The heat that arrives as a wave

An ordinary liquid has one velocity field and one wave equation, and the wave is sound: a compression that travels, carrying pressure and density.

Liquid helium below $2.17$ K has two velocity fields, because the two-fluid description gives it a superfluid component that carries no entropy and no viscosity and a normal component that carries both. Two velocity fields and two densities give two wave equations, and the second one has no analogue in any ordinary liquid.

The two modes #

Write down conservation of mass and of entropy, and the equations of motion of the two components — the superfluid driven by gradients of chemical potential, the normal one by pressure and by the entropy it carries. Linearise, and the result is a pair of coupled wave equations that decouple into two independent modes.

In the first mode the two components move in the same direction with the same velocity. The total density oscillates, the pressure oscillates, the entropy per unit mass does not. That is ordinary sound, travelling at $238$ m/s in helium II, and it behaves exactly as sound in any other liquid.

In the second mode they move in opposite directions with $\rho_s v_s + \rho_n v_n = 0$. The total mass flux is zero everywhere, so the density does not change and there is no pressure wave at all. What changes is the proportion of the two components — and since the proportion is a function of temperature, the oscillating quantity is the temperature.

That is second sound. It is not heat conduction dressed up: it is a genuine wave mode of the medium, with a wave equation, a speed, reflections at boundaries and standing modes in a resonator.

Why two components give two modes #

The counting is general and worth separating from helium.

A single fluid has two dynamical fields that matter for sound — a density and a velocity — related by one conservation law and one equation of motion. Eliminating between them gives one second-order wave equation and therefore one mode.

A two-component fluid has four: two densities and two velocities. But the two densities are not independent — the total is fixed by the equation of state at a given temperature and pressure, and the split between them is a function of temperature — so the independent variables are the total density, the temperature, and the two velocities. Four fields, four equations, and the system factorises into two wave equations.

That is why the second mode exists at all, and it is why the same thing happens in any system with two weakly coupled reservoirs of momentum and entropy. It also predicts what the second mode must be: since the first uses up the density oscillation, the second must leave the density alone, and the only remaining thing to oscillate is the temperature.

Nothing about helium entered that argument. What helium supplies is a case in which the coupling between the two components is weak enough for the second mode to propagate rather than being damped out within a wavelength, and the weakness of the coupling is exactly the absence of viscosity in one of them.

The speed, computed rather than fitted #

The two-fluid equations give the speed as

$$c_2^2 = \frac{\rho_s}{\rho_n}\cdot\frac{S^2 T}{C},$$

with $S$ the entropy per unit mass and $C$ the heat capacity. Every quantity in it is a property of the liquid, and below about six-tenths of a kelvin every one of them can be computed rather than measured, because there the normal component is nothing but a gas of phonons.

For a phonon gas the entropy density is $\tfrac43 u/T$ and the heat capacity is three times the entropy, both following from the energy density going as $T^4$. Substituting, and using the fact that at low temperature $\rho_s$ is essentially the whole density,

$$c_2 \to \frac{c_1}{\sqrt3}.$$

The generator computes the full expression at each temperature and checks that it approaches $238/\sqrt3 = 137.4$ m/s at twenty millikelvin, which it does to three figures.

That result is the reason to believe the two-fluid model rather than merely to use it. A factor of $\sqrt3$ is what a gas of massless excitations moving in three dimensions gives — the same factor that appears in the pressure of a photon gas and in the relation between the sound speed and the particle speed in any ultrarelativistic gas. Nothing was fitted; the number is a consequence of the excitations being phonons of the very medium they are propagating in.

The speed of ordinary sound is set by the properties of the medium, and second sound is the same kind of statement about a different medium — a gas of excitations moving through the liquid rather than the liquid itself. So the speed is computed rather than fitted, from the entropy and the two densities, and it comes out around 20 m/s against 240 for first sound. Two waves, one liquid, speeds differing by a factor of twelve.

Above a kelvin the phonons are joined by rotons, which are excitations with a minimum in their energy at finite momentum, and the arithmetic changes: the measured $c_2$ falls to about $20$ m/s over most of the temperature range where the effect is studied. The model here is the low-temperature one and is drawn only where it holds, which is what makes it a computation rather than a curve fitted to data.

A pulse with an arrival time #

The operational difference between a wave and a diffusion is the existence of an arrival, and it is what settled the matter experimentally.

Release a heat pulse at one end of a channel of helium II and a thermometer at the other end reads nothing at all for a while, then a sharp rise. The delay is the path length divided by $c_2$, and it is reproducible, and it changes when the temperature is changed because $c_2$ does.

In an ordinary liquid the same pulse produces a reading that begins to rise immediately — exponentially small, but nonzero — and peaks much later. A diffusive disturbance has no front, so there is no arrival to time, and the difference between the two curves in the figure is not one of degree.

Peshkov did exactly this in 1944 and measured the speed. That measurement is what turned the two-fluid model from an interpretive scheme into a theory with a checkable prediction, because Landau had predicted the speed from the excitation spectrum and the number agreed.

A helium-filled cavity driven by a heater at one end has resonances whose frequencies give the wave speed directly, and the mode structure is the ordinary one — nodes at the ends, a fundamental and its overtones. That is the cleanest evidence that the temperature oscillation is a genuine wave: it obeys the same boundary conditions and the same mode counting as anything else in a box.

Once there is a wave there is a resonator. A cavity filled with helium II, driven by a heater and monitored by a thermometer, shows resonances at the frequencies where a whole number of half-wavelengths fit, and the resulting speed measurement is far more precise than timing a pulse. That is now the standard way to measure $\rho_s/\rho_n$, which closes a satisfying loop: the model’s one parameter is measured by one of the model’s own predictions.

The numbers, side by side #

Putting the two modes’ properties next to each other makes the difference concrete.

Ordinary sound in helium II: speed $238$ m/s, oscillating quantity the pressure, amplitude generated by a moving piston, detected by a microphone, present in every liquid.

Second sound at $1.6$ K: speed about $20$ m/s, oscillating quantity the temperature, generated by a heater — a resistive film whose power is modulated — and detected by a thermometer, typically a carbon resistor whose resistance depends steeply on temperature. A piston generates almost no second sound and a heater generates almost no first sound, because the two modes are driven by different things.

That last point is the practical one. The two modes are orthogonal in the sense that a source which changes the density couples to the first and a source which changes the entropy couples to the second, so an experiment can excite one and ignore the other. It is the reason second sound was measurable at all in 1944 with the instruments of the time: nobody had to separate two overlapping signals, because the apparatus that made one made almost none of the other.

What is actually moving #

It is worth being careful about what the wave transports, because “a wave of temperature” invites a wrong mental picture.

Nothing is being conducted. In a steady heat flux the normal component streams away from the heater carrying entropy, and the superfluid streams back carrying none, so heat moves and mass does not — an entropy transported without a temperature gradient to drive it. That is the counterflow, and it is why helium II transports heat so effectively that it cannot be described by a conductivity: the transport rate depends on the geometry and the flux rather than on a material constant.

Second sound is that arrangement oscillating. The two components slosh past one another, the entropy piles up and thins out, and the temperature follows.

The most familiar demonstration of the whole business needs no apparatus. Liquid helium above the lambda point boils violently, with bubbles throughout, because heat reaches the surface by ordinary conduction and the interior superheats. Cool through $2.17$ K and the bubbling stops abruptly: the liquid goes still and clear, evaporating only from its free surface, because heat now reaches the surface by counterflow far faster than the interior can superheat. Anyone who has watched it has seen the transport mechanism change in a second.

Predicted, then measured, and the order mattered #

The history here is unusually clean and is worth the paragraphs, because it is a case where a theory was believed for a reason.

Tisza proposed the two-fluid picture in 1938, shortly after superfluidity was discovered, and it explained the qualitative facts: the viscosity that was zero in one experiment and not in another, the fountain effect, the enormous heat transport. Explaining known facts is cheap, and Landau was unconvinced by the picture’s foundation — Tisza had built it on Bose condensation, and helium is a strongly interacting liquid to which the ideal-gas calculation does not apply.

Landau rebuilt it in 1941 from an excitation spectrum instead, and in doing so predicted the second wave and computed its speed. The prediction was specific and testable and the number was not one anybody could have guessed: it depended on the entropy and heat capacity of the liquid in a particular combination.

Peshkov built the resonator and measured it in 1944, in wartime Moscow, and found the predicted speed and its predicted temperature dependence.

What that sequence bought was the model’s credibility for everything else. A framework that explains what is already known can always be adjusted; one that predicts a new wave at a new speed and is right has said something. Both Landau’s spectrum and the two-fluid equations were accepted after Peshkov’s measurement in a way they had not been before, and the subsequent programme — third sound, fourth sound, mutual friction, the critical velocity — was built on that acceptance.

Where it stops #

The two-fluid model is not two fluids. Every helium atom in the liquid is identical, and none of them is “a superfluid atom”. The two densities describe a single quantum fluid: the ground state, which carries no entropy, and the thermally excited quasiparticles, which carry all of it. The velocities $v_s$ and $v_n$ are the ground state’s flow and the excitation gas’s drift, and the model earns its keep by predicting several things from one measured ratio rather than by being a picture of anything.

The wave is damped, and by something interesting. In rotating or turbulent helium the superfluid contains quantised vortices, and the excitations scatter off them, coupling the two components that the second mode requires to move independently. That mutual friction damps second sound, and measuring the damping is the standard way of counting vortex line density — turning the wave into an instrument for the very thing that spoils it.

Second sound scattering off quantised vortices is the standard way of counting them, because the damping is proportional to their number. That is a measurement of a quantity with no classical counterpart, made with a temperature wave — and it is the sort of thing that makes second sound an instrument rather than a curiosity.

The amplitude has to be small. At large heat fluxes the counterflow velocity exceeds a critical value, turbulence sets in, and the linear analysis fails; the wave then steepens and can form a shock, since $c_2$ depends on temperature and a large-amplitude wave carries its own speed variation.

Below about half a kelvin it stops propagating. The wave needs the excitation gas to be collisional — the phonons must scatter off one another often enough within a wavelength for a hydrodynamic description to apply. As the temperature falls the phonon density falls as $T^3$ and the mean free path grows, and below the point where it exceeds the container the excitations simply stream ballistically from the heater to the wall. What arrives then is a pulse of phonons travelling at the first sound speed, not a temperature wave at $c_1/\sqrt3$, and the crossover between the two regimes is visible as the arrival time changing by a factor of root three as the sample is cooled.

And it is not unique to helium. Second sound is a general consequence of a system in which the entropy is carried by a gas of excitations that is only weakly coupled to the momentum, and it has been seen in solids — where the phonons themselves can propagate a temperature wave if their collisions conserve momentum well enough — in solid helium, in bismuth, and recently in graphite at surprisingly high temperature — wherever the collisions conserve momentum well enough that the excitation gas behaves hydrodynamically. It has also been seen in ultracold atomic gases, where the two-fluid description applies for the same reason and the speed can be computed from first principles.

The entropy second sound carries is the integral of the heat capacity, and it vanishes at absolute zero. So the wave itself must vanish there too: with no entropy to carry, there is nothing for the two components to oscillate in antiphase about. That is where the model stops, and it stops by running out of the quantity it transports rather than by breaking down.

What a wave of temperature does not mean #

Two readings of the phrase are worth blocking, because both are natural and both are wrong.

It does not mean that heat has a speed in the sense that a signal does. Second sound is a mode of a medium, and the medium has to be there: the wave exists in helium II and nowhere else in the apparatus. A pulse of heat delivered to a copper wire dipped in the helium does not travel along the wire at $20$ m/s; it diffuses, as heat does in copper, and only what enters the liquid becomes a wave.

Nor does it mean that the second law has been evaded. Second sound transports entropy back and forth reversibly in the same way ordinary sound transports momentum, and its damping — small but not zero — is where the irreversibility lives. Over a wavelength the process is very nearly reversible; over many, it is not, and the wave decays. The arrow of time is exactly where it always is, in the dissipation rather than in the equations.

What the phrase does mean is that a quantity usually governed by a parabolic equation is here governed by a hyperbolic one, and that is a change of kind rather than of degree. It is the same change that a relativistic correction to the heat equation is supposed to produce and never quite does, and helium II is the one place it happens for a mundane reason: there are two fluids, so there are two modes, and one of them has the temperature in it.

Where the entropy actually is #

One further point separates this wave from anything in an ordinary liquid, and it is the reason the two-fluid bookkeeping works at all.

All of the liquid’s entropy is carried by the excitations. The ground state — the condensate, loosely — has none, so the entropy per unit mass of the superfluid component is exactly zero rather than merely small. That is a strong statement and it is what makes the counterflow possible: a flow of superfluid carries mass and no entropy, so mass and heat can move in opposite directions without either being dragged by the other.

In every ordinary liquid the two are locked together. Moving heat means moving the molecules that carry it, so a temperature disturbance is inseparable from a density disturbance, and the two modes that helium II has collapse into one. The existence of a second sound is therefore a direct consequence of the entropy having been decoupled from the mass, and its speed going to zero at the lambda point is that decoupling disappearing as the superfluid fraction does.

The ladder from here #

Later rungs on this anchor: Landau’s excitation spectrum in full, with the roton minimum that changes the arithmetic above a kelvin and the critical velocity it implies; the derivation of both wave modes from the two-fluid equations with the coupling terms retained, giving the small mixing between them that a careful measurement can see; mutual friction quantified, and its use in counting quantised vortices; and third and fourth sound, which are what the same equations give in a film and in a superleak, where one of the two components is clamped by the geometry.

The neighbouring ladders are the liquid that will not slow down, where the two-fluid picture is set up; the fountain a lamp can drive, which is the steady version of this counterflow; and the summer that reaches the cellar in December, which is what a temperature disturbance does in every other material there is.