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Second Sound

How Heat Can Travel as a Wave Instead of Simply Diffusing

Quick Answer

Below the lambda transition, liquid helium-4 behaves as a superfluid described by a two-fluid hydrodynamic model: a superfluid component with no entropy and a normal component that carries entropy.

In ordinary first sound, the two components move largely together, producing a density and pressure wave. In ideal second sound, they move largely in opposite directions. Their mass currents nearly cancel while entropy and temperature oscillate strongly.

The result is a propagating thermal wave whose speed is set by superfluid density, normal density, entropy and specific heat rather than by the ordinary elastic compressibility alone.

Physical Review — Second Sound in Liquid Helium II →

What You Will Learn

• Why Fourier diffusion is not the only possible heat-transport law.
• What the superfluid two-fluid model means.
• Why the normal component carries entropy.
• How counterflow creates a temperature wave.
• Why first sound and second sound are different modes.
• What determines second-sound speed.
• Why the mode disappears at the lambda point.
• How vortices and mutual friction damp second sound.
• How second sound measures superfluid properties.
• Why heat pulses can reflect and form resonances.
• How ultracold Bose and Fermi gases extend two-fluid sound physics.
• Why phonon-hydrodynamic second sound in solids is related in behaviour but not identical in mechanism.

Part 1 — The Naive Model: Heat Only Diffuses

Fourier’s law describes heat current as proportional to the negative temperature gradient:

q = −κ∇T.

Combine this with energy conservation and temperature obeys a diffusion equation.

Diffusion smooths gradients. A localized pulse broadens as it spreads and has no single sharply defined propagation speed.

Second sound requires a regime where the relevant heat-carrying degrees of freedom behave collectively and inertially enough to support a wave equation instead.

Part 2 — Superfluid Helium Has Two Hydrodynamic Components

Below about 2.17 K at saturated vapour pressure, helium-4 enters the He II superfluid phase.

Landau’s two-fluid hydrodynamics describes the same liquid using two interpenetrating components:

• superfluid component: density ρ_s, velocity v_s, zero entropy in the ideal two-fluid model;
• normal component: density ρ_n, velocity v_n, carries entropy and thermal excitations.

They are not two liquids separated into different containers. They are two hydrodynamic degrees of freedom of the same quantum fluid.

Part 3 — First Sound Moves Mass Together

In ordinary acoustic first sound, pressure and density oscillate.

The normal and superfluid components move predominantly in phase, so there is a large net mass current.

This is the superfluid analogue of familiar pressure sound, though the exact thermodynamics differs from an ordinary liquid.

Part 4 — Second Sound Moves the Components Against Each Other

In ideal second sound, the two components counterflow:

ρ_sv_s + ρ_nv_n ≈ 0.

The total mass current can therefore be small while the normal component transports entropy in one direction and the superfluid component moves oppositely to maintain the mass balance.

Temperature and entropy oscillate strongly even when density oscillation is comparatively weak.

first sound is mainly a density wave; second sound is mainly an entropy/temperature wave.

Part 5 — Why a Temperature Difference Creates Counterflow

Heating one region creates an entropy and pressure response.

The normal excitations carry entropy away from the heated region. Superfluid can move oppositely, producing a counterflow with very little net mass movement.

Because the two components have inertia and are coupled by thermodynamics, the disturbance can overshoot and oscillate rather than simply relax diffusively.

Part 6 — The Speed Depends on Superfluid Fraction and Thermodynamics

In the limit of negligible thermal expansion, the second-sound speed can be written schematically as

u₂² ≈ (ρ_s/ρ_n) · (s²T/c_p)

where s is entropy per unit mass and c_p is specific heat per unit mass.

The exact formula depends on thermodynamic conventions and coupling to density fluctuations, but the message is clear: second sound is controlled by the relative amounts and thermodynamics of the two fluid components.

Part 7 — Why Second Sound Vanishes at the Lambda Point

As temperature approaches the lambda transition from below, the superfluid density ρ_s tends toward zero.

The two-component counterflow mode therefore loses its restoring structure and the second-sound speed falls toward zero at the transition.

Classic 1947 measurements found the speed rising from zero near the lambda point to around 20 m/s over part of the He II temperature range.

Part 8 — A Heater and Thermometer Can Make a Resonant Sound Experiment

A periodically driven heater can launch temperature oscillations into superfluid helium.

If the container length matches an integer number of half-wavelengths, standing second-sound resonances form.

Measuring resonant frequency and geometry gives the propagation speed just as an acoustic tube can measure ordinary sound speed.

Part 9 — Why Vortices Dampen the Wave

In superfluid helium, quantized vortex lines interact with the normal component.

This interaction creates mutual friction, which dissipates relative motion between normal and superfluid components.

Second sound is therefore strongly attenuated by vortex tangles and has become an important probe of quantum turbulence.

Part 10 — Second Sound Is a Measurement Tool, Not Just a Curiosity

Because its speed depends on entropy, specific heat and superfluid density, second-sound measurements can constrain thermodynamic properties of He II.

Attenuation can reveal vortex density, mutual-friction strength and dissipative dynamics.

A wave phenomenon therefore becomes a precision diagnostic of an invisible quantum fluid state.

Part 11 — First and Second Sound Can Couple

The clean textbook split—first sound equals pressure and second sound equals temperature—is an approximation.

When thermal expansion is non-negligible, pressure and temperature fluctuations couple. Each mode can contain some of both.

This is a useful model-limit lesson: names identify dominant character, not perfectly isolated physical variables under all conditions.

Part 12 — Quantum Gases Also Have Two-Fluid Sound

Two-fluid hydrodynamics is not unique to liquid helium.

Ultracold Bose gases and strongly interacting Fermi gases can support first- and second-sound modes when they enter a superfluid hydrodynamic regime.

Modern experiments use density imaging and controlled perturbations to watch these collective modes directly.

Physical Review Letters — First and Second Sound in a Compressible 3D Bose Fluid →

Part 13 — 2026: “Second Sound” Also Appears in Solids—but the Microscopic Mechanism Is Different

The term second sound is also used for wave-like heat propagation in certain crystals when phonon–phonon collisions conserve momentum strongly enough that the phonon gas behaves hydrodynamically.

In that regime, heat can propagate as a collective temperature wave rather than ordinary diffusion.

In 2026, room-temperature second-sound behaviour was reported in isotopically pure graphite.

Nature Communications (2026) — Room-Temperature Second Sound in Isotopically Pure Graphite →

That solid-state phenomenon is related by the common idea of hydrodynamic thermal waves, but it does not use the helium two-fluid superfluid/normal counterflow mechanism. The mechanisms must remain separately named.

Part 14 — Why This Is Not Ballistic Heat Transport

Ballistic transport occurs when heat carriers travel long distances with very few collisions.

Second sound is a collective hydrodynamic mode. Collisions can be essential because they establish local equilibrium and coordinated flow.

A sharp thermal pulse by itself therefore does not prove second sound; scientists must identify the propagation law and collision regime.

Failed Model → Better Model

Naive model	Why it fails	Better model
Heat always diffuses.	Collective hydrodynamic modes can carry entropy as waves.	Compare diffusion time with hydrodynamic collision and inertia scales.
Second sound is ordinary pressure sound.	The dominant oscillation is entropy/temperature counterflow.	Track normal and superfluid velocities separately.
The normal and superfluid components are two separate liquids.	They interpenetrate as hydrodynamic components of one quantum fluid.	Use Landau two-fluid variables.
Any wave-like heat pulse is the same second sound.	Superfluid and phonon-hydrodynamic mechanisms differ.	Identify the carriers and governing hydrodynamics.

How Do We Know?

• Launch a controlled heat pulse into He II.
• Measure arrival time with fast thermometry.
• Drive periodic heating and map resonance frequencies.
• Measure pressure and temperature amplitudes separately.
• Vary temperature toward the lambda transition.
• Compare measured velocity with two-fluid thermodynamics.
• Introduce controlled rotation or vortices and measure attenuation.
• For ultracold gases, image density and entropy-related perturbations.
• For solids, distinguish phonon hydrodynamics from ballistic and diffusive heat transport.

Observation vs Inference

• Observation: temperature disturbances propagate with a finite wave velocity in He II.
• Measurement: the speed changes strongly with temperature and vanishes near the lambda point.
• Inference: counterflow of entropy-carrying normal fluid and superfluid supports the mode.
• Measurement: vortices attenuate second sound through mutual friction.
• Boundary: solid-state phonon second sound shares wave-like heat propagation but not the same two-fluid microscopic mechanism.

Checkpoint Questions

1. How does diffusion differ from wave propagation?
2. What are the two components of He II hydrodynamics?
3. Which component carries entropy in the ideal model?
4. What distinguishes first from second sound?
5. Why can net mass current be small in second sound?
6. Why does second-sound speed fall near the lambda point?
7. How do vortices damp second sound?
8. Why is second sound useful as a measurement tool?
9. Why is solid-state phonon second sound not identical to helium second sound?
10. Why is ballistic heat transport not automatically second sound?

Answer Key

Open after attempting the questions

1. Diffusion broadens and smooths a pulse without one propagation speed; a wave propagates collectively with a characteristic velocity.
2. Superfluid and normal hydrodynamic components.
3. The normal component.
4. First sound is mainly density/pressure motion; second sound is mainly entropy/temperature counterflow.
5. The two component mass currents oppose and can nearly cancel.
6. The superfluid density tends to zero.
7. Mutual friction transfers and dissipates relative motion between vortices and the normal component.
8. Its speed and attenuation encode superfluid thermodynamics and vortex information.
9. In solids the carriers are hydrodynamic phonons rather than superfluid/normal counterflow.
10. Ballistic carriers undergo too few collisions; second sound is a collective hydrodynamic regime.

Primary Science Bridge

• heat moves from place to place;
• some materials behave differently at very low temperature;
• waves transport patterns without transporting all the matter with them;
• different parts of one material can move in different ways;
• temperature itself can participate in an oscillation.

Secondary and JC Bridge

Core idea	Higher-resolution route
Heat transfer	Diffusive versus hydrodynamic transport
Fluid motion	Two-fluid hydrodynamics
Sound	Coupled density and entropy modes
Superfluidity	Normal and superfluid density
Dissipation	Mutual friction and vortices
Solids	Phonon hydrodynamics

Unfamiliar Transfer Challenge

A crystalline sample shows a heat pulse travelling faster and with less broadening than Fourier diffusion predicts. The researcher calls it second sound.

What should be checked? Determine whether phonon collisions establish hydrodynamic collective flow, rule out simple ballistic propagation, measure wave velocity and attenuation, and test how isotopic disorder or temperature changes the regime.

Deep Science Window — Two Coupled Conserved Quantities

Wave modes emerge when conservation laws and restoring forces couple dynamical variables. In a superfluid, mass conservation, entropy transport and two independent velocity fields allow two sound eigenmodes. Diagonalising the coupled hydrodynamic equations produces first and second sound rather than one universal acoustic wave.

Deep Science Window — Heat Can Become Hydrodynamic

Fourier diffusion assumes heat current responds locally and almost instantaneously to a temperature gradient. In hydrodynamic regimes, the heat-carrying excitations have collective momentum and inertia. The constitutive law then acquires time dependence and can support propagating temperature waves.

Evidence Boundaries

• Second sound ≠ ordinary acoustic first sound.
• Temperature wave ≠ violation of heat flowing toward equilibrium.
• Two-fluid model ≠ two spatially separated liquids.
• Superfluid second sound ≠ every phonon-hydrodynamic heat wave.
• Wave-like pulse ≠ automatically hydrodynamic second sound.
• Ideal counterflow ≠ pressure and density fluctuations are exactly zero in every real experiment.

Manual Summary — KNOW → CONNECT → EXPLAIN → APPLY → CHECK

KNOW: superfluid, normal component, entropy, counterflow, first sound, second sound, mutual friction.

CONNECT: entropy transport to normal flow, normal flow to compensating superflow, and counterflow to a propagating temperature mode.

EXPLAIN: why heat can travel through a quantum fluid as a wave rather than only diffuse.

APPLY: distinguish superfluid second sound from ballistic or phonon-hydrodynamic thermal waves in an unfamiliar system.

CHECK: identify the carriers, hydrodynamic variables, collision regime and measured propagation law before using the label.

Research Sources and Further Reading

• Physical Review — Second Sound in Liquid Helium II
• Physical Review — The Theory of Liquid Helium and the Two-Fluid Model
• Physical Review Letters — First and Second Sound in a Compressible 3D Bose Fluid
• Nature Communications (2026) — Room-Temperature Second Sound in Isotopically Pure Graphite