The borrowed mass that goes negative

The borrowed mass the boundary decides put a circle beside two kinds of boundary. A solid wall, which forbids the water to cross it, raised the circle’s added mass. A surface held at constant pressure, which forbids the potential to change, lowered it. One radius clear of the boundary the two numbers were 1.1346 and 0.8833 times the mass of water the circle displaces. The essay ended by pointing out that the sea is neither boundary. Its surface is free to move, but gravity pulls it back, and a body shaken beneath it makes waves.

This essay puts gravity back. The first thing it finds is that the sea is both boundaries, one at each end of the frequency range. The second is that between the two ends the borrowed mass does something neither boundary can do: it runs outside the range they set. For a circle close enough to the surface it goes below zero, and the picture of added mass as carried water cannot survive that.

The picture that is taught #

The mass a body has to borrow gives the careful version of added mass: it is not a lump of water stuck to the body, but the kinetic energy the whole surrounding flow acquires, expressed as the mass that would carry that energy at the body’s own speed. That essay already warned that almost everything inferred from the carried-water picture is false. What survives in most minds anyway is a weaker version: that added mass is at least a property — something a body of a given shape, in a given place, simply has, like its displacement. Introductory tables give it that way: a half for a sphere, one for a circle, a number for each shape.

The two images of the previous essay did not disturb that. A wall gave one number and a free boundary gave another, and each was a property of the body and the boundary together. What they left out was the boundary that responds. A free surface under gravity is a boundary that moves in response to the body. How it moves depends on how fast it is asked to, so the added mass depends on the frequency, and a quantity that depends on the frequency is not a property. It is a response.

One number between two images #

The water is deep, inviscid and at rest. The circle has radius a, and its centre sits at depth f below the undisturbed surface. It oscillates up and down at frequency ω with a small amplitude. The flow is a potential flow, and on the surface the linearised conditions — the surface moves with the water, and its pressure is atmospheric — combine into one:

$$K\phi + \frac{\partial \phi}{\partial y} = 0, \qquad K = \frac{\omega^2}{g},$$

with y measured downwards. K is the only thing the frequency does. It is the wavenumber of the waves the circle makes, 2π divided by their length, and the dimensionless measure of how fast the circle is shaken is Ka.

The two limits are then immediate. When K is small the first term drops out, the condition becomes $\partial\phi/\partial y = 0$ — no flow through the surface — and the surface behaves as a rigid lid. A slowly shaken body sees the sea as a wall, because at long wavelengths gravity holds the surface almost flat. When K is large the condition becomes φ = 0, the constant-pressure boundary, which is the impact limit of the previous essay. At short wavelengths the surface has no time to respond by rising, and it simply lets the water go.

The computation reaches both. At K = 0 it gives 1.134576 and at K = ∞ 0.883294, and the panel method of the previous essay, a different technique altogether, gives 1.134606 and 0.883311. The two agree to three parts in a hundred thousand. That agreement is how an error in the previous essay’s published numbers came to light. It had quoted 1.1351 and 0.8836 from a coarser panel count, and it now quotes the converged values.

What the curve does between the limits is the finding. Starting from the lid’s 1.1346, the added mass rises, to 1.258 at Ka = 0.134. It then falls steeply through the free-space value of one, reaches 0.707 at Ka = 0.98 — well below the constant-pressure value — and climbs back towards 0.8833 only at wavenumbers of ten and more. A boundary that is partly a wall and partly a free surface does not give a number between the wall’s and the free surface’s. It gives numbers outside both, on both sides.

How the circle is solved #

The method is Ursell’s, from 1950. Around the circle’s centre the potential is expanded in multipoles — $\cos(n\alpha)/r^n$, α measured from the upward vertical. Each multipole is completed by a wave integral that makes it satisfy the surface condition and carry waves outwards and never inwards:

$$\phi_n = \frac{\cos n\alpha}{r^n} + \frac{1}{(n-1)!}\,\mathrm{PV}\!\int_0^\infty \frac{k+K}{k-K}\,k^{n-1}e^{-k(f+y)}\cos kx\,dk .$$

The integral has a pole at k = K, the wavenumber the surface allows. The way the path passes it decides that the far field is a wave travelling away, e^(iK|x|). Near the circle each wave integral is expanded again in powers of r, and the condition that the water follow the circle’s surface becomes one equation per Fourier mode for the multipole strengths. Forty or so strengths reproduce everything to machine precision for a circle down to a twentieth of a radius below the surface. Doubling their number changes no result by more than 4 × 10⁻¹⁵.

The force follows from the pressure on the circle, iωρφ. The part in phase with the acceleration is the added mass. The part in phase with the velocity is the damping — a force that takes energy out of the motion, although no viscosity is present.

Where the energy goes #

The damping is the other half of the answer, and it explains the first half. A circle under a still surface does work on the water every cycle, and the work does not come back: it leaves as two trains of waves travelling away on either side. The damping is that power, divided by the square of the velocity. The drag that is made of waves found the same thing for a body moving steadily under a surface. There, d’Alembert’s paradox failed with every one of its conditions still met, because energy was leaving in a wave train. Here it is the same failure, periodic instead of steady.

The band has definite edges. At small Ka the waves are long and their motion reaches deep, but a long wave takes a small slope to carry a lot of water, and a slowly shaken body makes very little of it. At large Ka the waves are short, and their motion decays with depth as e^(−Ky). The drift in a wave that has none draws that decay for the orbits under a passing wave. A circle deeper than a few short wavelengths cannot reach the surface to make them. So the damping peaks near Ka ≈ 0.4: at 2.14 for a circle with a tenth of a radius of water above it, 0.42 for one with a radius above, 0.19 for one with two radii.

The damping is computed twice, by two routes that share nothing after the solution. Once from the pressure on the circle’s surface. Once from the far field, where the potential becomes a pair of plane waves of amplitude P and the power they carry is $\tfrac12\rho\omega|P|^2$. The two agree to 1.3 × 10⁻¹⁵ over twelve combinations of depth and frequency. Conservation of energy is one of the few checks a wave calculation can fail silently, and here it does not.

Why the mass must overshoot #

The overshoot is not an accident of the circle. It is forced by the damping, and the reason is causality.

A force cannot respond to a motion before the motion happens. For any linear system that one fact makes the in-phase and out-of-phase parts of the response into a Hilbert pair: each is determined, at every frequency, by the other at all frequencies. For a body in waves, Kotik and Mangulis wrote it down in 1962. In the units here it reads

$$m(K) - m(\infty) = \frac{1}{\pi}\,\mathrm{PV}\!\int_0^\infty \frac{b(\kappa)}{\kappa - K}\,d\kappa ,$$

b being the damping. Read it at K = 0 and it says the rigid-lid added mass exceeds the constant-pressure one by the damping’s weighted area. The gap between the two images of the previous essay is made of waves — of the energy the circle can radiate in between. Read it at a frequency just above the damping band and the integrand is mostly negative, so the added mass there is below its high-frequency value. Just below the band it is mostly positive, and the added mass rises above its lid value. A peak of damping forces a rise before it and a trough after it. The shape of the added-mass curve is the shadow of the damping curve.

The computation checks this directly. The added mass is rebuilt from the damping curve alone, sampled at 384 frequencies with the pole removed by subtraction, and compared with the added mass computed from the pressure. With one radius of water above the circle the two agree to 10⁻⁹ everywhere. For the shallowest circle the worst of eight comparisons is 1.7 × 10⁻⁴, set by the quadrature. The dots in the limits figure are the rebuilt values.

That is the same relation that links the refractive index of glass to its absorption, and the reactance of a circuit to its resistance. Wherever something is absorbed, the response just beside the absorption is pushed out of line. The pulse that grows as it leaves the heart and the shock that passes without an echo both turn on impedance, which is the same pair of quantities under another name.

Negative water #

The trough deepens as the circle rises. With a radius of water above it, the least added mass is 0.707; with half a radius, 0.487; with a quarter, 0.223. It reaches zero when the water over the circle’s top is 0.149 of its radius, at Ka ≈ 0.85. A circle shallower than that has a band of frequencies in which its added mass is negative. With a tenth of a radius above it, the band runs from Ka = 0.57 to 1.11, and the least value is −0.182 at Ka = 0.76.

No amount of water can be negative, so the carried-water picture has nothing to say about this. The energy picture does not fail, because the kinetic energy of the flow is not what is being measured. The added mass is the force in phase with the acceleration, divided by it, and in a flow that radiates waves that force is not the rate of change of the flow’s kinetic energy. Some of the energy is going into the surface’s potential energy as it rises and falls. With the circle just under the surface, at frequencies just above the damping peak, the surface’s heave pushes back on the circle with its acceleration rather than against it. The body then behaves as though it were lighter than it is. A circle whose own mass is less than 0.182 of the water it displaces would, in that band, have a total inertia below zero.

McIver and Evans set out in 1984 when this can happen for submerged bodies. The causality relation says when it must be possible: a damping peak tall enough, over a band narrow enough, drives the trough after it below zero. The shallowest circles have the tallest peaks — 2.14 against 0.42 one radius down — and they are the ones that cross.

The consequence is not a curiosity. A structure’s natural frequency is set by its stiffness divided by its mass plus its added mass. A body with a negative added mass over a band of frequencies has a natural frequency higher than its dry mass alone would give, and a designer who took added mass to be a fixed positive amount would place the resonance on the wrong side of it. Submerged wave-energy converters work close under the surface at wavelengths of a few body sizes, which is this range.

The frequency at which it makes no waves #

Past the band the damping falls, as the depth factor says it should, and then it does something the depth factor does not predict: at one frequency it drops to nothing. For a circle 1.1 radii down this happens at Ka = 2.044, for 1.25 at 3.497, for 1.5 at 5.784, for 2 at 10.5 and for 3 at 21.2. At each of them the damping is below 10⁻⁹ of its peak, and after the decay factor is divided out it is still below 10⁻¹⁸. That is a zero, not a small number. The shallowest circle has a second one, at Ka = 9.0.

A point dipole — a vanishingly small circle — never does this: its wave amplitude is proportional to K·e^(−Kf), which has no zero. The zero belongs to the circle’s size. At these frequencies the waves are between a third of a radius and three radii long, shorter than the body. The waves the different multipoles send out then arrive at the far field in exact opposition and cancel. At its wave-free frequency the circle oscillates, does work on the water, gets all of it back, and sends nothing away. It is invisible to anything more than a wavelength off.

What the picture cannot show #

The waves are linear. The surface condition assumes a slope small everywhere. A circle a tenth of a radius below the surface, heaving with any real amplitude, lifts the water over it far more steeply than that, and it breaks. The negative band is a statement about infinitesimal motion. At finite amplitude it is where the linear theory is least trustworthy, which is also where the negative mass lives.

The circle is infinitely long and the water infinitely deep. Every number is per unit length of a two-dimensional body. A cylinder of finite length radiates waves in three dimensions, which spread and weaken with distance, and its coefficients differ from these. Finite depth adds a floor, whose image raises the added mass at the slow end.

Sway is not drawn, because it is the same. The side-to-side multipoles produce the same equations as the up-and-down ones, so a submerged circle’s sway added mass and damping equal its heave ones at every frequency — Ogilvie’s result of 1963, exact here by construction. The previous essay found the same equality for a circle beside a still boundary, for the same reason.

And the water is inviscid. A real circle sheds vortices when its amplitude exceeds a fraction of its radius, and that adds damping of a different kind, growing with the amplitude, which the potential flow has no place for.

The convention the numbers depend on #

The circle has radius a and its centre is at depth f. “Water above the circle” is f − a. The wavenumber is $K = \omega^2/g$ for deep water, and a frequency is quoted as Ka. Added mass is per unit length, divided by $\rho\pi a^2$, the displaced water. Damping is divided by $\rho\pi a^2\omega$, so that it is a mass too. Motion is e^(−iωt), waves are outgoing, and y is measured downwards from the undisturbed surface. The heave and sway coefficients are equal, and only heave is quoted.

How each number was checked #

The two limits are checked against a panel method with a different construction — sources on the outline and an image of either sign — at two depths. The damping is checked against the energy the waves carry. The count of multipoles is doubled. The added mass is rebuilt from the damping. Each check could have failed, and the calculation refuses a circle that breaks the surface, a negative frequency, and a causality tolerance of zero. The threshold depth of 0.149 radii is found by bisection on the least added mass. Each wave-free frequency is found by stepping up in Ka to the first minimum of the damping with its decay factor divided out, then closing on it.

Who found it, and when #

The multipole method is Ursell’s — for a half-immersed circle in 1949 and a submerged one in 1950. Ogilvie computed the submerged circle’s added mass and damping in 1963, and showed that heave and sway coincide. Dean had shown in 1948 that a submerged circle reflects no waves at all. Together these made the Bristol cylinder, Evans’s wave-energy absorber of the 1970s: a submerged cylinder moved in a circle, heave and sway a quarter-cycle apart, radiates waves to one side only. It can therefore absorb every wave arriving from the other side. The causality relation between added mass and damping for bodies in waves is Kotik and Mangulis’s, from 1962, and McIver and Evans’s 1984 paper is the account of when the added mass of a submerged body goes negative.

The wave-free frequency is the circle’s own version of an older idea. A body shaped so that its radiated waves cancel is a wave-free body, and designing one deliberately is a way to make a hull or a support that moves without making waves.

Still open: the hydrofoil under the surface, and a body that floats #

Two continuations follow directly. The first is the one the borrowed mass the boundary decides set beside this: a hydrofoil running steadily just beneath the surface. It is the steady counterpart of this problem. At high Froude number the surface is a constant-pressure boundary and the foil loses lift, as a wing over the ground gains it. At low Froude number the surface is a lid and the foil gains lift. Between, the lift passes through the same kind of overshoot, and the foil also pays a wave drag. The calculation uses the same multipoles with a travelling source in place of an oscillating one, and the question is where the lift loss sits in speed and depth for a foil of realistic proportions.

The second is a body that pierces the surface — Ursell’s original half-immersed circle, and every ship. Its waterline makes the surface condition apply right up to the body, the image picture has no clean limit at the slow end, and there are irregular frequencies at which the calculation fails without the physics doing so. How its added mass passes between the two images, and whether it too goes negative, is the version a naval architect actually uses. Beside both sits the steady ground-effect case from the cushion that is not there, now with a surface that can move: a wing skimming water whose surface its own pressure deforms, which the cushion that is there after all and a cushion that changes its physics both treated as rigid.