1. Overview — why wave making is treated as inviscid (§5.1)
Chapter 4 listed every resistance component; this chapter studies the physics behind one of the two main parts: the wave making of the ship and the wave resistance it produces. The companion viscous part is left for Chapter 6. The two are split because each maps onto a separate physical phenomenon — even though, physically, they happen at the same time and interact.
1.1 Three reasons to consider wave making inviscid
The whole chapter rests on one approximation: the wave making of a ship is treated as if viscosity did not exist. Three accepted facts justify it.
| Argument | What it says |
|---|---|
| Decomposition used | Total resistance is split into wave + viscous — the split most directly tied to distinct physical phenomena. |
| How model tests run | It is impossible to match both Reynolds number $Rn$ and Froude number $Fn$ for model and ship (§3.3.2). The proven fix is to test the model at the ship's Froude number: the wave pattern is then geometrically near-similar and the wave resistance scales up easily. A factor-of-100 difference in $Rn$ barely changes the wave making — so wave making is rather insensitive to viscosity. |
| Boundary-layer theory | At high $Rn$, viscous effects are confined to a thin boundary layer and a narrow wake (§6.2). Inside that layer the pressure equals the pressure just outside it, so to a first approximation the pressure field around the body — and hence its wave making — is independent of viscosity. |
So in the following the ship's wave making and wave pattern are treated as an inviscid phenomenon. The small real-world departures from this are collected later in §5.7. Up to §5.7 the water depth is assumed unlimited; §5.8–5.10 then add limited depth, §5.11 adds wash, and §5.12 adds channel width.
Fonte: LARSSON, Lars; RAVEN, Hoyte C. Ship Resistance and Flow. The Principles of Naval Architecture Series. J. R. Paulling (ed.). Jersey City: SNAME, 2010. Chapter 5 — Inviscid Flow Around the Hull, Wave Making, and Wave Resistance.
Edital: Anexo 2-A, Área I, item 5 (escoamento inviscido em torno do casco, geração de ondas e resistência de ondas; padrão de ondas / cunha de Kelvin; humps e hollows; wave breaking; efeitos de águas rasas e canal) e item 1.2/1.4 (resistência às ondas e wave-breaking) · Anexo 2-B, Área I, item 3 (Larsson & Raven), Chapter 5.
2. Inviscid flow around a body, no free surface (§5.2)
As a warm-up to wave resistance, consider inviscid flow around a body deeply submerged — no free water surface yet. Use a coordinate system fixed to the ship (Fig. 2.1): an incoming flow in the $+x$ direction with velocity $U_\infty$, equal and opposite to the ship speed, undisturbed far ahead.
2.1 Euler equations, Bernoulli, and the velocity potential (§5.2.1)
For inviscid flow, every viscous term in the Navier–Stokes equations is dropped, leaving the balance of convective and pressure-gradient terms — the Euler equations. Taking the inner product with the velocity vector and assuming steady flow yields a quantity that stays constant along each streamline:
$$ \tfrac{1}{2}(u^2+v^2+w^2) + \frac{p}{\rho} + gz = \text{constant along a streamline} \tag{5.3} $$
Two further simplifications convert the flow into a potential flow. First, the flow is irrotational — its vorticity (the curl of velocity) is zero everywhere:
$$ \vec{\omega} = \nabla \times \vec{v} = 0 \tag{5.4} $$
This is acceptable because the inflow is uniform far upstream (hence irrotational), and in an incompressible inviscid flow no vorticity is generated at the walls (there is no wall friction). By Kelvin's theorem vorticity is merely convected with the flow, so an irrotational inflow stays irrotational everywhere. Second, for any irrotational flow the velocity can be written as the gradient of a scalar, the velocity potential $\phi$:
$$ \vec{v}(x,y,z) = \nabla\phi \tag{5.5} $$
Because the curl of a gradient always vanishes, (5.5) guarantees irrotationality automatically. Substituting it leads to two governing equations. From Bernoulli (for steady potential flow the constant is now the same for the whole field, not just along a streamline) the pressure follows directly:
$$ p = -\rho g z + \tfrac{1}{2}\rho\left(U_\infty^2 - \nabla\phi\cdot\nabla\phi\right) \tag{5.10} $$
The continuity equation, with $\vec v=\nabla\phi$, becomes the Laplace equation for the potential:
$$ \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2} = 0 \qquad\text{or}\qquad \nabla^2\phi = 0 \tag{5.11} $$
2.2 Flow around a 2-D body: curvature sets the pressure (§5.2.2)
Take a 2-D body in a parallel inviscid flow (Fig. 5.1) — think of the waterplane of an infinite-draft ship, still ignoring the surface. Approaching the body, the straight streamlines bend sideways to pass it; for a bluff body there are four regions of strong curvature: the front end, the fore shoulder, the aft shoulder, and the aft end.
A simple force balance links a streamline's curvature to the pressure gradient normal to it. For a fluid element following a path of radius $r$ at local speed $u$, a centripetal force is needed, and in inviscid flow only a pressure gradient can supply it:
$$ \frac{\partial p}{\partial r} = \rho\,u^2/r \tag{from §5.2.2} $$
By Bernoulli, pressure and velocity are linked: high pressure means low velocity (around the bow and stern stagnation regions), low pressure means high velocity (at the shoulders, where the local speed exceeds the ship speed). The streamline running exactly along the symmetry line cannot curve to port or starboard; it ends right at the bow at a stagnation point where the velocity drops to zero and the hydrodynamic pressure reaches its maximum:
$$ p_{\max} = \tfrac{1}{2}\rho U_\infty^2 \tag{stagnation pressure} $$
This stagnation pressure is the highest the hydrodynamic pressure can reach in a steady flow. It is convenient to non-dimensionalise pressure as a pressure coefficient:
$$ C_p = \frac{p + \rho g z}{\tfrac{1}{2}\rho U_\infty^2} = 1 - \frac{\nabla\phi\cdot\nabla\phi}{U_\infty^2} \tag{5.12} $$
At a stagnation point $C_p = 1$ (its maximum); at the shoulders $C_p$ is negative.
2.3 Flow around a 3-D hull and d'Alembert's paradox (§5.2.3)
For a 3-D body such as a ship (surface still assumed flat), the curvature rule still holds, but now two kinds of curvature matter: curvature in planes normal to the surface (a pressure deviation across the flow, as in 2-D) and curvature in planes parallel to the surface (pressure variation along the girth). Fig. 5.5 shows the inviscid streamlines and isobars on the KVLCC2 tanker, a standard test case.
- At least one stagnation point at the bow (e.g. at the bulb tip), where a streamline impinges and the velocity drops to zero.
- High pressure on the forebody (streamlines bend outward); low pressure at the fore shoulder (sharpest, so lowest), and low again at the aft shoulder; high pressure back at the stern.
- On the bilge the normal curvature is large, giving a strong outward pressure gradient and a low hull pressure ($C_p \approx -0.3$).
- On the parallel middle body there is no surface curvature, so pressure gradients are small; a slight outward curvature of the streamlines just off the hull still gives a slightly negative pressure.
- The flow ends in one or more stagnation points at the stern.
This pressure distribution exerts a force on the hull. But because we have a closed body in an infinite inviscid fluid and are ignoring the free surface, d'Alembert's paradox applies: the longitudinal pressure force integrates exactly to zero (Prandtl & Tietjens, 1957). No resistance arises without the free surface — yet the pressure distribution itself stays useful (§11). (Larsson & Raven, §5.2.3)
3. Free-surface waves and their properties (§5.3)
The inviscid flow without a free surface gives a pressure field but no wave pattern and no wave resistance. The next step adds the free surface explicitly. A ship wave pattern (Fig. 5.6) has a clear, regular structure that points to a mathematical basis; understanding it allows hull design for minimum wave resistance (§11.5).
Far from the ship the pattern can be seen as a superposition of sinusoidal wave components, generated by different parts of the hull and travelling in various directions. Superposition is allowed only if the governing equations and the surface boundary conditions are (nearly) linear — which requires small wave amplitude.
3.1 Deriving a sinusoidal deep-water wave (§5.3.1)
For a wave in still water of unlimited depth, use an earth-fixed frame (the wave moves, so the flow is unsteady). Dropping surface tension ($p=p_a$) and using the unsteady Bernoulli equation, the surface elevation $\zeta$ follows from the potential. The full dynamic and kinematic surface conditions are nonlinear and would forbid superposition; but for small wave steepness (wave height ≪ wave length) they linearise to a single, linear, homogeneous condition for the potential:
$$ \frac{\partial^2\phi}{\partial t^2} + g\,\phi_z = 0 \tag{5.16} $$
A sinusoidal trial potential that also satisfies the Laplace equation, and decays with depth (so the disturbance vanishes far below the surface), takes the form
$$ \phi(x,z,t) = C\,e^{kz}\,\sin(kx-\omega t) \tag{5.19} $$
where the wave number $k = 2\pi/\lambda$ and the radial frequency $\omega = 2\pi/T$. Forcing it to satisfy the Kelvin condition gives the deep-water dispersion relation:
$$ \omega^2 = g\,k \tag{5.20} $$
Since one wavelength passes in one period, $c=\lambda/T=\omega/k$, so
$$ k = \frac{g}{c^2} \qquad\text{or}\qquad \lambda = \frac{2\pi c^2}{g} \tag{5.21} $$
The energy stored in a 2-D wave splits evenly between kinetic and potential parts; per unit area the total comes to
$$ E_{\text{wave}} = \tfrac{1}{2}\rho g A^2\,\lambda \quad\text{(per wavelength)} \tag{5.23} $$
and the time-averaged energy flux through a fixed vertical plane is
$$ |\dot{E}| = \tfrac{1}{4}\rho g A^2\,c \tag{5.24} $$
where $A$ is the wave amplitude.
3.2 The five key properties of sinusoidal waves (§5.3.2)
| Property | What it means |
|---|---|
| Linearity | For small amplitude the surface conditions linearise; sinusoidal waves satisfying Laplace + Kelvin can be superposed to build new solutions. |
| Dispersion | Wave speed depends on wave length (not amplitude): $c=\sqrt{g\lambda/2\pi}$. Unlike sound or light, water waves are dispersive — a general disturbance disintegrates into components moving at different speeds. |
| Vertical decay | The disturbance decays exponentially with depth, $\propto e^{2\pi z/\lambda}$. Half a wavelength down it is only 4% of the surface value — short waves are confined to a thin surface layer. |
| Orbital motion | The $x$ and $z$ velocity components oscillate with equal amplitude and a 90° phase difference, so fluid particles describe circles of radius $A\,e^{gz/c^2}$ (Fig. 5.8). The mean fluid velocity is zero in linear theory — only the wave form travels at $c$. |
| Group velocity | Comparing energy (5.26) and energy flux (5.24): the energy travels at half the crest speed. |
$$ c = \sqrt{\dfrac{g\lambda}{2\pi}} \qquad\text{(dispersion relation)} \tag{5.25} $$
$$ c_g = \frac{|\dot E|}{E} = \tfrac{1}{2}c \qquad\text{(deep-water group velocity)} \tag{5.27} $$
4. Ship waves and the Kelvin pattern (§5.4)
4.1 The 2-D case: one wave, only astern (§5.4.1)
Take a 2-D object (e.g. a submerged cylinder across the flow) moving at constant speed $V$ through still water (Fig. 5.9). After a steady state is reached, the pattern is steady in the body frame: every wave follows the body, so its phase speed equals $V$. The dispersion relation then fixes the wave length to the fundamental value:
$$ \lambda = 2\pi V^2/g = 2\pi\,Fn^2\,L \tag{2-D wave length} $$
Far aft there is a single wave component (length known, amplitude and phase not). Crucially, waves occur only downstream — a consequence of the group velocity being less than the phase velocity: the trailing wave energy, moving at $c_g=\tfrac12 V$, is lost through the downstream boundary and must be supplied continuously by the body. That energy supply, per unit span, is the wave resistance:
$$ R_W = (E\!\cdot\!V - \dot E)/V = \tfrac{1}{4}\rho g A^2 \tag{5.28} $$
4.2 The 3-D case: transverse and diverging waves (§5.4.2)
In 3-D, a single disturbance (e.g. the bow) generates a continuous set of wave components propagating in directions $\theta$ from $-\pi/2$ to $+\pi/2$ (Fig. 5.10). Each must be steady in the ship frame, so its phase speed is $c = V\cos\theta$, and its length follows from the dispersion relation:
$$ \lambda(\theta) = 2\pi\,c^2/g = 2\pi\,Fn^2\,L\cos^2\theta \tag{5.29} $$
As long as amplitudes are small, the far-field pattern is just the sum of all these components, written as an integral over $\theta$ with amplitude functions $A(\theta)$, $B(\theta)$ — the free-wave spectrum (eq. 5.30).
4.3 The Kelvin wedge — the 19°28′ envelope (§5.4.3)
A naïve construction (continuous crests for every $\theta$) would fill the whole area behind the first transverse crest. But group velocity changes this drastically: energy supplied to a crest at the bow lags behind it (just as in the pond), so each crest dies out and a new one emerges behind it. The result is short crests in a fan-shaped pattern, not infinite crest lines.
Following where the energy of component $\theta$ actually travels (it moves at $c_g=\tfrac12 c$ relative to the moving bow) defines a ray. The ray direction is
$$ \frac{y_E}{x_E} = \frac{\sin\theta\cos\theta}{2-\cos^2\theta} \tag{ray direction} $$
This has a maximum at $\sin\theta = \sqrt{1/3}$, i.e. $\theta = 35°$, giving a ray direction of $\tan^{-1}(y_E/x_E) = 19°28'$.
All wave energy generated by the ship is contained in a sector originating at the bow with a half-top angle of 19°28′. This sector is the Kelvin wedge (Lord Kelvin, ~1900). Inside it a system of transverse waves follows the ship, with diverging waves on the sides; transverse waves curve back and meet the diverging waves in cusps at the wedge edge. (Larsson & Raven, §5.4.3)
Because the rays diverge linearly with distance, the wave-energy density falls as $1/(\text{distance})$, and since energy is quadratic in amplitude, the amplitude inside the wedge decays as $(\text{distance})^{-1/2}$. But the waves right at the wedge edge ($\theta\approx 35°$) decay only as $(\text{distance})^{-1/3}$ (Lighthill, 1980) — so far from the ship they decay more slowly and eventually dominate, which is why the outer edge is the most conspicuous part of the pattern.
4.4 How a real ship pattern differs from Kelvin's (§5.4.4)
- The 19°28′ envelope still holds at sufficient distance, where all waves behave linearly.
- A ship is not one isolated pressure point: bow, stern, and shoulders each generate their own wave system in overlapping, interfering Kelvin wedges.
- The wave-making features have finite size and spacing, so some components are generated more strongly than others — the ship wave spectrum differs from Kelvin's. Crests are often short and straight rather than the long concave crests of the ideal pattern (Fig. 5.15).
- A near-field disturbance exists in addition to the far-field waves (the sinusoidal waves alone cannot satisfy the hull boundary condition); it deflects, accelerates and decelerates the flow, much like §5.2.3. Waves first propagate over this curved near-field flow, so their final direction/length/amplitude are seen only once clear of the near field — which is why the bow-wave cusp line appears displaced outward/forward.
4.5 Interference: the five wave systems (§5.4.5)
The pressure distribution (§5.2.3) gives high pressure at bow and stern (waves starting with a crest) and low pressure at the two shoulders (waves starting with a trough). Wigley (1931) computed the waves of a double-wedge hull with a parallel midbody and found the centreline profile to be the sum of five contributions:
| # | System | Starts with |
|---|---|---|
| 1 | Near-field (local) disturbance, the "Bernoulli wave" — dies out fore and aft, radiates no energy | peaks at ends, troughs at shoulders |
| 2 | Bow wave system | crest |
| 3 | Fore-shoulder wave | trough |
| 4 | Aft-shoulder wave | trough |
| 5 | Stern wave system | crest (for this hull) |
The primary crests/troughs are fixed in position, but as speed rises each wave length grows, so the second crest/trough shifts downstream and the systems pass through one another. When crests coincide, high waves result; when a crest meets a trough, they cancel. For this hull the interference is set uniquely by the ratio $\lambda_0/L = 2\pi Fn^2$ (e.g. bow and stern waves reinforce when this ratio is one) — strictly only for the transverse waves, but those dominate. For ship-shaped hulls the same five-component picture usually holds, with shoulder systems no longer tied to fixed points; for a given hull the interference is fixed by the Froude number. The Froudes demonstrated this experimentally by testing the same bow and stern separated by different lengths of parallel body.
4.6 The ship wave spectrum (§5.4.6)
Far aft, the pattern is a superposition of sinusoidal components, each obeying the dispersion relation and the steadiness condition $c=V\cos\theta$. So for each direction $\theta$ the length and speed are known; only the amplitude and phase remain. The functions $A(\theta)$, $B(\theta)$ are the free-wave spectrum — a compact one-dimensional description of the far-field pattern (Fig. 5.17). For reference, the Kelvin point-source spectrum, $\propto 1/\cos^2\theta$, has a large diverging-wave contribution. The spectrum is measured by wave pattern analysis — solving $A(\theta)$, $B(\theta)$ from longitudinal or transverse cuts through the measured wave field (§8.4.6).
5. Wave resistance, humps and hollows (§5.5)
If the body travels on or near the surface, a wave pattern alters the pressure distribution, and the net longitudinal pressure force no longer cancels. That force is the wave-making resistance:
$$ R_W = -\iint_S p\,n_x\,dS \tag{5.31} $$
Equivalently, the energy that the radiated wave system carries away must be supplied by the ship. From the far-field spectrum, Havelock (1934) obtained
$$ R_W = \tfrac{1}{2}\pi\rho V^2 \int_{-\pi/2}^{+\pi/2}\!\left(A(\theta)^2+B(\theta)^2\right)\cos^3\theta\,d\theta \tag{5.32} $$
Because resistance is quadratic in amplitude, the resistance contributions of components do not superpose — so interference is decisive. Constructive interference (systems reinforcing) gives high wave resistance; destructive interference (cancelling) gives low. As speed varies, the interference oscillates with $Fn$, producing peaks called humps and valleys called hollows in the resistance curve. This is the root of the link between economical speed and ship length — a designer aims to run the ship at a favourable (hollow) speed.
5.1 Locating humps: the wedge hull and the z-method
For the wedge hull the wave-resistance coefficient takes the form
$$ C_W \equiv \frac{R_W}{\tfrac{1}{2}\rho V^2 S} = V^4 \times(\text{constant term}+\text{four oscillating terms}) \tag{5.33} $$
the oscillating terms coming from interference (Fig. 5.18). Above $Fn\approx 0.48$ most systems cancel and $C_W$ falls. Wigley (1942) showed that up to $Fn\approx 0.4$ the transverse waves set the hump positions; above that the diverging waves matter more (Fig. 5.19). For an arbitrary ship the empirical z-method uses just two systems — the bow (crest) and aft-shoulder (trough) — separated by the wave-making length $z$. A maximum occurs when $z=(2k+1)\lambda/2$ and a minimum when $z=k\lambda$. Baker & Kent (1919) proposed $z = C_P\,L + \tfrac14\lambda$ (for $k=1,2,3$), giving humps at
$$ Fn = \sqrt{\frac{C_P}{2\pi(k+1/4)}} \tag{hump Froude numbers} $$
For Wigley's wedge ($C_P = 0.637$) this predicts humps at $Fn = 0.177,\ 0.212,\ 0.285$, agreeing fairly well with the computed $0.173,\ 0.205,\ 0.269,\ 0.476$. The method still cannot be precise because hull shape (where the shoulder/stern systems sit, which components prevail) varies a lot even for a given $C_P$ — which is exactly why model testing at corresponding speeds (equal $Fn$) is used.
5.2 A tour of the resistance curve from low to planing speed
Fig. 5.20 plots typical resistance curves (Todd, 1963) using the older coefficient $\mathbb{C}=125\,R_T/(\tfrac12\pi\rho\nabla^{2/3}V^2)$. Reading from low to high speed:
| Froude number | What happens |
|---|---|
| $Fn < 0.10$ | Waves negligible; resistance almost entirely viscous; $C_T$ nearly constant (slightly falling). |
| $Fn \approx 0.15$ | Clear bow wave and forward-shoulder depression on fuller ships; transverse system develops. Service speed of VLCCs — wave resistance small, of the same order as still-air resistance. |
| $Fn \approx 0.20$ | Pattern more developed; stern transverse wave matters more. Coastal/merchant ships of moderate fullness, made slenderer to cut wave resistance. |
| $Fn \approx 0.25$ | Wave resistance can be important. Sea-going ferries, cruise liners, containerships. Fine trawlers reach $Fn\approx 0.24$ before the rise; often overdriven to $0.30$. |
| $Fn \approx 0.30$ | Bow–stern interference important; an unwanted hump often appears. Fishing vessels, free-running tugs, frigate cruising speed. Length (high $L/B$) delays heavy wave-making. |
| $Fn \approx 0.40$ | Transverse wave length equals ship length → sharp rise toward the main hump near $Fn\approx 0.5$. The "hull speed" range. |
| $Fn \approx 0.50$ | Ship sails in its own wave (main hump). Only transom-type flat-afterbody ships pass it easily; V-section afterbodies meet ever-higher resistance. |
| $Fn > 0.5$ | Semi-planing: dynamic lift on the flat afterbody lifts the ship and flattens the curve; transverse waves lose importance. |
| $Fn \approx 1.0$ | Planing range (no sharp limit, unlike Mach 1). Dynamic lift carries most of the weight; wave resistance relatively less important; the pattern shrinks to a narrow V whose angle decreases with speed (components at $\theta > 35°$ are no longer generated). |
6. Wave breaking and spray (§5.6)
The sinusoidal-wave theory above assumes small amplitude. Close to the ship the steepness can be large, giving mild nonlinear corrections — and, where the steepness is large enough, a completely different behaviour: wave breaking. Steep slopes collapse forward and a breaker forms. Two types are distinguished:
| Type | Behaviour |
|---|---|
| Plunging (Tulin & Landrini, 2000) | A steep crest forms a forward-directed jet at its tip that falls onto the wave surface ahead. |
| Spilling (Cointe & Tulin, 1994) | A patch of aerated water rides on top of the flow, sliding down the forward face. A plunging breaker often becomes spilling later. |
- At a sharp bow (high speed): bow wave climbs the hull, falls over → plunging breaker, crest overturning sideways.
- At a blunt bow: spilling breaker at the front of the wide bow-wave crest.
- At a pronounced fore or aft shoulder: spilling breaker on the aft side of the trough.
- Aft of the stern: spilling breaker on the first crest behind the transom; sometimes breaking of the diverging waves from the transom corners.
The inception mechanism is not fully settled. For bow-wave breaking, a free-surface boundary layer ahead of the ship may cause separation (thickening it artificially by towing a plastic sheet increased the breaking); alternatively the bow wave goes unstable and forms a necklace vortex (Fig. 5.21), where notable vorticity has been measured. A breaking wave is still steady in a time-averaged sense; the ordered orbital motion becomes turbulent, and the forward motion leaves a wake with momentum loss — Baba (1969) measured two wake lobes outside the main wake of bluff hulls.
Wave breaking converts wave energy into turbulence and other kinetic energy — a loss tied to a wave breaking resistance (as opposed to the wave pattern resistance of the radiated waves). Baba and others, from the wake momentum loss, found it could reach 15% of total resistance for a full hull. But the conversion also reduces the trailing-wave amplitude, so the sum of the two is not necessarily very different from the no-breaking case. (Larsson & Raven, §5.6)
When scaling to full scale, both wave components (pattern + breaking) are scaled up directly, because both relate to wave generation. The split between them may be off — the Weber number is much smaller at model scale, so surface tension is relatively more important and there is less breaking with less white water — but this does not noticeably affect the total wave-resistance extrapolation. Today it is not customary to separate the two parts of wave resistance, as it gives little advantage.
At high speed an extra component arises: spray drag. A sharp bow can throw a thin sheet of water that detaches and breaks into spray; spray rails deflect the sheet away from the hull and reduce it. For planing craft a low deadrise and aft trim give a blunt waterline entrance, and spray is essential to the attachment of the water surface to the hull bottom (Savitsky, DeLorme & Datla, 2007).
7. Viscous effects on the wave pattern (§5.7)
Treating wave making as wholly inviscid is an approximation; real viscous effects exist but are usually small, and only recent accurate free-surface viscous methods (§9.8) can quantify them.
- Viscous effects on the propagation of deep-water waves (viscous attenuation) are entirely negligible on ship scale. The effects are on the wave generation, plus a wave propagating through the viscous wake changing its length and direction.
- The main effect is a pronounced reduction of the stern wave system, caused by the boundary layer and wake changing the pressure field at the stern (Fig. 9.11). This reduction is larger at model scale (thicker boundary layer) and grows with hull fullness but depends on hull-form details.
- Viscosity is crucial in the flow off an immersed transom stern (wetted-transom recirculation) and its regime transitions. Flow separation at the waterline (sharp aft shoulder, or a bulbous bow at too low a draft) can also change the pattern substantially. In most cases these are secondary effects.
8. Shallow water — effect on wave properties (§5.8)
So far the depth and width were unlimited. In coastal regions the water can be shallow, which strongly affects flow, waves and resistance. First, how limited depth $h$ changes the wave properties. The vertical distribution $F(z)$ that in deep water gave $C\,e^{kz}$ must now satisfy a no-normal-flow condition at the bottom, $dF/dz=0$ at $z=-h$, giving $F\propto\cosh k(z+h)$. Substituting into the Kelvin condition yields the finite-depth dispersion relation:
$$ c = \sqrt{\frac{g\lambda}{2\pi}\,\tanh\!\left(\frac{2\pi h}{\lambda}\right)} \tag{5.38} $$
| Property | Change in shallow water |
|---|---|
| Dispersion | The extra factor $\tanh(2\pi h/\lambda)$ → 1 as $h\to\infty$ (recovers deep water). Once $\lambda$ exceeds 2–3 times $h$, waves of equal length are slower than in deep water. As $h$ shrinks, $\tanh\to 2\pi h/\lambda$ and the speed approaches a limit $c=\sqrt{gh}$ — an upper bound on wave speed. |
| Loss of dispersion | The constant-speed range is reached when $h$ is less than ~7% of $\lambda$; then all waves longer than ~14 h run at essentially the same speed. "Shallow" means small $h/\lambda$ — the same waterway is shallow for long waves and deep for short ones. |
| Orbital motion | Circular particle paths become elliptical (larger horizontal, smaller vertical component); at the bottom the vertical motion vanishes and the motion is purely horizontal (Fig. 5.23). For equal amplitude and length, horizontal velocities are substantially larger than in deep water (Fig. 5.24). |
| Group velocity | The ratio $c_g/c$ rises from $\tfrac12$ (deep) toward 1 as $\lambda/h\to\infty$ (Fig. 5.25) — in the shallow limit energy travels at the wave speed. |
$$ c = \sqrt{gh} \quad\text{(shallow-water speed limit)} \tag{5.40} $$
$$ \frac{c_g}{c} = \tfrac{1}{2}\!\left(1 + \frac{4\pi h/\lambda}{\sinh(4\pi h/\lambda)}\right) \tag{5.41} $$
9. Shallow water — effect on the wave pattern (§5.9)
The steadiness relation $c=V\cos\theta$ still holds in shallow water, but the speed–length relation changes. The key new parameter is the depth Froude number, the ratio of ship speed to the maximum wave speed:
$$ Fn_h \equiv \frac{V}{\sqrt{gh}} \tag{depth Froude number} $$
The pattern geometry now depends on both the length Froude number and $Fn_h$. Four regimes follow.
9.1 Low subcritical, $Fn_h \lesssim 0.7$ (§5.9.1)
While the depth exceeds ~one-third of the transverse wave length ($h > \tfrac{2}{3}\pi Fn^2 L$, i.e. $Fn_h < 0.7$), wave lengths and directions are essentially unaffected. But shallow-water effects already appear: with small keel clearance the bottom forces the flow along the sides rather than under the hull, giving a more horizontal path with larger curvature, larger pressure gradients, lower pressure and increased dynamic sinkage. Both raise the wave amplitudes even where the lengths are unchanged. The primary disturbance (overspeed beside the hull) extends farther — felt as stronger "suction" in shallow water or a channel.
The viscous flow is affected too: the overspeed raises friction, and the boundary layer, driven by the outer flow and subjected to larger gradients, may thicken near the stern and even separate — increasing viscous resistance. For the KVLCC2 case at $h/T = 1.20$ the calculated sinkage rose by a factor 4 and the bow-down trim by a factor 3.
9.2 High subcritical, $0.7 \lesssim Fn_h \lesssim 0.9$ (§5.9.2)
As depth decreases or speed rises, the pattern starts to change (Havelock, 1908, for a point pressure impulse — Fig. 5.27). The longest components, the transverse waves, pass the $h<\lambda/3$ limit first; held to the ship speed, they lengthen. Diverging components may still be unaffected. At the same time the Kelvin-wedge half-angle widens: the ratio $c_g/c$, which is 0.5 in deep water, rises in shallow water, pushing the energy ray outward, until the half-top angle reaches 90° at $Fn_h=1$. Both physical effects (shape change from $Fn_h$, amplitude increase from $h/T$, $h/B$) act at once, each on its own parameters.
9.3 (Trans)critical, $0.9 \lesssim Fn_h \lesssim 1.1$ (§5.9.3)
Around $Fn_h=1$ the effects can be dramatic. At $Fn_h=1$ the transverse waves move at the maximum speed $\sqrt{gh}$ and become pure shallow-water waves — nondispersive, their length no longer set by the ship speed. The group velocity equals the wave speed, so wave energy does not lag behind into a trailing system but stays in the wave where it is generated, which is continuously fed by the ship's pressure field and can grow strongly. The Kelvin wedge ceases to exist; the transverse crests extend ever farther laterally.
- If the critical waves are substantial, a large mass of water is pushed ahead, with a very large resistance rise plus large trim and sinkage. Most displacement ships cannot pass critical speed (lack of power or hitting the bottom).
- Only the transverse waves are critical; the shorter, slower diverging waves are still subcritical and may be barely affected — so a fan of diverging crests can be seen together with critical transverse waves (Fig. 5.28, middle).
- Severity depends on how strong the ship's transverse waves are: very slender ships (or fast ships well beyond the primary hump, $Fn_L > 0.5$) generate mostly diverging waves and may hardly notice critical speed. Worst case: critical speed coinciding with strong transverse-wave generation (e.g. $Fn_L\approx 0.4$, around $L/h\approx 6.25$).
- If the waterway is also laterally restricted (canal or towing tank), transverse waves can build up and move ahead as a series of solitary waves, causing periodic variations in measured quantities; rarely important full scale.
9.4 Supercritical, $Fn_h > 1$ (§5.9.4)
Above critical speed the ship outruns the maximum wave speed, so transverse waves cannot exist — only diverging waves remain. Since $c=V\cos\theta$ and $c\le\sqrt{gh}$, the limiting angle is
$$ \theta = \cos^{-1}\!\left(1/Fn_h\right) \tag{supercritical limit angle} $$
The wave at this limiting angle is a nondispersive shallow-water wave forming a single continuous crest/trough — the outer boundary of the whole system. The pattern is again triangular, but this is not a Kelvin wedge: its top angle is set by the ship-speed/wave-speed ratio, not the group/phase ratio. The higher $Fn_h$, the narrower the pattern. The outer waves resemble the critical transverse waves but build up less (energy moves outward along the crest). Passing critical speed is usually accompanied by a substantial drop in trim, sinkage and resistance. Fig. 5.29 is a full-scale wash measurement from an airplane (Bolt, 2001) of a catamaran ferry at $Fn_L = 0.41$, $Fn_h = 1.5$.
10. Shallow water — effect on resistance (§5.10)
Havelock (1908) computed the resistance of a pressure disturbance over depth $h$ (Fig. 5.30). Each curve, marked by $h/l$, shows a marked peak at the critical speed $Fn_h=1$, where resistance is much greater than in deep water (especially for small $h/l$); beyond the peak (supercritical) it falls below the deep-water value.
Model data for a destroyer (Rota, 1900; Figs 5.31–5.32) confirm this: approaching critical speed, stern trim and resistance both rise rapidly. The percentage rise at the peak is greater for smaller draft/depth ratio — for the shallowest case an almost threefold resistance. Past critical speed, trim and the shallow-water resistance increase both fall off quickly; well above it the resistance is slightly less than in deep water. The large critical peak is why nearly all displacement ships stay subcritical, while planing craft and multihulls often run supercritical.
10.1 Estimating the increase: Schlichting, Lackenby, Jiang
For the subcritical regime only, simple methods exist. In shallow water a given wave length is generated at a lower speed, so the resistance curve's humps and hollows shift to lower speeds. Schlichting (1934) built on this (Fig. 5.33): at deep-water speed $V_\infty$ the transverse waves have length $\lambda_0$ from $V_\infty=\sqrt{g\lambda_0/2\pi}$; in depth $h$ the same $\lambda_0$ is generated at a lower speed $V_I$, with
$$ \frac{V_I}{V_\infty} = \left(\tanh\frac{2\pi h}{\lambda_0}\right)^{1/2} = \left(\tanh\frac{gh}{V_\infty^2}\right)^{1/2} \tag{5.44} $$
Schlichting assumed the wave-making resistance in shallow water at $V_I$ equals that at $V_\infty$ in deep water (the first speed correction, on wave resistance). A second speed correction $\Delta V_P$ accounts for the increased viscous resistance, controlled mainly by $\sqrt{A_M}/h$ ($A_M$ = maximum cross-sectional area). The total speed loss $\Delta V = \Delta V_C + \Delta V_P$ gives the shallow-water resistance (Fig. 5.35). The method is not rigorous — it ignores diverging waves and the amplitude rise (which probably partly cancel in his data), and the viscous part is hull-form dependent — but it is a useful first estimate.
Lackenby (1963) gave a simpler correction (Fig. 5.36) to adjust speed-trial results in moderately limited depth, using $\sqrt{A_M}/h$ and $Fn_h$:
$$ \frac{\Delta V_s}{V_s} = 0.1242\!\left(\frac{A_M}{h^2}-0.05\right) + 1.0 - \left(\tanh\frac{gh}{V_s^2}\right)^{1/2} \tag{5.45} $$
Jiang (2001) showed that plotting resistance against an effective speed based on the mean dynamic sinkage $z_v$ collapses data from different depths onto one curve:
$$ V_e = \sqrt{\frac{V_s^2 + 2gz_v}{1 - z_v/h}} \tag{5.46} $$
All these semi-empirical corrections are useful indications in the subcritical regime only; the effects are complex and hull-form dependent, so precise estimates from one or two parameters should not be expected.
11. Far-field waves and wash (§5.11)
Since 1990, the detrimental effects of ship waves — "wave wash" — have drawn attention: bank/bottom erosion, damage to moored or small craft, danger to bathers, and harm to natural environments, often from fast-ferry operation or growing conventional traffic.
11.1 Far-field amplitudes by regime (§5.11.2)
| Regime | Decay with distance |
|---|---|
| Subcritical | Most components $A(y)\sim y^{-1/2}$; waves near the Kelvin-wedge edge ($\theta\approx 35°$) $A(y)\sim y^{-1/3}$. Kelvin-edge waves dominate only at very large distance (beyond $a^6 y$ if inside waves are $a$ times larger), so $y^{-1/2}$ is often seen in practice. |
| Critical | No simple theory. Far-field amplitudes increase, and decay rate decreases, with time spent at critical speed; for prolonged operation a conservative estimate assumes no decay. Simply extrapolating from a model run can mislead. |
| Supercritical | Leading waves nondispersive but steady; measured decay between $y^{-0.2}$ (very shallow) and $y^{-0.33}$ — very slow; smaller $h/L$ means slower decay and more far-field energy. |
The shape of a distant wave cut cannot be obtained by simple scaling of a nearby cut — it changes substantially with distance, needing spectral methods (Raven, 2000). Variable depth complicates the evolution further.
11.2 Far-field periods (§5.11.3)
Far-field periods matter mostly because long waves grow much more in height and steepness when entering shallow water, and cause large plunging breakers and beach run-up (unlike the spilling breakers of conventional ships) — happening unexpectedly when the fast ship is already out of sight. Subcritical periods are set by the near-field spectrum; the critical regime is poorly documented. The supercritical regime gave surprises: full-scale measurements showed apparent periods of 20–40 s at 3 km with heights of 0.4–0.7 m, because the leading critical wave and the second wave diverge at a small angle (10–12° near the vessel, ~2° far away), so their separation — and the apparent period — grows with distance, then magnifies by shoaling. Wash limits being imposed in some countries should therefore be based not just on wave height but also on period or energy.
12. Channel effects and blockage (§5.12)
If the waterway is also restricted in width (canal, river, towing tank), more effects appear. Wave propagation speeds are unaffected by width, so $Fn_h$-governed effects are unchanged to first order. But the bottom-proximity effect is magnified by the blockage — the ratio of midship sectional area to the waterway cross-section — which raises the overspeed beside the hull, the sinkage, the pressure gradients, the wave amplitudes and the viscous resistance.
For moderate effects, a Schlichting-like correction works for the wave-making part; the viscous speed correction $\Delta V_P$ must be modified. Landweber (1939) introduced the channel width via the hydraulic radius $R_H$ = (cross-sectional area)/(wetted perimeter):
$$ R_H = \frac{bh - A_M}{b + 2h + p} \tag{5.48} $$
where $b$ is channel width, $h$ depth, $A_M$ the hull's maximum sectional area and $p$ the wetted girth. For very large $b$, $R_H\to h$ (shallow water of unlimited width). He deduced a single curve for $V_h/V_I$ in terms of $\sqrt{A_M}/R_H$ (Fig. 5.34).
12.1 Kreitner's theory — the inaccessible region
Channel effects can be far more severe, explained by Kreitner (1934). For a ship at low $Fn_L$ with a long parallel midbody, in restricted water all the inflow (rate $V_s\,b\,h$) must squeeze past the ship, forcing an overspeed. With blockage $\beta \equiv A_M/(bh)$ and overspeed ratio $\gamma \equiv (V_s+v)/V_s$, at low speed
$$ \gamma = \frac{1}{1-\beta} \tag{5.49} $$
But the overspeed lowers the water surface beside the ship (Bernoulli), causing an equal sinkage and reducing the cross-section by $\Delta h = \tfrac12 h\,b\,Fn_h^2(1-\gamma^2)$. Requiring all flow to pass the now-smaller area gives a third-degree equation:
$$ \gamma\left(1-\beta-\tfrac{1}{2}Fn_h^2\left[\gamma^2-1\right]\right) = 1 \tag{5.51} $$
For increasing $Fn_h$ the overspeed rises, then increases faster (water-level depression), until at a certain $Fn_h$ it grows indefinitely: it becomes impossible to let all fluid pass. This is the lower limit of an "inaccessible region" around $Fn_h=1$ where no steady solution exists — wider for higher blockage. There the solution is unsteady: a wave of translation moves ahead and a trough propagates aft. (Larsson & Raven, §5.12)
So in a restricted-width waterway, critical-speed-like phenomena (large sinkage, big resistance rise, violent flow) set in at a depth Froude number well below 1 — the steep resistance rise shifts to lower $Fn_h$. This is something to watch in model testing at high $Fn_h$, as it can cause gross errors when predicting the unlimited-width curve. Kreitner's assumption of uniform overspeed limits the method to ships with long parallel midbody in narrow channels, so its quantitative results need caution.