves and pretty much all aspects of physics.


 If you’re interested in waves (apart from just surfing) check out this video by DanielPalacios;


Watch it in full screen if you can (preferably not on your phone). It’s one of the staff picks on Vimeo. It would seem that Daniel has his feet planted equally in the arts and in science and the 5:20 video of a spinning rope is absolutely fascinating.

Daniel Palacio's Rotating Rope
Daniel Palacio's Rotating Rope