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.
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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.
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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.
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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.
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As the main hull emerges through the surface, the wave pattern becomes more like that due to a normal surface-piercing vessel.
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In the above figure, the waves created at the stern can be seen clearly, as can their interference with waves created further upstream.
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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.
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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.)
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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.
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In general, diverging waves are more difficult to cancel. The degree of cancellation depends on hull shape and, for multihulls, the lateral hull spacing.
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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).
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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.