4. SHIP WAVES

The wave elevation patterns in this note illustrate (very briefly!) some of the terminology used in ship wave theory. For a comprehensive graduate text see Marine Hydrodynamics, Newman (1977). The bible is Surface Waves by Wehausen and Laitone (1962).

Data for the wave pattern contour plots in this section that show waves very close to the ship were prepared by the author using the program SWPE. SWPE is not included in the Michlet package of progams. Enquiries regarding SWPE should be sent to Dr. David Scullen of Scullen and Tuck, Pty Ltd.
Email: dscullen@maths.adelaide.edu.au

We define the length-based Froude number of a vessel by FL=U/sqrt(gL), where U is the ship speed in metres/second, g is gravitational acceleration in metres/second/second, and L is the length of the ship in metres.

The depth-based Froude number is Fh=U/sqrt(gh), where h is the depth in metres. In infinitely deep water, Fh=0. The subcritical range is defined as Fh<1; the supercritical range is Fh>1.

Near-field, far-field, transverse and diverging waves

Figure 4.1: Wave pattern of DDG51 destroyer travelling at 15.4 m/sec
which corresponds to length-based Froude number 0.4136.

The near-field is the region close to the ship. In figure 4.1, near-field effects are most evident near the bow, and it can be seen that they die away very quickly as the distance from the bow increases. Note that Michlet does not calculate near-field effects: data for near-field plots in this section were created using SWPE.

Far-field waves are those waves far behind the ship where near-field effects are negligible. Wave resistance is the energy that is needed to sustain the far-field wave pattern. Transverse waves are those waves travelling roughly perpendicular to the ship's track (the roughly vertical bands in the figure above); diverging waves are those travelling diagonally outwards.

Submergence effects

The size of the waves that a body makes depends on how much of its volume is near the surface. Wave effects decay exponentially with depth.

Figure 4.2: US LA class submarine travelling at 5.15 m/sec.

In the figure above, the conning tower of the submarine is just protruding through the surface. The tower contains only a small part of the total volume, however it is much closer to the surface and thus makes the largest contribution to the wave system.

Figure 4.3: Same as above, but the main hull is about to protrude.

As the main hull approaches close to the surface, it makes larger waves. Near field effects are very evident in the above figure, especially near the bow of the submerged main hull.

Figure 4.4: Same as above, but most of the main hull has emerged.

As the main hull emerges through the surface, the wave pattern becomes more like that due to a normal surface-piercing vessel.

Figure 4.5: Same as above, but with fully-emerged main hull.

In the above figure, the waves created at the stern can be seen clearly, as can their interference with waves created further upstream.

Water depth effects

Water depth can have a significant effect on the far-field waves created by ships and consequently on their wave resistance. The figures in this section were created using Michlet.

Figure 4.6: Far-field wave pattern for a DDG51 destroyer travelling at
15.4 m/sec in infinitely deep water. The length-based Froude
number is 0.41. The depth-based Froude number is 0.

Figure 4.7: As above, but with Fh=0.8.
The enveloping wedge is wider, and the transverse
wavelength is longer.

As Fh increases towards 1 (the critical Froude number), the wave pattern changes dramatically. The angle of the enveloping wedge (the Kelvin angle) widens, until at Fh=1 it is perpendicular to the ship's track.

Figure 4.8: As above, but with Fh=0.95.
The enveloping wedge is significantly wider; the transverse
wavelength is almost twice as long as for the infinitely-deep situation.

Figure 4.9: As above, but with Fh=1.05.
Transverse waves have almost disappeared and the Kelvin angle is very large.

Figure 4.10: As above, but with Fh=1.4.
Transverse waves are still absent; the Kelvin angle is narrower.

Multihulls and fleets

The wave patterns created by a multihulled vessel or by several ships travelling together can be extremely complicated. Interference between hulls can reinforce waves in one part of the pattern and cancel waves in another part. The degree of wave cancellation depends on the size and spacing of the hulls, and on their individual shapes.

The figures in this section were created using from data using SWPE: (Michlet can only be used to predict the waves behind the aftmost vessel in the ensemble.)

Figure 4.11: Wave pattern produced by a DDG51 destroyer, a US LA class
submarine and a Wigley-hulled catamaran, each travelling at 15.4 m/sec.
The submarine (uppermost vessel) is just submerged.

Transverse wave cancellation depends on the length of the hulls and, for multihulls, the longitudinal spacing between hulls. If the longitudinal spacing between hull centres is 1/2 (or 3/2 or 5/2...) wavelength, there can be almost total cancellation as shown in Figure 4.12. Maximum reinforcement of transverse waves occurs when the hull centres are a separated by a whole number of transverse wavelengths as shown in Figure 4.13.

Figure 4.12: Wave pattern of two identical DDG51 destroyers each
of length 142m, travelling in tandem at a speed of 15.4 m/sec. The
hull centres are separated by 1.5 transverse wavelengths,
(approximately 229m). Note the very small transverse waves.

Figure 4.13: As above, but the separation length between hull centres
is two transverse wavelengths (approximately 306m). Transverse waves
are reinforcing.

In general, diverging waves are more difficult to cancel. The degree of cancellation depends on hull shape and, for multihulls, the lateral hull spacing.

Figure 4.14: Wave pattern created by a catamaran travelling at
a speed of 15.4 m/sec, equivalent to a length-based
Froude number of 0.53.

For standard side-by-side catamarans, diverging waves shed at particular propagation angles can be cancelled by using a lateral spacing that can be calculated using a fairly simple formula. See Optimum Hull Spacing of a Family of Multihulls, Tuck and Lazauskas (1998).

Figure 4.15: Wave pattern created by two SWATH-like hulls in a
Weinblum arrangement travelling at a speed of 15.4 m/sec.

It is possible to almost completely eliminate waves on one side of a two-hulled ship using two identical hulls if asymmetric hull placement is allowed. In the above figure, transverse waves have been cancelled by choosing the appropriate longitudinal hull spacing; diverging waves on the starboard side have almost been eliminated by a judicious choice of lateral spacing, and by using a hull shape that makes very small diverging waves. Using four identical hulls in a diamond arrangement can reduce significantly waves shed on both sides of the vessel, with an attendant large reduction in wave resistance. However, the increased surface area of the four hulls leads to a much larger frictional resistance.


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