A body moving on an otherwise undisturbed water surface creates a varying pressure field which manifests itself as waves because the pressure at the surface must be constant and equal to atmospheric pressure. From observation when the body moves at a steady speed, the wave pattern seems to remain the same and move with the body. With a ship the energy for creating and maintaining this wave system must be provided by the ship's propulsive system. Put another way, the waves

cause a drag force on the ship which must be opposed by the propulsor if the ship is not to slow down. This drag force is the wave-making resistance,

A submerged body near the surface will also cause waves. It is in this way that a submarine can betray its presence. The waves, and the associated resistance, decrease in magnitude quite quickly with increasing depth of the body until they become negligible at depths a little over half the body length.

The wave pattern

The nature of the wave system created by a ship is similar to that which Kelvin demonstrated for a moving pressure point. Kelvin showed that the wave pattern had two main features: diverging waves on each side of the pressure point with their crests inclined at an angle to the direction of motion and transverse waves with curved crests intersecting the centreline at right angles. The angle of the divergent waves to the centreline is sin"1!, that is just under 20°, Figure 8.2.

A similar pattern is clear if one looks down on a ship travelling in a calm sea. The diverging waves are readily apparent to anybody on board. The waves move with the ship so the length of the transverse waves must correspond to this speed, that is their length is 2nV1/'g, The pressure field around the ship can be approximated by a moving pressure field close to the bow and a moving suction field near the stern.

Both the forward and after pressure fields create their own wave system as shown in Figure 8.3. The after field being a suction one creates a trough near the stern instead of a crest as is created at the bow. The angle of the divergent waves to the centreline will not be exactly that of the Kelvin wave field. The maximum crest heights of the divergent waves do lie on a line at an angle to the centreline and the local crests at the maxima are at about twice this angle to the centreline. The stern generated waves are less clear, pardy because they are weaker, but mainly because of the interference they suffer from the bow system.

In addition to the waves created by the bow and stern others may be created by local discontinuities along the ship's length. However the qualitative nature of the interference effects in wave-making resistance are illustrated by considering just the bow and stern systems. The transverse waves from the bow travel aft relative to the ship, reducing in height. When they reach the stern-generated waves they interact with them. If crests of the two systems coincide the resulting wave is of greater magnitude than either because their energies combine. If the crest of one coincides with a trough in the other the resultant energy

will be less. Whilst it is convenient to picture two wave systems interacting, in fact the bow wave system modifies the pressure field around the stern so that the waves it generates are altered. Both wave systems are moving with the ship and will have the same lengths. As ship speed increases the wavelengths increase so there will be times when crests combine and others when crest and trough become coincident.

The ship will suffer more or less resistance depending upon whether the two waves augment each other or partially cancel each other out. This leads to a series of humps and hollows in the resistance curve, relative to a smoothly increasing curve, as speed increases. This is

This effect was shown experimentally by Froude3 by testing models with varying lengths of parallel middle body but the same forward and after ends. Figure 8.5 illustrates some of these early results. The residuary resistance was taken as the total measured resistance less a calculated skin friction resistance.

Now the distance between the two pressure systems is approximately 0.9L. The condition therefore that a crest or trough from the bow system should coincide with the first stern trough is:

The troughs will coincide when JVis an odd integer and for even values of N a crest from the bow coincides with the stern trough. The most pronounced hump occurs when N = 1 and this hump is termed the main hump. The hump at N = 3 is often called the prismatic hump as it is greatly affected by the ship's prismatic coefficient.

It has been shown that for geometrically similar bodies moving at corresponding speeds, the wave pattern generated is similar and the wave-making resistance can be taken as proportional to the displacements of the bodies concerned. This assumes that wave-making was unaffected by the viscosity and this is the usual assumption made in studies of this sort. In fact there will be some viscosity but its major effects will be confined to the boundary layer. To a first order then, the effect of viscosity on wave-making resistance can be regarded as that of modifying the hull shape in conformity with the boundary layer

addition. These effects are relatively more pronounced at model scale than the full scale which means there is some scale effect on wave making resistance. For the purposes of this book this is ignored.

4.1 General. The wave-making resistance of a ship is the net fore-and-aft force upon the ship due to the fluid pressures acting normally on all parts of the hull, just as the frictional resistance is the result of the tangential fluid forces. In the case of a deeply submerged body, travelling horizontally at a steady speed far below the surface, no waves are formed, but the normal pressures will vary along the length. In a nonviscous fluid the net fore-and-aft force due to this variation would be zero, as previously noted.

If the body is travelling on or near the surface, however, this variation in pressure causes waves which alter the distribution of pressure over the hull, and the resultant net fore-and-aft force is the wave-making resistance. Over some parts of the hull the changes in pressure will increase the net sternward force, in others decrease it, but the overall effect must be a resistance

of such magnitude that the energy expended in moving the body against it is equal to the energy necessary to maintain the wave system. The wave making resistance depends in large measure on the shapes adopted for the area curve, waterlines and

transverse sections, and its determination and the methods by which it can be reduced are among the main goals of the study of ships' resistance. Two paths have been followed in this study—experiments with models in towing tanks and theoretical research into wave-making phenomena. Neither has yet led to a complete solution, but both have contributed greatly to a

better understanding of what is a very complicated problem. At present, model tests remain the most important tool available for reducing the resistance of specific ship designs, but theory lends invaluable help in interpreting model results and in guiding model research.

4.2 Ship Wave Systems.

The earliest account of the way in which ship waves are formed is believed to be that due to Lord Kelvin (1887, 1904). He considered a single pressure point travelling in a straight line over the surface of the water, sending out waves which

combine to form a characteristic pattern. This consists of a system of transverse waves following behind the point, together with a series of divergent waves radiating from the point, the whole pattern being contained within two straight lines starting from the pressure point and making angles of 19 deg 28 min on each side of the line of motion.The heights of successive transverse-wave crests along the middle line behind the pressure point diminish going aft. The waves are curved back some distance out from the centerline and meet the diverging waves in cusps, which are the highest points in the system. The heights of these cusps decrease less rapidly with distance from the point than do those of the transverse waves, so that eventually well astern of the point the divergent waves become the more prominent.

The Kelvin wave pattern illustrates and explains many of the features of the ship-wave system. Near the bow of a ship the most noticeable waves are a series of divergent waves, starting with a large wave at the bow, followed by others arranged on each side along a diagonal line in such a way that each wave is stepped back behind the one in front in echelon (Fig. 8) and is of quite short length along its crest line.

Between the divergent waves on each side of the ship, transverse waves are formed having their crest lines normal to the direction of motion near the hull, bending back as they approach the divergent-system waves and finally coalescing with them. These transverse waves are most easily seen along the middle portion of a ship or model with parallel body or just behind a ship running at high speed. It is easy to see the general Kelvin pattern in such a bow system.

Similar wave systems are formed at the shoulders, if any, and at the stern, with separate divergent and transverse patterns, but these are not always so clearly distinguishable because of the general disturbance already present from the bow system. Since the wave pattern as a whole moves with the ship, the transverse waves are moving in the same direction as the ship at the same speed V, and might Fig 7 (a) Pattern of diverging waves Fig. 7(b) Typical ship wave pattern be expected to have the length appropriate to free waves running on surface at that speed Actually, the waves in the immediate vicinity of a

model are found to be a little shorter, but they attain the length Lw about two wave lengths astern.

The divergent waves will have a different speed along the line normal to their crests (Fig. 9). In this case, the component of speed parallel to the line of the ship's motion must be equal to the ship's speed in order to retain the fixed pattern relative to the ship. If the line normal to the crest of a divergent wave makes an angle with the ship's course, the speed in that direction will be Vcos , and the corresponding wave length 4.3 Wave-Making Resistance of Surface Ships. At low speeds, the waves made by the ship are very small, and the resistance is almost wholly viscous in character.

Since the frictional resistance varies at a power of the speed a little less than the square, when the coefficient of total resistance CT = plotted to a base of Froude number Fn (or of at first the value of CT decreases with increase of speed (Fig.10). With further increase in speed, the value of CT begins to increase more and more rapidly, and at Froude numbers approaching 0.45

the resistance may vary at a power of V of 6 or more.

However, this general increase in CT is usually accompanied by a number of humps and hollows in the resistance

curve. As the speed of the ship increases, the wave pattern must change, for the length of the waves

will increase and the relative positions of their crests and troughs will alter. In this process there will be a

succession of speeds when the crests of the two sys- Fig 10 Typical resistance curve, showing interference effects

tems reinforce one another, separated by other speeds at which crests and troughs tend to cancel one another.

The former condition leads to higher wave heights, the latter to lower ones, and as the energy of the systems depends upon the square of the wave heights, this means alternating speeds of higher and lower than average resistance. The humps and hollows in the curve are due to these interference effects between the wave systems, and it is obviously good design

practice to ensure whenever possible that the ship will be running under service conditions at a favorable

speed. As will be seen later, it is the dependence of these humps and hollows on the Froude number that

accounts for the close relationship between economic speeds and ship lengths.

The mechanism by which wave-making resistance is generated is well illustrated by experiments made by Eggert (1939). He measured the normal pressure distribution over the ends of a model and plotted resulting pressure contours on a body plan (Fig. 11). By integrating the longitudinal components of these pressure forces over the length, he showed that the resulting

resistance agreed fairly well with that measured on the model after the estimated frictional resistance had been subtracted. Fig. 12a shows curves of longitudinal force per meter length; Fig. 12b shows form resistance derived from pressure experiments and residuary resistance model tests. One important point brought out by these experiments is that a large proportion of the wave-making resistance is generated by the upper part of the hull near the still waterline.

4.4 Theoretical Calculation of Wave-Making Resistance.

Much research has been devoted to theoretical methods of calculating wave-making resistance and to their experimental verification (Lunde, 1957).One method is to determine the flow around the hull and hence the normal pressure distribution, and then to integrate the fore-and-aft components of these pressures over the hull surface. This method was developed

by Michell (1898) for a slender hull moving over the surface of a nonviscous fluid. It corresponds to the experimental technique employed by Eggert. The pioneer work of Michell was unfortunately overlooked and neglected for many years until rescued from obscurity by Havelock (1951).

A second method is to calculate the wave pattern generated by the ship at a great distance astern, as done by Havelock, the wave-making resistance then being measured by the flow of energy necessary to maintain the wave system. This method has been used experimentally by Gadd, et al (1962), Eggers (1962), Ward (1962) and many others.

Both methods lead to the same final mathematical expression, and in each case the solution is for a nonviscous and incompressible fluid, so that the ship will experience only wave-making resistance (Timman, et al, 1955).

Michell obtained the mathematical expression for the flow around a "slender" ship of narrow beam when placed in a uniform stream. From the resultant velocity potential the velocity and pressure distribution over the hull can be obtained, and by integrating the fore and-aft components of the pressure an expression can be derived for the total wave-making resistance.11

The theory as developed by Michell is valid only for certain restrictive conditions:

(a) The fluid is assumed to be nonviscous and the flow irrotational. Under these circumstances the motion can be specified by a velocity potential ; which in addition must satisfy the necessary boundary conditions.

(b) The hull is narrow compared with its length, so that the slope of the surface relative to the centerline plane is small.

(c) The waves generated by the ship have heights small compared with their lengths, so that the squares of the particle velocities can be neglected compared with the ship speed.

(d) The ship does not experience any sinkage or trim. The boundary conditions to be satisfied by the velocity

potential are:

(a) At all points on the surface of the hull, the normal velocity relative to the hull must be zero.

(b) The pressure everywhere on the free surface of the water must be constant and equal to the atmospheric pressure.

To make the problem amenable to existing mathematical methods, Michell assumed that the first boundary condition could be applied to the centerline plane rather than to the actual hull surface, so that the results applied strictly to a vanishingly thin ship, and that the condition of constant pressure could be applied to the original flat, free surface of the water, the distortion

of the surface due to the wave pattern being neglected.

The alternative method developed by Havelock, in which the wave-making resistance is measured by the energy in the wave system, makes use of the idea of sources and sinks.12 This is a powerful tool with which to simulate the flow around different body shapes and so to find the wave pattern, pressure distribution, and resistance. A "thin" ship, for example, can be simulated by a distribution of sources on the centerline plane of the forebody and of sinks in the afterbody,

the sum of their total strength being zero. The re- The velocity potential has the property that the velocity

of the flow in any given direction is the partial derivative of with respect to that direction.

Residuary resistance derived from towing experiments to be everywhere proportional to the slope of the hull surface, this will result in a total strength of zero, and the total velocity potential will be the sum of those due to the individual sources and sinks and the uniform flow.

Each source and sink when in motion in a fluid, on or near the surface, gives rise to a wave system, and by summing these up the total system for the ship can be obtained. Havelock by this method found the wave pattern far astern, and from considerations of energy obtained the wave-making resistance.

Much of the research into wave-making resistance has been done on models of mathematical form, having sections and waterlines defined by sine, cosine, or parabolic functions. When the calculations are applied to actual ship forms, the shape of the latter must be expressed approximately by the use of polynomials (Weinblum, 1950, Wehausen, 1973); or by considering

the hull as being made up of a number of elementary wedges (Guilloton, 1951).

In recent years, a great deal of work on the calculation of wave-making resistance has been carried out

in Japan by Professor Inui and his colleagues (Inui, 1980). They used a combination of mathematical and

experimental work and stressed the importance of observing the wave pattern in detail as well as simply

measuring the resistance. Instead of starting with a given hull geometry, Professor Inui began with an

assumed source-sink distribution, with a view to obtaining better agreement between the measured and

calculated wave systems, both of which would refer to the same hull shape. The wave pattern and the wavemaking

resistance were then calculated from the amplitudes of the elementary waves by using Havelock's

concept.

Professor Inui tried various distributions of sources and sinks (singularities) by volume over the curved

surface, in a horizontal plane and over the vertical middle-line plane. For displacement ships at Froude

numbers from 0.1 to 0.35, he found the geometry of the ends to be most important, and these could be

represented quite accurately by singularities on the middle-line plane. For higher Froude numbers, the disstriction

to a "thin" ship can be removed if the sources and sinks are distributed over the hull surface itself.

If the strengths of the sources and sinks are assumed 12 A source may be looked upon as a point in a fluid at which new

fluid is being continuously introduced, and a sink is the reverse, a point where fluid is being continuously abstracted. The flow out of a source or into a sink will consist of radial straight stream lines,

Fig. 13. If a source and an equal sink be imagined in a uniform stream flow, the axis of the source and sink being parallel to the

flow, the streamlines can be combined as shown in Fig. 14, and there will be one completely closed streamline ABCD. Since the source and sink are of equal strength, all fluid entering at +s will be removed at — s, and no fluid will flow across ABCD, and the space inside this line could be replaced by a solid body.

Fig 13 Flow patterns for source and sink

Fig 14 Flow patterns for a source and sink in a uniform stream tribution of sources along the whole length becomes

important. In summary, the method is to choose a singularity distribution which will give good resistance qualities, obtain the corresponding hull geometry, carry out resistance and wave-observation tests and modify the hull to give a more ship-shape form amidships.

In this way Inui has been able to obtain forms with considerably reduced wave-making resistance, usually associated with a bulb at the stem and sometimes at the stern also.

Recent developments in wave-making resistance theory can be divided into four main categories. The first concerns applications of linearized potential flow theory, either with empirical corrections to make it more accurate, or uncorrected for special cases where the errors due to linearization are not serious. The second concerns attempts to improve on linearized potential flow theory, by analysis of non-linear effects on the free-surface condition, or by an assessment of the effects

of viscosity. Thirdly, attempts have been made to apply wave resistance theory to hull form design.

Fourthly there has been an increase in the number of primarily numerical approaches to ship wave resistance estimation. In the second category (non-linear calculations) the work of Daube (1980), (1981) must be mentioned. He uses an iterative procedure where at each step a linear problem is solved. To this end an initial guess of the location of the free surface is made which is subsequently changed to fulfill a free surface condition. In the computation of the free surface elevation the assumption is made that the projection of the free surface streamlines on the horizontal plane, always agree with the double model streamlines.

This is in fact a low-speed assumption. The non-linear calculation method has been applied to a Wigley hull and a Series 60 ship. Comparison with measurements show a qualitatively satisfactory agreement, and quantitatively the calculations are better than with linear theory for these cases. Part of the discrepancies between measurements and calculations (at least for the higher speed range) can be ascribed to trim and sinkage effects which have not been properly included in Daube's method.

An interesting development has been the determination of pure wave-making resistance from measurements of model wave patterns. The attempts to improve hull forms using the data of wave pattern measurement combined with linearized theory are particularly interesting. For example, Baba (1972) measured the difference in wave pattern when a given hull was modified according to the insight gained from wave resistance theory and thereby gained an improvement.

To a certain degree, the hull forms of relatively high speed merchant ships have improved because of the application of wave resistance theory. Pien et al (1972) proposed a hull-form design procedure for high-speed displacement ships aided by wave resistance theory.

Inui and co-workers have applied the streamline tracing method to practical hull forms with flat bottoms and a design method for high-speed ships with the aid of minimum wave resistance theory has been proposed by Maruo et al (1977). The development of special types of hull forms for drastically reduced wave making have also been guided to a certain extent by wave resistance theory. One of these is the small waterplane area twin-hull (SWATH) ship, discussed in Section 9. The accuracy and usefulness of wave resistance theory was recently demonstrated at a workshop organized by the DTRC (Bai, et al, 1979).

The results of theoretical work would therefore seem at present to be most useful in giving guidance in the choice of the secondary features of hull shape for given proportions and fullness, such as the detail

Fig 15 Wave systems for a simple wedge-shaped form shapes of waterlines and sections, and choice of size

and location of bulbs.

The calculation of resistance cannot yet be done with sufficient accuracy to replace model experiments, but it is a most valuable guide and interpreter of model work. The advent of the computer has placed new power in the hands of the naval architect, however, and has brought much closer the time when theory can overcome its present limitations and begin to give meaningful numerical answers to the resistance problem.

4.5 Interference Effects. The results of mathematical research have been most valuable in providing an insight into the effects of mutual wave interference upon wave-making resistance. A most interesting example is that of a double-wedge-shaped body with parallel inserted in the middle, investigated by Wigley (1931). The form of the hull and the calculated and

measured wave profiles are shown in Fig. 15. He showed that the expression for the wave profile along the hull contained five terms:

(a) A symmetrical disturbance of the surface, which has a peak at bow and stern and a trough along the center, dying out quickly ahead and astern of the hull. It travels with the hull and because of its symmetry does not absorb any energy at constant speed, and four wave systems, generated at

Fig 16 Resistance curves for wedge-shaped model

Dimensions: 4.8 X 0.46 X 0.30 m, prismatic coefficient 0.636.

(b) the bow, beginning with a crest;

Fig 17 Analysis of wave-making resistance into components for wedge-shaped model shown above

(c) the forward shoulder, starting with a trough;

(d) the after shoulder also starting with a trough;

(e) the stern, beginning with a crest.

These five systems are shown in Fig. 15. Considerably aft of the form, all four systems become sine curves of continuously diminishing amplitude, of a length appropriate to a free wave travelling at the speed of the model, this length being reached after about two waves.

The calculated profile along the model is the sum of these five systems, and the measured profile was in general agreement with it so far as shape and positions of the crests and troughs were concerned, but the heights of the actual waves towards the stern were considerably less than those calculated (Fig. 15).

This simple wedge-shaped body illustrates clearly the mechanism of wave interference and its effects upon wave-making resistance. Because of the definite sharp corners at bow, stern, and shoulders, the four free-wave systems have their origins fixed at points along the hull. As speed increases, the wave lengths of each of the four systems increase. Since the primary

crests and troughs are fixed in position, the total wave profile will continuously change in shape with speed

as the crests and troughs of the different systems pass through one another. At those speeds where the interference

is such that high waves result, the wave making resistance will be high, and vice-versa.

In this simple wedge-shaped form the two principal types of interference are between two systems of the same sign, e.g., bow and stern, or the shoulder systems, and between systems of opposite sign, e.g., bow and forward shoulder. The second type is the most important in this particular case, because the primary hollow of the first shoulder system can coincide with the first trough of the bow system before the latter has been materially reduced by viscous effects.

Wigley calculated the values of for minima and maxima of the wave-making resistance coefficient Cw for this form, and found them to occur at the following points:

The mathematical expression for the wave-making resistance Rw is of the form (constant term + 4 oscillating terms) so

that the wave-making resistance coefficient Cw is (constant term + 4 oscillating terms)

The curve is thus made up of a steady increase varying as due to the constant term and four oscillating curves due to the interference between the different free-wave systems (Fig. 17). These latter ultimately, at very high speeds, cancel both each other and the steady increase in and there is no further hump beyond that occurring at a value of about 0.45 after which the value of continuously decreases with further increase in speed. However, at these high speeds the hull will sink bodily and change trim so much that entirely new phenomena arise. For more ship-shaped forms, where the waterlines are curved and have no sharp discontinuities, the wave pattern still consists of five components—a symmetrical disturbance and four free-wave systems (Wigley,1934). Two systems begin with crests, one at the bow and one at the stern, and are due to the change in the

angle of the flow at these points. The other two systems, like the shoulder systems in the straight line form, begin with hollows, but are no longer tied to definite points, since the change of slope is now gradual and spread over the whole entrance and run. They commence at the bow and after shoulder, respectively, as shown in Fig. 18, much more gradually than in the

case of the wedge-shaped form. The one due to entrance curvature, for example, may be looked upon as

a progressive reduction of that due to the bow angle as the slope of the waterline gradually becomes less

in going aft.

Wigley also made calculations to show the separate contributions to the wave-making resistance of the transverse and divergent systems (Wigley, 1942). Up to a Froude number of 0.4 the transverse waves are mainly responsible for the positions of the humps and hollows, Fig. 19. Above this speed the contribution from the divergent waves becomes more and more

important, and the interference of the transverse waves alone will not correctly determine the position of the higher humps, particularly the last one at Fn = 0.5.

The existence of interference effects of this kind was known to naval architects long before such mathematical analysis was developed. The Froudes demonstrated them in a striking way by testing a number of models consisting of the same bow and stern separated by different lengths of parallel body (Froude, W., 1877 and Froude, R.E., 1881). W. Froude's sketch of the bow wave system is shown in Fig. 20. As the ship advances but the water does not, much of the energy given to the water by the bow is carried out laterally and away from the ship. This outward spreading of the energy results in a decrease in the height

of each succeeding wave of each system with no appreciable change in wave length. Fig. 21 shows a series of tests made at the EMB, Washington, and the corresponding curve of model residuary resistance plotted against length of parallel body (Taylor, 1943). The tests were not extended to such a length of parallel body that the bow system ceased to affect that at the stern.

It is clear, however, that its effect is decreasing and would eventually died out, as suggested by the dotted extension of the resistance curve.

Fig. 22 shows a series of curves for the same form at various speeds. In this chart the change of parallel middle-body length which results in successive humps on any one curve is very nearly equal to the wave length for the speed in question, as shown for speeds of 2.6 and 3.2 knots. This indicates that ship waves do have substantially the lengths of deep-sea waves of

the same speed.

If all the curves in Fig. 22 are extended in the direction of greater parallel-body length until the bow system ceases to affect the stern system, as was done in Fig. 21, the mean residuary resistances for this form, shown by the dashed lines at the left of the

chart, are found to increase approximately as the sixth power of the speed. They are, in fact, the actual resistances

stripped of interference effects and represent the true residuary resistances of the two ends. This rate of variation with speed is the same as that given by theory for the basic wave-making resistance before taking into account the interference effects (Fig. 17).

The mathematical theory indicates that the wave resistance is generated largely by those parts of the

hull near the surface, which is in agreement with the experimental results obtained by Eggert. This suggests

that from the point of view of reducing wavemaking resistance the displacement should be kept as

low down as possible. The relatively small effect of the lower part of the hull on the wave systems also means

that the wave-making resistance is not unduly sensitive to the midship section shape (Wigley, et al, 1948).

4.6 Effects of Viscosity on Wave-Making Resistance.

Calculations of wave-making resistance have so far been unable to take into account the effects of viscosity, the role of which has been investigated by Havelock (1923), (1935) and Wigley (1938). One of these effects is to create a boundary layer close to the hull, which separates the latter from the potential-flow pattern with which the theory deals. This layer grows thicker from stem to stern, but outside of it the fluid behaves very much in accordance with the potential flow theory. Havelock (1926) stated that the direct influence of viscosity on the wave motion is comparatively small, and the "indirect effect might possibly be allowed for later by some adjustment of the effective form of the ship." He proposed to do this by assuming that the after body was virtually lengthened and the , aft end waterlines thereby reduced in slope, so reducing the after-body wavemaking. Wigley (1962) followed up this suggestion by comparing calculated and measured wave-making resistance for 14 models of

mathematical forms, and deriving empirical correction factors. He found that the remaining differences in resistance were usually within 4 percent, and that the virtual lengthening of the hull due to viscosity varied between 2 and 8 percent.

The inclusion of a viscosity correction of this nature also explains another feature of calculated wave-making resistance. For a ship model which is unsymmetrical fore and aft, the theoretical wave-making resistance in a nonviscous fluid is the same for both directions of motion, while the measured resistances are different. With the viscosity correction included, the calculated resistance will also be different. Professor Inui (1980) in his wave-making resistance work also allows for viscosity by means of two empirical coefficients, one to take care of the virtual lengthening of the form, the other to allow for the effect of viscosity on wave height.

4.7 Scale Effect on Wave-Making Resistance.

Wigley (1962) has investigated the scale effect on Cw due to viscosity, pointing out that the calculated curves of are usually higher than those measured in experiments and also show greater oscillations. These differences he assigned to three major causes:

(a) Errors due to simplifications introduced to make the mathematical work possible.

(b)Errors due to neglect of the effects of viscosity on

(c) Errors due to the effects of wave motion on Errors under (a) will decrease with increasing speed, ! since they depend on the assumption that the velocities de to the wave motion are small compared with the speed of the model, which is more nearly fulfilled at high speed.

Errors under (b) will depend on Reynolds number, and therefore on the size of the model, decreasing as size increases. From experiments on unsymmetrical models tested moving in both directions, these errors cease to be important for Fn greater than 0.45.

At low speeds errors under (c) are negligible, but become important when Fn exceeds 0.35, 1.15) as evidenced by the sinkage

and trim, which increase very rapidly above this speed. A practical conclusion from this work is the effect on the prediction of ship resistance from a model. In a typical model the actual wave resistance is less than that calculated in a perfect fluid for Froude numbers less than about 0.35. This difference is partly due to viscosity, the effect of which will decrease with increased size, and will increase with scale instead of being constant as assumed in extrapolation work. Wigley made estimates of the

difference involved in calculating the resistance of a 121.9 m ship from that of a 4.88 m model at a Froude number of 0.245 and found that the resistance of the ship would be underestimated, using the usual calculations, by about 9 percent, the variation with speed being approximately as shown in Fig. 23. The effect disappears at low speeds and for values of Fn

above 0.45.

4.8 Comparison Between Calculated and Observed Wave-Making Resistance. Many comparisons have been made between the calculated and measured wave-making resistances of models. Such a comparison is difficult to make, however. All that can be measured on the model is the total resistance and the value of can only be obtained by making assumptions as to the amount of frictional resistance, viscous pressure drag and eddy-making resistance, all quantities subject to considerable doubt. The wave making resistance has been measured directly by observing the shape of the wave system astern of the

model and computing its energy and the total viscous drag has been measured by a pitot tube survey behind the model (Wehausen, 1973). Both of these methods are relatively new, and there are problems in interpreting the results. In the meantime it is perhaps best to be content with comparing the differences between the calculated and measured resistances for pairs of models of the same overall proportions and coefficients but differing in those features which are likely to affect wave-making resistance.

A comparison of much of the available data has been made by Lunde, the measured being derived from on Froude's assumption and using his skin friction coefficients, the calculated being empirically corrected for viscosity (Lunde, 1957).

At low Froude numbers, less than 0.18, it is difficult to determine CR with any accuracy. At higher speeds, the humps at Fn of 0.25 and 0.32 and the intervening hollow are much exaggerated in the calculated curves, and any advantage expected from designing a ship to run at the "hollow" speed would not be fully achieved in practice (Fig. 24). The general agreement in level

of the curves over this range depends to some extent upon the form of the model, theory overestimating the resistance for full ships with large angles of entrance.

Just above a speed of Fn = 0.32 the model becomes subject to increasing sinkage and stern trim, effects

which are not taken into account in the calculations.

The last hump in the curve occurs at a Froude number of about 0.5, and here the calculated value of is less than the measured again possibly due to the neglect of sinkage and trim. In all cases the humps and hollows on the measured curves occur at higher values of Fn than those given by theory, by amounts varying from 2 to 8 percent. In other words, the model behaves as though it were longer than its actual length, and this is undoubtedly due mostly to the virtual lenghening of the form due

to the viscous boundary layer. At very low speeds, Fn = 0.1, the wave-making resistance varies approximately

as the square of the tangent of the half-angle of entrance, but its total value in terms of RT is very small. At high speeds, with Fn greater than 1.0, the wave-making resistance varies approximately as the square of the displacement, illustrating the well-known fact that at very high speeds shape is relatively unimportant, the chief consideration being the displacement

carried on a given length.

cause a drag force on the ship which must be opposed by the propulsor if the ship is not to slow down. This drag force is the wave-making resistance,

A submerged body near the surface will also cause waves. It is in this way that a submarine can betray its presence. The waves, and the associated resistance, decrease in magnitude quite quickly with increasing depth of the body until they become negligible at depths a little over half the body length.

The wave pattern

The nature of the wave system created by a ship is similar to that which Kelvin demonstrated for a moving pressure point. Kelvin showed that the wave pattern had two main features: diverging waves on each side of the pressure point with their crests inclined at an angle to the direction of motion and transverse waves with curved crests intersecting the centreline at right angles. The angle of the divergent waves to the centreline is sin"1!, that is just under 20°, Figure 8.2.

A similar pattern is clear if one looks down on a ship travelling in a calm sea. The diverging waves are readily apparent to anybody on board. The waves move with the ship so the length of the transverse waves must correspond to this speed, that is their length is 2nV1/'g, The pressure field around the ship can be approximated by a moving pressure field close to the bow and a moving suction field near the stern.

Both the forward and after pressure fields create their own wave system as shown in Figure 8.3. The after field being a suction one creates a trough near the stern instead of a crest as is created at the bow. The angle of the divergent waves to the centreline will not be exactly that of the Kelvin wave field. The maximum crest heights of the divergent waves do lie on a line at an angle to the centreline and the local crests at the maxima are at about twice this angle to the centreline. The stern generated waves are less clear, pardy because they are weaker, but mainly because of the interference they suffer from the bow system.

**Interference effects**In addition to the waves created by the bow and stern others may be created by local discontinuities along the ship's length. However the qualitative nature of the interference effects in wave-making resistance are illustrated by considering just the bow and stern systems. The transverse waves from the bow travel aft relative to the ship, reducing in height. When they reach the stern-generated waves they interact with them. If crests of the two systems coincide the resulting wave is of greater magnitude than either because their energies combine. If the crest of one coincides with a trough in the other the resultant energy

will be less. Whilst it is convenient to picture two wave systems interacting, in fact the bow wave system modifies the pressure field around the stern so that the waves it generates are altered. Both wave systems are moving with the ship and will have the same lengths. As ship speed increases the wavelengths increase so there will be times when crests combine and others when crest and trough become coincident.

The ship will suffer more or less resistance depending upon whether the two waves augment each other or partially cancel each other out. This leads to a series of humps and hollows in the resistance curve, relative to a smoothly increasing curve, as speed increases. This is

This effect was shown experimentally by Froude3 by testing models with varying lengths of parallel middle body but the same forward and after ends. Figure 8.5 illustrates some of these early results. The residuary resistance was taken as the total measured resistance less a calculated skin friction resistance.

Now the distance between the two pressure systems is approximately 0.9L. The condition therefore that a crest or trough from the bow system should coincide with the first stern trough is:

The troughs will coincide when JVis an odd integer and for even values of N a crest from the bow coincides with the stern trough. The most pronounced hump occurs when N = 1 and this hump is termed the main hump. The hump at N = 3 is often called the prismatic hump as it is greatly affected by the ship's prismatic coefficient.

**Scaling wave-making resistance**It has been shown that for geometrically similar bodies moving at corresponding speeds, the wave pattern generated is similar and the wave-making resistance can be taken as proportional to the displacements of the bodies concerned. This assumes that wave-making was unaffected by the viscosity and this is the usual assumption made in studies of this sort. In fact there will be some viscosity but its major effects will be confined to the boundary layer. To a first order then, the effect of viscosity on wave-making resistance can be regarded as that of modifying the hull shape in conformity with the boundary layer

addition. These effects are relatively more pronounced at model scale than the full scale which means there is some scale effect on wave making resistance. For the purposes of this book this is ignored.

4.1 General. The wave-making resistance of a ship is the net fore-and-aft force upon the ship due to the fluid pressures acting normally on all parts of the hull, just as the frictional resistance is the result of the tangential fluid forces. In the case of a deeply submerged body, travelling horizontally at a steady speed far below the surface, no waves are formed, but the normal pressures will vary along the length. In a nonviscous fluid the net fore-and-aft force due to this variation would be zero, as previously noted.

If the body is travelling on or near the surface, however, this variation in pressure causes waves which alter the distribution of pressure over the hull, and the resultant net fore-and-aft force is the wave-making resistance. Over some parts of the hull the changes in pressure will increase the net sternward force, in others decrease it, but the overall effect must be a resistance

of such magnitude that the energy expended in moving the body against it is equal to the energy necessary to maintain the wave system. The wave making resistance depends in large measure on the shapes adopted for the area curve, waterlines and

transverse sections, and its determination and the methods by which it can be reduced are among the main goals of the study of ships' resistance. Two paths have been followed in this study—experiments with models in towing tanks and theoretical research into wave-making phenomena. Neither has yet led to a complete solution, but both have contributed greatly to a

better understanding of what is a very complicated problem. At present, model tests remain the most important tool available for reducing the resistance of specific ship designs, but theory lends invaluable help in interpreting model results and in guiding model research.

4.2 Ship Wave Systems.

The earliest account of the way in which ship waves are formed is believed to be that due to Lord Kelvin (1887, 1904). He considered a single pressure point travelling in a straight line over the surface of the water, sending out waves which

combine to form a characteristic pattern. This consists of a system of transverse waves following behind the point, together with a series of divergent waves radiating from the point, the whole pattern being contained within two straight lines starting from the pressure point and making angles of 19 deg 28 min on each side of the line of motion.The heights of successive transverse-wave crests along the middle line behind the pressure point diminish going aft. The waves are curved back some distance out from the centerline and meet the diverging waves in cusps, which are the highest points in the system. The heights of these cusps decrease less rapidly with distance from the point than do those of the transverse waves, so that eventually well astern of the point the divergent waves become the more prominent.

The Kelvin wave pattern illustrates and explains many of the features of the ship-wave system. Near the bow of a ship the most noticeable waves are a series of divergent waves, starting with a large wave at the bow, followed by others arranged on each side along a diagonal line in such a way that each wave is stepped back behind the one in front in echelon (Fig. 8) and is of quite short length along its crest line.

Between the divergent waves on each side of the ship, transverse waves are formed having their crest lines normal to the direction of motion near the hull, bending back as they approach the divergent-system waves and finally coalescing with them. These transverse waves are most easily seen along the middle portion of a ship or model with parallel body or just behind a ship running at high speed. It is easy to see the general Kelvin pattern in such a bow system.

Similar wave systems are formed at the shoulders, if any, and at the stern, with separate divergent and transverse patterns, but these are not always so clearly distinguishable because of the general disturbance already present from the bow system. Since the wave pattern as a whole moves with the ship, the transverse waves are moving in the same direction as the ship at the same speed V, and might Fig 7 (a) Pattern of diverging waves Fig. 7(b) Typical ship wave pattern be expected to have the length appropriate to free waves running on surface at that speed Actually, the waves in the immediate vicinity of a

model are found to be a little shorter, but they attain the length Lw about two wave lengths astern.

The divergent waves will have a different speed along the line normal to their crests (Fig. 9). In this case, the component of speed parallel to the line of the ship's motion must be equal to the ship's speed in order to retain the fixed pattern relative to the ship. If the line normal to the crest of a divergent wave makes an angle with the ship's course, the speed in that direction will be Vcos , and the corresponding wave length 4.3 Wave-Making Resistance of Surface Ships. At low speeds, the waves made by the ship are very small, and the resistance is almost wholly viscous in character.

Since the frictional resistance varies at a power of the speed a little less than the square, when the coefficient of total resistance CT = plotted to a base of Froude number Fn (or of at first the value of CT decreases with increase of speed (Fig.10). With further increase in speed, the value of CT begins to increase more and more rapidly, and at Froude numbers approaching 0.45

the resistance may vary at a power of V of 6 or more.

However, this general increase in CT is usually accompanied by a number of humps and hollows in the resistance

curve. As the speed of the ship increases, the wave pattern must change, for the length of the waves

will increase and the relative positions of their crests and troughs will alter. In this process there will be a

succession of speeds when the crests of the two sys- Fig 10 Typical resistance curve, showing interference effects

tems reinforce one another, separated by other speeds at which crests and troughs tend to cancel one another.

The former condition leads to higher wave heights, the latter to lower ones, and as the energy of the systems depends upon the square of the wave heights, this means alternating speeds of higher and lower than average resistance. The humps and hollows in the curve are due to these interference effects between the wave systems, and it is obviously good design

practice to ensure whenever possible that the ship will be running under service conditions at a favorable

speed. As will be seen later, it is the dependence of these humps and hollows on the Froude number that

accounts for the close relationship between economic speeds and ship lengths.

The mechanism by which wave-making resistance is generated is well illustrated by experiments made by Eggert (1939). He measured the normal pressure distribution over the ends of a model and plotted resulting pressure contours on a body plan (Fig. 11). By integrating the longitudinal components of these pressure forces over the length, he showed that the resulting

resistance agreed fairly well with that measured on the model after the estimated frictional resistance had been subtracted. Fig. 12a shows curves of longitudinal force per meter length; Fig. 12b shows form resistance derived from pressure experiments and residuary resistance model tests. One important point brought out by these experiments is that a large proportion of the wave-making resistance is generated by the upper part of the hull near the still waterline.

4.4 Theoretical Calculation of Wave-Making Resistance.

Much research has been devoted to theoretical methods of calculating wave-making resistance and to their experimental verification (Lunde, 1957).One method is to determine the flow around the hull and hence the normal pressure distribution, and then to integrate the fore-and-aft components of these pressures over the hull surface. This method was developed

by Michell (1898) for a slender hull moving over the surface of a nonviscous fluid. It corresponds to the experimental technique employed by Eggert. The pioneer work of Michell was unfortunately overlooked and neglected for many years until rescued from obscurity by Havelock (1951).

A second method is to calculate the wave pattern generated by the ship at a great distance astern, as done by Havelock, the wave-making resistance then being measured by the flow of energy necessary to maintain the wave system. This method has been used experimentally by Gadd, et al (1962), Eggers (1962), Ward (1962) and many others.

Both methods lead to the same final mathematical expression, and in each case the solution is for a nonviscous and incompressible fluid, so that the ship will experience only wave-making resistance (Timman, et al, 1955).

Michell obtained the mathematical expression for the flow around a "slender" ship of narrow beam when placed in a uniform stream. From the resultant velocity potential the velocity and pressure distribution over the hull can be obtained, and by integrating the fore and-aft components of the pressure an expression can be derived for the total wave-making resistance.11

The theory as developed by Michell is valid only for certain restrictive conditions:

(a) The fluid is assumed to be nonviscous and the flow irrotational. Under these circumstances the motion can be specified by a velocity potential ; which in addition must satisfy the necessary boundary conditions.

(b) The hull is narrow compared with its length, so that the slope of the surface relative to the centerline plane is small.

(c) The waves generated by the ship have heights small compared with their lengths, so that the squares of the particle velocities can be neglected compared with the ship speed.

(d) The ship does not experience any sinkage or trim. The boundary conditions to be satisfied by the velocity

potential are:

(a) At all points on the surface of the hull, the normal velocity relative to the hull must be zero.

(b) The pressure everywhere on the free surface of the water must be constant and equal to the atmospheric pressure.

To make the problem amenable to existing mathematical methods, Michell assumed that the first boundary condition could be applied to the centerline plane rather than to the actual hull surface, so that the results applied strictly to a vanishingly thin ship, and that the condition of constant pressure could be applied to the original flat, free surface of the water, the distortion

of the surface due to the wave pattern being neglected.

The alternative method developed by Havelock, in which the wave-making resistance is measured by the energy in the wave system, makes use of the idea of sources and sinks.12 This is a powerful tool with which to simulate the flow around different body shapes and so to find the wave pattern, pressure distribution, and resistance. A "thin" ship, for example, can be simulated by a distribution of sources on the centerline plane of the forebody and of sinks in the afterbody,

the sum of their total strength being zero. The re- The velocity potential has the property that the velocity

of the flow in any given direction is the partial derivative of with respect to that direction.

Residuary resistance derived from towing experiments to be everywhere proportional to the slope of the hull surface, this will result in a total strength of zero, and the total velocity potential will be the sum of those due to the individual sources and sinks and the uniform flow.

Each source and sink when in motion in a fluid, on or near the surface, gives rise to a wave system, and by summing these up the total system for the ship can be obtained. Havelock by this method found the wave pattern far astern, and from considerations of energy obtained the wave-making resistance.

Much of the research into wave-making resistance has been done on models of mathematical form, having sections and waterlines defined by sine, cosine, or parabolic functions. When the calculations are applied to actual ship forms, the shape of the latter must be expressed approximately by the use of polynomials (Weinblum, 1950, Wehausen, 1973); or by considering

the hull as being made up of a number of elementary wedges (Guilloton, 1951).

In recent years, a great deal of work on the calculation of wave-making resistance has been carried out

in Japan by Professor Inui and his colleagues (Inui, 1980). They used a combination of mathematical and

experimental work and stressed the importance of observing the wave pattern in detail as well as simply

measuring the resistance. Instead of starting with a given hull geometry, Professor Inui began with an

assumed source-sink distribution, with a view to obtaining better agreement between the measured and

calculated wave systems, both of which would refer to the same hull shape. The wave pattern and the wavemaking

resistance were then calculated from the amplitudes of the elementary waves by using Havelock's

concept.

Professor Inui tried various distributions of sources and sinks (singularities) by volume over the curved

surface, in a horizontal plane and over the vertical middle-line plane. For displacement ships at Froude

numbers from 0.1 to 0.35, he found the geometry of the ends to be most important, and these could be

represented quite accurately by singularities on the middle-line plane. For higher Froude numbers, the disstriction

to a "thin" ship can be removed if the sources and sinks are distributed over the hull surface itself.

If the strengths of the sources and sinks are assumed 12 A source may be looked upon as a point in a fluid at which new

fluid is being continuously introduced, and a sink is the reverse, a point where fluid is being continuously abstracted. The flow out of a source or into a sink will consist of radial straight stream lines,

Fig. 13. If a source and an equal sink be imagined in a uniform stream flow, the axis of the source and sink being parallel to the

flow, the streamlines can be combined as shown in Fig. 14, and there will be one completely closed streamline ABCD. Since the source and sink are of equal strength, all fluid entering at +s will be removed at — s, and no fluid will flow across ABCD, and the space inside this line could be replaced by a solid body.

Fig 13 Flow patterns for source and sink

Fig 14 Flow patterns for a source and sink in a uniform stream tribution of sources along the whole length becomes

important. In summary, the method is to choose a singularity distribution which will give good resistance qualities, obtain the corresponding hull geometry, carry out resistance and wave-observation tests and modify the hull to give a more ship-shape form amidships.

In this way Inui has been able to obtain forms with considerably reduced wave-making resistance, usually associated with a bulb at the stem and sometimes at the stern also.

Recent developments in wave-making resistance theory can be divided into four main categories. The first concerns applications of linearized potential flow theory, either with empirical corrections to make it more accurate, or uncorrected for special cases where the errors due to linearization are not serious. The second concerns attempts to improve on linearized potential flow theory, by analysis of non-linear effects on the free-surface condition, or by an assessment of the effects

of viscosity. Thirdly, attempts have been made to apply wave resistance theory to hull form design.

Fourthly there has been an increase in the number of primarily numerical approaches to ship wave resistance estimation. In the second category (non-linear calculations) the work of Daube (1980), (1981) must be mentioned. He uses an iterative procedure where at each step a linear problem is solved. To this end an initial guess of the location of the free surface is made which is subsequently changed to fulfill a free surface condition. In the computation of the free surface elevation the assumption is made that the projection of the free surface streamlines on the horizontal plane, always agree with the double model streamlines.

This is in fact a low-speed assumption. The non-linear calculation method has been applied to a Wigley hull and a Series 60 ship. Comparison with measurements show a qualitatively satisfactory agreement, and quantitatively the calculations are better than with linear theory for these cases. Part of the discrepancies between measurements and calculations (at least for the higher speed range) can be ascribed to trim and sinkage effects which have not been properly included in Daube's method.

An interesting development has been the determination of pure wave-making resistance from measurements of model wave patterns. The attempts to improve hull forms using the data of wave pattern measurement combined with linearized theory are particularly interesting. For example, Baba (1972) measured the difference in wave pattern when a given hull was modified according to the insight gained from wave resistance theory and thereby gained an improvement.

To a certain degree, the hull forms of relatively high speed merchant ships have improved because of the application of wave resistance theory. Pien et al (1972) proposed a hull-form design procedure for high-speed displacement ships aided by wave resistance theory.

Inui and co-workers have applied the streamline tracing method to practical hull forms with flat bottoms and a design method for high-speed ships with the aid of minimum wave resistance theory has been proposed by Maruo et al (1977). The development of special types of hull forms for drastically reduced wave making have also been guided to a certain extent by wave resistance theory. One of these is the small waterplane area twin-hull (SWATH) ship, discussed in Section 9. The accuracy and usefulness of wave resistance theory was recently demonstrated at a workshop organized by the DTRC (Bai, et al, 1979).

The results of theoretical work would therefore seem at present to be most useful in giving guidance in the choice of the secondary features of hull shape for given proportions and fullness, such as the detail

Fig 15 Wave systems for a simple wedge-shaped form shapes of waterlines and sections, and choice of size

and location of bulbs.

The calculation of resistance cannot yet be done with sufficient accuracy to replace model experiments, but it is a most valuable guide and interpreter of model work. The advent of the computer has placed new power in the hands of the naval architect, however, and has brought much closer the time when theory can overcome its present limitations and begin to give meaningful numerical answers to the resistance problem.

4.5 Interference Effects. The results of mathematical research have been most valuable in providing an insight into the effects of mutual wave interference upon wave-making resistance. A most interesting example is that of a double-wedge-shaped body with parallel inserted in the middle, investigated by Wigley (1931). The form of the hull and the calculated and

measured wave profiles are shown in Fig. 15. He showed that the expression for the wave profile along the hull contained five terms:

(a) A symmetrical disturbance of the surface, which has a peak at bow and stern and a trough along the center, dying out quickly ahead and astern of the hull. It travels with the hull and because of its symmetry does not absorb any energy at constant speed, and four wave systems, generated at

Fig 16 Resistance curves for wedge-shaped model

Dimensions: 4.8 X 0.46 X 0.30 m, prismatic coefficient 0.636.

(b) the bow, beginning with a crest;

Fig 17 Analysis of wave-making resistance into components for wedge-shaped model shown above

(c) the forward shoulder, starting with a trough;

(d) the after shoulder also starting with a trough;

(e) the stern, beginning with a crest.

These five systems are shown in Fig. 15. Considerably aft of the form, all four systems become sine curves of continuously diminishing amplitude, of a length appropriate to a free wave travelling at the speed of the model, this length being reached after about two waves.

The calculated profile along the model is the sum of these five systems, and the measured profile was in general agreement with it so far as shape and positions of the crests and troughs were concerned, but the heights of the actual waves towards the stern were considerably less than those calculated (Fig. 15).

This simple wedge-shaped body illustrates clearly the mechanism of wave interference and its effects upon wave-making resistance. Because of the definite sharp corners at bow, stern, and shoulders, the four free-wave systems have their origins fixed at points along the hull. As speed increases, the wave lengths of each of the four systems increase. Since the primary

crests and troughs are fixed in position, the total wave profile will continuously change in shape with speed

as the crests and troughs of the different systems pass through one another. At those speeds where the interference

is such that high waves result, the wave making resistance will be high, and vice-versa.

In this simple wedge-shaped form the two principal types of interference are between two systems of the same sign, e.g., bow and stern, or the shoulder systems, and between systems of opposite sign, e.g., bow and forward shoulder. The second type is the most important in this particular case, because the primary hollow of the first shoulder system can coincide with the first trough of the bow system before the latter has been materially reduced by viscous effects.

Wigley calculated the values of for minima and maxima of the wave-making resistance coefficient Cw for this form, and found them to occur at the following points:

The mathematical expression for the wave-making resistance Rw is of the form (constant term + 4 oscillating terms) so

that the wave-making resistance coefficient Cw is (constant term + 4 oscillating terms)

The curve is thus made up of a steady increase varying as due to the constant term and four oscillating curves due to the interference between the different free-wave systems (Fig. 17). These latter ultimately, at very high speeds, cancel both each other and the steady increase in and there is no further hump beyond that occurring at a value of about 0.45 after which the value of continuously decreases with further increase in speed. However, at these high speeds the hull will sink bodily and change trim so much that entirely new phenomena arise. For more ship-shaped forms, where the waterlines are curved and have no sharp discontinuities, the wave pattern still consists of five components—a symmetrical disturbance and four free-wave systems (Wigley,1934). Two systems begin with crests, one at the bow and one at the stern, and are due to the change in the

angle of the flow at these points. The other two systems, like the shoulder systems in the straight line form, begin with hollows, but are no longer tied to definite points, since the change of slope is now gradual and spread over the whole entrance and run. They commence at the bow and after shoulder, respectively, as shown in Fig. 18, much more gradually than in the

case of the wedge-shaped form. The one due to entrance curvature, for example, may be looked upon as

a progressive reduction of that due to the bow angle as the slope of the waterline gradually becomes less

in going aft.

Wigley also made calculations to show the separate contributions to the wave-making resistance of the transverse and divergent systems (Wigley, 1942). Up to a Froude number of 0.4 the transverse waves are mainly responsible for the positions of the humps and hollows, Fig. 19. Above this speed the contribution from the divergent waves becomes more and more

important, and the interference of the transverse waves alone will not correctly determine the position of the higher humps, particularly the last one at Fn = 0.5.

The existence of interference effects of this kind was known to naval architects long before such mathematical analysis was developed. The Froudes demonstrated them in a striking way by testing a number of models consisting of the same bow and stern separated by different lengths of parallel body (Froude, W., 1877 and Froude, R.E., 1881). W. Froude's sketch of the bow wave system is shown in Fig. 20. As the ship advances but the water does not, much of the energy given to the water by the bow is carried out laterally and away from the ship. This outward spreading of the energy results in a decrease in the height

of each succeeding wave of each system with no appreciable change in wave length. Fig. 21 shows a series of tests made at the EMB, Washington, and the corresponding curve of model residuary resistance plotted against length of parallel body (Taylor, 1943). The tests were not extended to such a length of parallel body that the bow system ceased to affect that at the stern.

It is clear, however, that its effect is decreasing and would eventually died out, as suggested by the dotted extension of the resistance curve.

Fig. 22 shows a series of curves for the same form at various speeds. In this chart the change of parallel middle-body length which results in successive humps on any one curve is very nearly equal to the wave length for the speed in question, as shown for speeds of 2.6 and 3.2 knots. This indicates that ship waves do have substantially the lengths of deep-sea waves of

the same speed.

If all the curves in Fig. 22 are extended in the direction of greater parallel-body length until the bow system ceases to affect the stern system, as was done in Fig. 21, the mean residuary resistances for this form, shown by the dashed lines at the left of the

chart, are found to increase approximately as the sixth power of the speed. They are, in fact, the actual resistances

stripped of interference effects and represent the true residuary resistances of the two ends. This rate of variation with speed is the same as that given by theory for the basic wave-making resistance before taking into account the interference effects (Fig. 17).

The mathematical theory indicates that the wave resistance is generated largely by those parts of the

hull near the surface, which is in agreement with the experimental results obtained by Eggert. This suggests

that from the point of view of reducing wavemaking resistance the displacement should be kept as

low down as possible. The relatively small effect of the lower part of the hull on the wave systems also means

that the wave-making resistance is not unduly sensitive to the midship section shape (Wigley, et al, 1948).

4.6 Effects of Viscosity on Wave-Making Resistance.

Calculations of wave-making resistance have so far been unable to take into account the effects of viscosity, the role of which has been investigated by Havelock (1923), (1935) and Wigley (1938). One of these effects is to create a boundary layer close to the hull, which separates the latter from the potential-flow pattern with which the theory deals. This layer grows thicker from stem to stern, but outside of it the fluid behaves very much in accordance with the potential flow theory. Havelock (1926) stated that the direct influence of viscosity on the wave motion is comparatively small, and the "indirect effect might possibly be allowed for later by some adjustment of the effective form of the ship." He proposed to do this by assuming that the after body was virtually lengthened and the , aft end waterlines thereby reduced in slope, so reducing the after-body wavemaking. Wigley (1962) followed up this suggestion by comparing calculated and measured wave-making resistance for 14 models of

mathematical forms, and deriving empirical correction factors. He found that the remaining differences in resistance were usually within 4 percent, and that the virtual lengthening of the hull due to viscosity varied between 2 and 8 percent.

The inclusion of a viscosity correction of this nature also explains another feature of calculated wave-making resistance. For a ship model which is unsymmetrical fore and aft, the theoretical wave-making resistance in a nonviscous fluid is the same for both directions of motion, while the measured resistances are different. With the viscosity correction included, the calculated resistance will also be different. Professor Inui (1980) in his wave-making resistance work also allows for viscosity by means of two empirical coefficients, one to take care of the virtual lengthening of the form, the other to allow for the effect of viscosity on wave height.

4.7 Scale Effect on Wave-Making Resistance.

Wigley (1962) has investigated the scale effect on Cw due to viscosity, pointing out that the calculated curves of are usually higher than those measured in experiments and also show greater oscillations. These differences he assigned to three major causes:

(a) Errors due to simplifications introduced to make the mathematical work possible.

(b)Errors due to neglect of the effects of viscosity on

(c) Errors due to the effects of wave motion on Errors under (a) will decrease with increasing speed, ! since they depend on the assumption that the velocities de to the wave motion are small compared with the speed of the model, which is more nearly fulfilled at high speed.

Errors under (b) will depend on Reynolds number, and therefore on the size of the model, decreasing as size increases. From experiments on unsymmetrical models tested moving in both directions, these errors cease to be important for Fn greater than 0.45.

At low speeds errors under (c) are negligible, but become important when Fn exceeds 0.35, 1.15) as evidenced by the sinkage

and trim, which increase very rapidly above this speed. A practical conclusion from this work is the effect on the prediction of ship resistance from a model. In a typical model the actual wave resistance is less than that calculated in a perfect fluid for Froude numbers less than about 0.35. This difference is partly due to viscosity, the effect of which will decrease with increased size, and will increase with scale instead of being constant as assumed in extrapolation work. Wigley made estimates of the

difference involved in calculating the resistance of a 121.9 m ship from that of a 4.88 m model at a Froude number of 0.245 and found that the resistance of the ship would be underestimated, using the usual calculations, by about 9 percent, the variation with speed being approximately as shown in Fig. 23. The effect disappears at low speeds and for values of Fn

above 0.45.

4.8 Comparison Between Calculated and Observed Wave-Making Resistance. Many comparisons have been made between the calculated and measured wave-making resistances of models. Such a comparison is difficult to make, however. All that can be measured on the model is the total resistance and the value of can only be obtained by making assumptions as to the amount of frictional resistance, viscous pressure drag and eddy-making resistance, all quantities subject to considerable doubt. The wave making resistance has been measured directly by observing the shape of the wave system astern of the

model and computing its energy and the total viscous drag has been measured by a pitot tube survey behind the model (Wehausen, 1973). Both of these methods are relatively new, and there are problems in interpreting the results. In the meantime it is perhaps best to be content with comparing the differences between the calculated and measured resistances for pairs of models of the same overall proportions and coefficients but differing in those features which are likely to affect wave-making resistance.

A comparison of much of the available data has been made by Lunde, the measured being derived from on Froude's assumption and using his skin friction coefficients, the calculated being empirically corrected for viscosity (Lunde, 1957).

At low Froude numbers, less than 0.18, it is difficult to determine CR with any accuracy. At higher speeds, the humps at Fn of 0.25 and 0.32 and the intervening hollow are much exaggerated in the calculated curves, and any advantage expected from designing a ship to run at the "hollow" speed would not be fully achieved in practice (Fig. 24). The general agreement in level

of the curves over this range depends to some extent upon the form of the model, theory overestimating the resistance for full ships with large angles of entrance.

Just above a speed of Fn = 0.32 the model becomes subject to increasing sinkage and stern trim, effects

which are not taken into account in the calculations.

The last hump in the curve occurs at a Froude number of about 0.5, and here the calculated value of is less than the measured again possibly due to the neglect of sinkage and trim. In all cases the humps and hollows on the measured curves occur at higher values of Fn than those given by theory, by amounts varying from 2 to 8 percent. In other words, the model behaves as though it were longer than its actual length, and this is undoubtedly due mostly to the virtual lenghening of the form due

to the viscous boundary layer. At very low speeds, Fn = 0.1, the wave-making resistance varies approximately

as the square of the tangent of the half-angle of entrance, but its total value in terms of RT is very small. At high speeds, with Fn greater than 1.0, the wave-making resistance varies approximately as the square of the displacement, illustrating the well-known fact that at very high speeds shape is relatively unimportant, the chief consideration being the displacement

carried on a given length.