IFST · Parabolic ↔ Hyperbolic · Heat

Fourier’s heat, Cattaneo’s heat

τ ∂²θ∂t² + ∂θ∂t = κ ∂²θ∂x²

τ = relaxation time of the heat flux τ = 1.00

Time t t = 3.00

Fourier reference (τ = 0)

TypeHYPERBOLIC

b² − 4ac = 4κτ4.00

Second-sound speed c = √(κ/τ)1.000

Regime at this tt/τ = 4.0

Temperature excess θ(x, t)bar of length 40, insulated ends (q = 0)

Maxwell–Cattaneo Fourier (τ = 0) fronts x₀ ± (w + ct)

Space–time mapcolour ∝ √|u|, so weak tails show

characteristics dx/dt = ±c current t

τ = 0: Fourier’s law

The flux follows the gradient instantly, q = −k ∂θ/∂x. A hot spot that is zero outside |x − x₀| < w warms the whole bar at t = 0⁺: heat travels at infinite speed.

τ > 0: second sound

The flux lags the gradient by τ. Heat now moves as two damped fronts at c = √(κ/τ), and outside the cone |x − x₀| < w + ct the bar is still at its initial temperature.

t ≫ τ: Fourier returns

After many relaxation times the fronts have died out and the profile follows Fourier’s law. The ends are insulated, so total heat is conserved and the bar ends at a uniform temperature.

How to use it in class

One knob, τ, takes heat conduction from Fourier’s law (parabolic) to heat that travels as a damped wave (hyperbolic). The diffusivity, the initial hot spot and the insulated ends never change; only the lag between flux and gradient does. Each “Show this” button sets the demo to the state described and switches back to it.

Start with Fourier

At τ = 0 the hot spot spreads by diffusion and the whole bar warms at once. Ask the class whether that is physically acceptable.
Add a short relaxation time

At τ = 0.1 the type becomes hyperbolic with fronts at c ≈ 3.2, yet at t = 3 the curve is almost identical to Fourier. This is the situation in ordinary solids, where τ is of the order of picoseconds: the correction exists but cannot be seen.
Heat as a wave

At τ = 10, press Play from t = 0. Two temperature pulses travel at c ≈ 0.32 and damp slowly as e^−t/2τ. This is second sound.
Insulated ends

At τ = 0.1 the fronts reach the ends near t ≈ 6 and reflect without changing sign, because no heat can leave. On the cable page the grounded ends invert the pulse instead.
Back to Fourier

Set τ = 1 and drag t well past τ. The curve settles onto the Fourier one and both approach the same uniform temperature, set by the heat in the initial spot.

Points to draw out

Type is about characteristics. The discriminant b² − 4ac = 4κτ jumps from 0 to positive as soon as τ > 0, while the solution changes continuously.
Carriers and collisions. In the two-stream picture, heat carriers move at ±c and reverse direction at random, with mean time 2τ between reversals. In 1D this gives the equation exactly, with κ = c²τ. Fourier’s law is the limit of many collisions per unit time; few collisions mean near-ballistic transport.
Where second sound is seen. In superfluid helium (predicted by Tisza and Landau, observed by Peshkov in 1944), in very pure crystals such as NaF and Bi at a few kelvin, and in graphite above 100 K (2019). In rocks τ is of the order of picoseconds, so geophysical heat flow is Fourier.
Limits of the model. The equation is exact for 1D two-stream transport but only an approximation in 2D and 3D, where it concentrates too much energy in a sharp front. Its consistency with the second law has also been debated.
Reading the panels. Left: Maxwell–Cattaneo (yellow), Fourier (blue dashed) and the fronts x₀ ± (w + ct) (coral). Right: θ over (x, t), coloured by √|θ|, with the characteristics dx/dt = ±c.

Where the extra term comes from

Energy conservation in a bar, with θ the temperature excess and q the heat flux:

ρc∂θ∂t=−∂q∂x

Fourier sets q = −k ∂θ/∂x. Maxwell and Cattaneo let the flux relax towards that value over a time τ:

τ∂q∂t+q=−k∂θ∂x

Eliminating q gives exactly the equation above, with

τ = flux relaxation timeκ = k/(ρc)c = √(κ/τ)

So the slider holds the diffusivity fixed and lets the flux lag the gradient.

1822
J. Fourier, Théorie analytique de la chaleur: q = −k ∂θ/∂x, so heat diffuses at infinite speed. This is τ = 0.
1867
J. C. Maxwell, “On the dynamical theory of gases”, obtains a relaxation term for the heat flux and drops it as negligible for gases. The same paper introduces the Maxwell viscoelastic body.
1938–41
L. Tisza and L. Landau predict temperature waves, second sound, in superfluid helium.
1944
V. Peshkov observes second sound in He II.
1948 · 1958
C. Cattaneo, “Sulla conduzione del calore” (Modena), restores the term and removes the infinite speed; P. Vernotte reaches the same law independently.
1951 · 1974
S. Goldstein and M. Kac: a particle moving at speed c that reverses direction at random obeys this equation, the two-stream picture of heat carriers.
1970s
Second sound is measured in very pure NaF and Bi crystals at a few kelvin.
2019
Second sound is observed in graphite above 100 K.

Model: κ = 1, bar 0 ≤ x ≤ 40 with insulated ends (q = 0, hence ∂θ/∂x = 0), initial hot spot θ = cos²(π(x − x₀)/2w) for |x − x₀| < w (x₀ = 20, w = 1) and zero elsewhere, no initial flux (q = 0, hence θ_t(x,0) = 0). Exact solution by expansion in 400 cosine modes; the k = 0 mode carries the conserved total heat, and each other mode is a damped oscillator τ a″ + a′ + κ k² a = 0. Keys: Space play/pause, ←→ on a slider to step.

τ = 0: Fourier’s law

τ > 0: second sound

t ≫ τ: Fourier returns

How to use it in class

Start with Fourier

Add a short relaxation time

Heat as a wave

Insulated ends

Back to Fourier

Points to draw out

Where the extra term comes from