The drag that is made of waves

No drag at all is one of the strongest results in this collection and one of the most carefully bounded. A body moving steadily through an unbounded, incompressible, inviscid fluid in irrotational flow feels no force in the direction of motion: the pressure integral vanishes identically, and the vanishing is exact rather than approximate.

Every clause in that sentence is load-bearing, and the one that gets least attention is unbounded.

Put a free surface above the body and the flow is still incompressible, still inviscid, still steady in the body’s frame and still irrotational everywhere. And there is drag — a great deal of it, for a ship; more than half the total for many hulls at cruising speed. It is not a failure of the theorem. It is the theorem’s hypothesis being read properly.

Where the energy goes, which is the whole answer in one line #

The mechanism is stated before any algebra because it is the thing to carry away.

A body under a free surface leaves a train of waves behind it. Those waves carry energy, at a rate set by their amplitude and their group velocity. The energy has to come from somewhere, and the only thing supplying it is the body. So the body does work against a force, and that force is the wave resistance.

D’Alembert’s argument never contemplated this because it took the fluid to fill all space, in which case a disturbance cannot leave: everything the body does to the fluid is done again in reverse somewhere else, and the pressure integral closes. A free surface is a place where energy walks out and does not come back.

The energy leaving is not a small correction to the pressure integral. On a ship at its design speed it is the largest single term in the resistance, and the reason it is not in ideal against real’s comparison of the exact theory with a viscous computation is that both of those flows are unbounded.

The linearised problem, and the pole in it #

The computation is Havelock’s and it is short enough to follow.

Put a two-dimensional doublet of moment $M$ at depth $d$ below the undisturbed surface, in a stream $U$. Linearise the free-surface condition — small waves, small slopes — and it becomes

$$U^2\phi_{xx} + g\phi_y = 0 \quad\text{on } y = 0.$$

Fourier transform in $x$. The doublet’s own potential has amplitude $-\tfrac{i}{2}M\,\mathrm{sgn}(k)\,e^{-|k|d}$ at the surface; add a correction $C(k)e^{|k|y}$ that decays downward and impose the condition, and

$$C(k) = a_D(k)\,\frac{U^2|k| + g}{g - U^2|k|}.$$

There is the whole physics, in a denominator. It vanishes at

$$|k| = k_0 = \frac{g}{U^2},$$

which is the wavenumber whose free deep-water wave travels at exactly $U$ — and therefore stands still relative to the body. Everything about wave resistance is the residue at that pole.

The amplitude, and a check that is not circular #

Taking the residue with the radiation condition — waves behind, not in front — gives a far-downstream elevation of amplitude

$$A = \frac{4\pi k_0 M e^{-k_0 d}}{U}.$$

That is a closed form obtained by a contour argument, and closed forms obtained by contour arguments are exactly the kind that come out with the wrong factor. So it is checked against a completely different calculation: the whole Fourier integral, evaluated numerically with a small Rayleigh damping to impose the radiation condition, at four thousand points in $k$ and a hundred and twenty in $x$.

The two agree to 1.9 per cent, and the residual is the damping: it falls linearly with it, from 5.3 per cent at $\mu = 0.02$ to 1.9 at $\mu = 0.005$.

The picture carries two further checks worth naming. The amplitude behind the body is 121 times the amplitude ahead of it, which is the radiation condition working rather than being asserted. And the wavelength behind is 6.2813 against $2\pi U^2/g = 6.2832$ — the wave that stands still, to three parts in ten thousand.

Two accounts of the force, and why only one of them is evidence #

The resistance follows from the amplitude by an energy argument. The train behind carries energy $\tfrac12\rho g A^2$ per unit area and transports it at the group velocity, which in deep water is half the phase velocity. In the body’s frame the train is being created at a rate $U - c_g = U/2$, so

$$R\,U = \tfrac12\rho gA^2 \cdot \tfrac{U}{2} \quad\Longrightarrow\quad R = \frac{\rho g A^2}{4} = 4\pi^2\rho\,k_0^3 M^2 e^{-2k_0 d}.$$

The last equality is worth being honest about. $\rho gA^2/4$ and $4\pi^2\rho k_0^3M^2e^{-2k_0d}$ are the same expression rewritten, so verifying that they agree — which the machinery does, to $4\times10^{-16}$ — checks the algebra and not the physics. The check that is evidence is the one in the previous section, where an independent numerical integral reproduces the amplitude.

That distinction matters more than it sounds, and this collection has been caught by the other side of it: an identity dressed as a confirmation is the most comfortable kind of wrong result to have.

Depth, and the exponential that never arrives at zero #

Now take the limits, which is what these essays are for.

Send the body deeper. The resistance falls as $e^{-2k_0 d} = e^{-2gd/U^2}$, and the decay rate measured back out of the sweep is $2k_0$ to 3.6 parts in $10^{16}$.

So d’Alembert’s paradox is recovered — in the limit, and only in the limit. At every finite depth the drag is strictly positive, and the approach is exponential rather than algebraic, which is why a submarine at three diameters is nearly free of wave drag and one at half a diameter is not.

That is the rule these essays run on, in its cleanest form. The limit removes the drag; the residue at any finite depth is an exponential that is never zero.

What the body feels, as against what it radiates #

There is a second force in this problem and it is worth separating from the first, because the two are often run together.

A body under a free surface feels a steady vertical force as well as a horizontal one: the surface above it is deformed, the pressure distribution on the body is no longer the symmetric one, and there is a suction towards the surface. That force is not the wave resistance and does not come from radiation; it is the near field, and it exists even when the body is deep enough that the wave train is negligible. It is the reason a submarine near the surface is pulled towards it.

The horizontal force is different in kind. It is not a property of the near field at all — it is fixed entirely by the amplitude of the train at infinity, which is why an energy argument gets it exactly right without ever looking at the body. That is the same division of labour where the lift reaction is makes for a lifting body: the total force can be computed on any contour whatever, and the split between pressure and momentum flux depends entirely on which contour is chosen, running from three per cent to ninety-seven.

Forces that can be computed at infinity are the robust ones. Forces that cannot are the ones that need the body.

Speed, and a hump at an exact number #

The other limit is the speed, and it is the one a naval architect cares about.

$k_0 = g/U^2$, so going slowly makes $k_0$ large and the exponential kills the drag; going fast makes $k_0$ small and the $k_0^3$ kills it. In between there is a maximum, and it is the wave-resistance hump every ship’s resistance curve has.

Where the maximum is depends on what is held fixed, and the two cases give two exact numbers.

Hold the doublet moment fixed and $R \propto k_0^3 e^{-2k_0d}$, whose maximum is at $k_0 d = 3/2$ and therefore at a depth Froude number of $\sqrt{2/3} = 0.8165$.

Hold the body fixed — a cylinder of radius $a$ is a doublet of moment $Ua^2$, so the moment rises with speed — and $R \propto k_0 e^{-2k_0 d}$, whose maximum is at $k_0 d = 1$ and therefore at

$$\mathrm{Fr}_d = \frac{U}{\sqrt{gd}} = 1, \quad\text{exactly.}$$

The second is the physical case, and the golden-section search finds it at 1.00000000 with no fitted constant anywhere in the derivation. It does not move when the body is made smaller or the fluid lighter, which is the check that it is a property of the mechanism.

Why the wave that matters is the one that stands still #

The pole is at the wavenumber whose free wave travels at $U$, and it is worth saying why that is the only wavenumber the body can excite in steady motion.

A body moving steadily presents a disturbance that is stationary in its own frame. In that frame a free wave of wavenumber $k$ has a phase speed $c(k) - U$ relative to the body, and only the wave with $c(k) = U$ is stationary. Every other wave sweeps past and averages away over a long enough time; the resonant one accumulates.

That is the same statement the angle that does not care makes about the Kelvin wake, from a two-dimensional angle rather than a one-dimensional one: the wedge behind a ship has a half-angle of $\arcsin\tfrac13$ for every ship at every speed on every planet, because deep-water dispersion makes the group velocity half the phase velocity and stationary phase does the rest. A two-dimensional body has no wedge — there is nowhere for the pattern to spread sideways — and what it has instead is a single wavelength, which is that essay’s dispersion relation evaluated at one point.

Where the analogy with the paradox breaks, and where it does not #

It would be easy to read this as “d’Alembert’s paradox has an exception”, and that is not quite the statement.

The paradox is a theorem with hypotheses, and one hypothesis is that the fluid is unbounded so that no disturbance can propagate to infinity carrying energy. A free surface violates it. So does a compressible fluid above the speed of sound — which is why the exact theory can have wave drag in supersonic flow while having none in subsonic — and so does a stratified fluid, which radiates internal waves.

In every one of those cases the drag is an energy flux to infinity, and in every one it is proportional to the square of a wave amplitude. That is a much better generalisation than “the theory sometimes fails”. The theorem is right; the domain decides whether its hypothesis holds.

The added mass, which is the other thing a free surface changes #

The force of getting going computes the other force an ideal fluid can exert on a body: the added mass, which appears when the body accelerates and which is exactly the displaced mass of fluid for a cylinder. That force is not a drag — it is in phase with the acceleration rather than the velocity — and d’Alembert’s paradox has nothing to say about it, because the paradox is a statement about steady motion.

A free surface changes the added mass too, and by an amount that depends on the frequency: a body oscillating near a surface radiates waves, and the radiated waves produce a force with a component in phase with the velocity as well as one in phase with the acceleration. The first is a damping and it is the unsteady relative of everything in this essay; the second is a modified added mass.

That pair — added mass and radiation damping, both functions of frequency, both computed from the same free-surface Green’s function — is the whole of linear ship motions, and it is the same physics as the steady wave resistance seen at a different frequency. What makes the steady case simple is that only one wavenumber is excited.

Why the depth Froude number, and not the length one #

A practical note, because the number computed here is not the number in a naval architecture textbook.

A ship’s resistance curve is plotted against the length Froude number $U/\sqrt{gL}$, and its humps are at values like 0.5 and 0.3, set by interference between the wave systems of the bow and the stern — which is to say by the ratio of the wavelength $2\pi U^2/g$ to the ship’s length.

The number here is a depth Froude number, $U/\sqrt{gd}$, and it is set by the ratio of the wavelength to the submergence. There is no interference in this problem because there is only one singularity, so the only length in it is the depth.

Both are consequences of the same dispersion relation and both are exact statements about their own problem. Which one a reader wants depends on whether the body is long or deep, and counting what matters is the essay about how many such numbers a problem has in the first place: with a length, a depth, a speed and gravity there are two independent groups, and choosing one of them to call “the” Froude number is a convention rather than a discovery.

The other exception, and how this one differs #

The collection already has one drag in the exact theory: drag in the theory that forbids it computes Kirchhoff’s free-streamline flow past a plate, which has a drag coefficient of $2\pi/(\pi+4)$ and no viscosity in it anywhere.

The two are not the same mechanism and the difference is instructive. Kirchhoff’s drag comes from a wake — a region of constant-pressure fluid extending to infinity, which is a change to the topology of the flow rather than to the domain, and which is put in by hand. The wave drag here comes from radiation: nothing about the flow’s topology is unusual, and the energy leaves through a boundary the problem always had.

One is a modelling choice about what the flow looks like. The other is a consequence of the boundary conditions.

What a real ship pays, and what this does not compute #

The apparatus above is linear, two-dimensional and submerged, and a ship is none of those.

The body is a doublet. Representing a hull by a singularity is Michell’s thin-ship approximation taken to its extreme, and it is good for a slender submerged body and poor for a surface-piercing one. A real hull’s wave resistance is computed by distributing sources over its own surface, and the interference between the bow and stern systems produces the humps and hollows that make a real resistance curve wavy rather than single-peaked.

The waves are linear. The free-surface condition used here has been linearised, so wave slopes are small and breaking is impossible. A real bow wave at the hump condition is neither.

And nothing is three-dimensional. A ship’s waves spread into a wedge and their amplitude falls with distance; a two-dimensional body’s do not spread at all. That changes the exponents in every scaling above, and the exact number $\mathrm{Fr}_d = 1$ is a two-dimensional number.

What survives all three is the mechanism, the exponential dependence on depth, and the existence of a hump.

The invariance of the peak’s location under changes of size and density is worth a sentence, because it is what distinguishes a mechanism from a fit. The resistance itself moves by a factor of fifty between those two figures. The Froude number at which it is largest does not move at all, to eight decimal places, and it would not move for a different fluid or a different planet — which is the same kind of statement the number that really is one makes about $\mathrm{Fr} = 1$ in a channel, and the same kind of evidence that a dimensionless group is doing real work rather than absorbing a coefficient.

Limits recorded rather than smoothed over #

The radiation condition is imposed by a damping. The Fourier integral is evaluated with a small imaginary part that pushes the pole off the real axis, which is the standard device and which also decays the computed train slowly with distance. The 1.9 per cent residual between the integral and the residue is that decay, and it is reported rather than tuned away.

The energy account and the momentum account are the same algebra. They are quoted together above because both are useful ways to remember the result, and the fact that they agree is not evidence.

And the amplitude is measured near the body. Far downstream the damping has eaten the train; near the body the linear theory is weakest. The window used is a compromise, and the number to trust is “about two per cent” rather than the five figures the arithmetic prints.

The residue, again #

Two limits, two residues.

Deep water: the drag goes to zero as the depth goes to infinity, and at every finite depth it is $e^{-2gd/U^2}$ times something — small, positive, and never zero. The paradox is recovered asymptotically and never exactly.

Slow or fast: the drag goes to zero at both ends of the speed range, and in between there is a maximum at a depth Froude number of exactly one. That number is the residue of a limit nobody took: it is what is left when the exponential and the power law are made to balance, and it has no fitted constant in it at all.