A drop rings like a bell

The drop that is not a tear draws a falling raindrop’s shape as the flattened ball its pressure field makes of it, and says plainly what that shape leaves out: real raindrops of a millimetre and more oscillate continuously between oblate and prolate, and the shape drawn is the mean of that oscillation. It also repeats the usual reason for the oscillation, which is that the drop is being driven by its own wake.

This essay computes the oscillation and then tests the reason. The oscillation turns out to be one of the cleanest calculations in fluid mechanics, known since 1879 and recoverable two independent ways. The reason does not survive the numbers.

The pitch #

Pull a drop of radius $a$ slightly out of round, into the shape $r = a[1 + \varepsilon P_l(\cos\theta)]$ with $P_l$ a Legendre polynomial, and let go. Surface tension pulls the surface back towards the sphere, which has the least area for its volume, and the liquid inside overshoots because it has momentum. For small $\varepsilon$, no viscosity and air too light to matter, Rayleigh found in 1879

$$\omega_l^2 = l(l-1)(l+2)\,\frac{\sigma}{\rho a^3}.$$

The mode $l = 0$ would change the volume and $l = 1$ is a translation, so the fundamental is $l = 2$ — the lemon-and-lentil swing of the first figure — at $\omega_2^2 = 8\sigma/\rho a^3$. Every mode above it rings at a fixed ratio to it, $\sqrt{l(l-1)(l+2)/8}$: 1.94 for $l = 3$, 3.00 for $l = 4$, 5.48 for $l = 6$.

The formula contains no length but the drop’s own and no speed at all. The time scale $\sqrt{\rho a^3/\sigma}$ is the capillary time — the same one that sets how fast a jet breaks into drops — and the pitch falls as the drop’s size to the minus three halves. A drop two millimetres across rings at 122 Hz, a drop of half a millimetre at 972 Hz. Air changes this by a factor involving its density relative to water’s, a correction of a few parts in ten thousand that the figures include and the argument can ignore.

The pitch from two energies #

Rayleigh derived the frequency by balancing energies, and the balance can be done here numerically on the shape itself, which is a genuinely independent route to the formula rather than a restatement of it.

The potential energy is the surface energy: $\sigma$ times the extra area the deformed shape has over a sphere of the same volume. The same volume matters. A shape $1 + \varepsilon P_l$ has more volume than the sphere, by an amount of order $\varepsilon^2$, so the radius is corrected by a constant $c$ of the same order, found by solving for it so that the volume, integrated over the deformed shape, equals the sphere’s. The surface area is then integrated over the shape, and its excess over the sphere’s — measured with the same quadrature, so that the quadrature’s own error cancels — gives a stiffness.

The kinetic energy is that of the flow inside, which for a small swing is a potential flow with $\phi \propto r^l P_l$, the one harmonic function that moves the surface in the shape of the mode. Its energy per unit of surface speed squared is an effective mass. The frequency squared is stiffness over mass.

One detail of that calculation is worth stating, because it is the kind of thing a formula hides. For $l = 2$ the prolate half of the swing and the oblate half do not cost the same area: the surface energy has a term cubic in $\varepsilon$, which is why a large-amplitude drop spends longer on one side of its swing than the other. Taking the stiffness from one sign alone converges only as $\varepsilon$. Averaging the two signs cancels every odd power and leaves the $\varepsilon^2$ convergence in the figure. With that done, the energy route agrees with Rayleigh for modes 2 to 5 to within $5\times10^{-6}$ at an amplitude of 0.2 per cent.

The ring-down #

Viscosity turns the swing into heat. Lamb worked out how fast in 1881, by a route that is itself a revealing shortcut: he kept the inviscid potential flow as the motion and asked what it dissipates. A potential flow has no vorticity but plenty of strain, and it is the strain a viscosity acts on, so the flow loses energy at the rate $2\mu\int S{:}S\,dV$ integrated over the drop. The amplitude then decays with time constant

$$\tau_l = \frac{a^2}{(l-1)(2l+1)\,\nu}.$$

That is what was done here too: the strain of the mode’s potential flow was differenced from the potential on a grid over the sphere and its square integrated, and the ratio of dissipation to energy gives the decay. For $l = 2, 3, 4$ it reproduces $(l-1)(2l+1)$ — 5, 14 and 27 — to $5\times10^{-6}$.

For a two-millimetre drop the time constant is 0.2 seconds and the period is 8.2 milliseconds, so the amplitude falls by a factor $e$ only after 24 full swings. The quality factor — $\omega\tau/2$, the number of radians of swing per e-folding of the energy — is 76. A raindrop is a lightly damped oscillator, in the same class as a tuning fork rather than a shock absorber, and it rings for a long time once struck.

The shortcut has a stated range. It assumes that viscosity is weak enough not to change the flow, only to drain it, which is the statement that the Ohnesorge number $\mu/\sqrt{\rho\sigma a}$ is small. For a two-millimetre water drop it is 0.0037. The full viscous problem — Chandrasekhar’s and Reid’s, which solves for a flow that has vorticity in a boundary layer at the surface — gives a correction that grows as the square root of the Ohnesorge number and, at an Ohnesorge number of order one, a drop that no longer oscillates at all but creeps back to round. That is a borrowed result here; the weak-damping limit is the one computed, and for water drops it is the one that applies.

Put the pitch and the ring-down together and a drop has exactly two properties that matter to it, one from each formula, and one dimensionless group that relates them. The quality factor is, up to a number, the reciprocal of the Ohnesorge number: $Q = \sqrt2/(5\,\mathrm{Oh})$ for the fundamental. A drop of honey the same size has an Ohnesorge number thousands of times larger and does not ring at all; a drop of mercury rings longer than water. Nothing else about the liquid enters.

The higher modes damp much faster, as $(l-1)(2l+1)$: thirteen times faster for $l = 6$ than for the fundamental, while ringing only 5.5 times higher. Their quality factors are therefore smaller by more than two, and they die in a few cycles. A swing in a fluid loses its energy where the strain is, and a mode with more lobes has its strain packed into a thinner shell near the surface.

A drop as an instrument #

The two formulas run backwards, and that is how they are most used. Measure a drop’s pitch and its size, and Rayleigh gives the surface tension, $\sigma = \rho a^3\omega_2^2/8$; measure how fast the ring dies, and Lamb gives the viscosity, $\nu = a^2/5\tau_2$. No container touches the liquid and nothing flows through a tube, which makes it the method of choice for liquids nothing can hold — molten metals at two thousand kelvin, levitated by electromagnetic or acoustic fields, and on orbiting laboratories where there is no gravity to flatten the drop. The levitating field perturbs the pitch slightly and its corrections are a subject of their own; the principle is the one computed here.

It also shows why the ring-down is a delicate measurement and the pitch is not. The pitch goes as the square root of the surface tension, so a per cent in frequency is two per cent in tension. The damping measures the viscosity directly, but it measures every other loss too — a trace of surfactant, a surface that is not quite clean, energy radiated into the surrounding gas — and a ring that dies too fast reads as a viscosity that is too high. The formula is exact for the model; the instrument is only as clean as the drop.

What could strike it #

A raindrop at terminal speed has a wake, and a wake that sheds vortices pushes on the body periodically. The standard account is that this is what keeps raindrops oscillating: the wake strikes the drop at, or near, the drop’s own pitch, and a lightly damped oscillator driven at resonance swings hard.

That is a claim about two frequencies, and both can be put on one graph. The drop’s pitch is Rayleigh’s. The shedding frequency is $f = \mathrm{St}\,U/d$, with $U$ the terminal speed — taken here from a sphere-drag calculation for a drop of the same diameter — and a Strouhal number of 0.2, the textbook value for a bluff body’s wake and borrowed rather than computed. A sphere’s wake does not shed at all below a Reynolds number of about 270, another borrowed number, so shedding is drawn only for drops above it, which is drops of about a millimetre and more.

The two lines do not cross. At every size that sheds, the wake is faster than the fundamental: 2.3 times at a millimetre and a half, 5.4 times at two millimetres, 18 times at six. The pitch falls steeply with size and the shedding falls slowly, because a bigger drop falls faster, so the gap widens as the drops get bigger. The shedding meets the drop’s spectrum not at the fundamental but at the fifth and sixth modes, near one and a half to two millimetres — and those are exactly the modes that damp in a few cycles.

The sphere-drag terminal speeds are too high for the largest drops, which flatten and fall slower — a four-millimetre drop really falls at about 8.8 m/s rather than the 10.2 used here — and the Strouhal number of a sphere’s wake is itself uncertain by a few tens of per cent across this range. Neither moves the conclusion: a slower drop sheds more slowly by the same factor, and to bring the shedding down onto the fundamental at two millimetres would take a Strouhal number near 0.04, five times below the value that defines the wake.

A bell does not need to be struck at its own note #

The oscillation is real and observed. If the wake does not strike at the fundamental, what keeps it going?

The answer is in the quality factor. A lightly damped oscillator responds to any forcing whose spectrum has energy at its own frequency, and it responds there in proportion to its quality factor — which is 76. A bell struck with a hammer rings at its own pitch, not at the hammer’s, because the blow is short and contains every frequency. The wake of a falling drop is not a clean periodic street; above a Reynolds number of a few hundred it is unsteady at many frequencies at once, and whatever share of that buffeting lies near 122 Hz is amplified seventy-six-fold into the fundamental while the rest is ignored. A resonator this sharp picks its own note out of broadband noise. It does not need the noise to be tuned.

That is an argument, not a calculation: nothing here computes the wake’s spectrum at 122 Hz, and the real excitation may be something else entirely. Collisions between drops are the other candidate, and in heavy rain a drop meets smaller drops often enough to be struck again before it has stopped ringing — at a quality factor of 76 and a period of 8 milliseconds, the ring lasts a fifth of a second. What the numbers do settle is the negative: the fundamental is not driven at resonance by shedding at the wake’s own Strouhal number, and an account that says it is has not put the two frequencies on one axis.

A second place the same pitch appears #

The capillary time sets more than the ring. A drop that lands on a surface it does not wet — a water drop on a lotus leaf, or on a coated plate — spreads, recoils and leaves, and the time it spends in contact was measured by Richard, Clanet and Quéré in 2002 to be $2.6\sqrt{\rho a^3/\sigma}$, independent of how fast the drop arrived. That number is borrowed. Rayleigh’s period for the fundamental is $2\pi\sqrt{\rho a^3/8\sigma} = 2.22\sqrt{\rho a^3/\sigma}$. The bounce lasts about one swing of the drop’s own fundamental, and it does not depend on the impact speed for the same reason a bell’s pitch does not depend on how hard it is struck.

That is the surprising connection this subject offers: a contact time on a surface, a ring-down in the air, the time a jet takes to break, and the size a hanging drop reaches before it falls are all set by the same balance of surface tension against inertia, and most of them share its pitch.

What the picture cannot show #

The mode figure draws the swings at a fifth of the radius, which is far outside the range the formula covers — a swing that large has a pitch lower by several per cent and is visibly asymmetric in time. The frequencies and the damping belong to swings of a per cent or less, and the drawing is there to show the shapes, not to be measured.

Nor does any figure show a falling drop oscillating. A raindrop is flattened on average by its own pressure field and swings about that flattened shape, not about a sphere, and its modes are therefore slightly split — the axisymmetric and transverse versions of the fundamental ring at slightly different frequencies once the drop is not round. The calculation here is the round drop’s, which is the limit the split modes approach as the drop gets small.

Where the model stops #

Small swings. Every result is linear in the amplitude. Large oscillations have a pitch that falls with amplitude and a shape that spends longer oblate than prolate, and they can break the drop, which is one of the routes by which a drop is torn apart.

A clean surface. A trace of surfactant makes the surface resist being stretched, which adds a damping mechanism far stronger than the bulk viscosity for small drops. The damping computed here is for clean water, and measured ring-downs of real drops are often faster for that reason.

Borrowed wake numbers. The Strouhal number and the shedding threshold are measurements for rigid spheres, and a drop is neither rigid nor, at the larger sizes, a sphere. The comparison is robust to both by the margins stated above; it would not survive an error of a factor of five in the Strouhal number.

The convention the numbers depend on #

The mode number $l$ counts the Legendre polynomial describing the shape, so $l = 2$ is the fundamental and has two lobes; some accounts count the fundamental as the first mode. Frequencies are in hertz, $f = \omega/2\pi$. The quality factor is $\omega\tau/2$ with $\tau$ the amplitude’s time constant, which is the number of radians of oscillation per e-folding of the energy. The Strouhal number is built on the drop’s diameter and its terminal speed.

Who found it, and when #

Rayleigh derived the frequencies in 1879, in the same paper series in which he explained the breakup of jets, and by exactly the energy argument recomputed above. Lamb added the damping in 1881, by the dissipation argument used here, and it stayed in every edition of his Hydrodynamics. Chandrasekhar solved the full viscous problem in 1959, and Reid, independently, in 1960. That raindrops oscillate was established by photographing them, and the suggestion that eddy shedding drives the oscillation dates from Gunn’s measurements of 1949; the contact time of a bouncing drop is Richard, Clanet and Quéré’s, from 2002.

Still open: what the wake puts at the fundamental #

The argument above replaces resonance with broadband excitation and does not compute the broadband part. What it needs is the spectrum of the unsteady pressure a sphere’s wake exerts on the sphere, at a Reynolds number of several hundred, evaluated at the drop’s fundamental — a frequency a fifth of the shedding frequency. Multiplied by the quality factor it would give the rms amplitude a raindrop should ring at, and that can be compared with photographs.

The calculation that would follow takes a measured or simulated wake-pressure spectrum for a sphere, filters it through the drop’s l = 2 response with the damping computed here, and asks whether the amplitude that results is the few per cent observed for two-millimetre drops, or whether collisions are needed to account for it. Beside it is the question the split modes raise: whether the transverse version of the fundamental, which a wake that sheds from alternating sides pushes more directly, is the one that rings — and at what frequency, once the flattening has split it from the axisymmetric one.