IFST · Parabolic ↔ Hyperbolic

Kelvin’s cable, Heaviside’s cable

τ ∂²u∂t² + ∂u∂t = D ∂²u∂x²

τ = L/R (inductance on) τ = 1.00

Time t t = 3.00

pure diffusion reference

TypeHYPERBOLIC

b² − 4ac = 4Dτ4.00

Front speed v = √(D/τ)1.000

Regime at this tt/τ = 4.0

Snapshot u(x, t)line of length 40, ends grounded (u = 0)

telegraph solution τ = 0 reference fronts x₀ ± (w + vt)

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

characteristics dx/dt = ±v current t

τ = 0: parabolic

The discriminant is zero and the only characteristics are the lines t = const. The initial pulse is zero for |x − x₀| > w, yet at t = 0⁺ the blue curve is already non-zero at every x. Only u(x,0) can be prescribed.

τ > 0: hyperbolic

Any τ, however small, gives two real characteristics dx/dt = ±√(D/τ). Outside the cone |x − x₀| < w + vt, u is exactly zero. Now u_t(x,0) is also needed; here it is zero, so the pulse splits into two halves that leave a wake behind them.

t ≫ τ: diffusion returns

The ∂u/∂t term damps the waves as e^−t/2τ. At late times the yellow curve approaches the blue one. This is how strongly scattered wave energy, such as coda, ends up obeying a diffusion law.

How to use it in class

One knob, τ, takes the same equation from the parabolic row of the classification table to the hyperbolic row. The initial pulse and boundary conditions never change. Each “Show this” button sets the demo to the state described and switches back to it.

Start with Kelvin’s cable

At τ = 0 the equation is pure diffusion. The initial pulse is zero outside |x − x₀| < w, yet a moment later u is non-zero along the whole line: the signal is felt everywhere instantly. This is the “spreading flow” row of the table.
Add a little inductance

Go to τ = 0.1. The type jumps to hyperbolic and a light cone appears with v = √(D/τ) ≈ 3.2, yet at t = 3 the yellow curve is almost on top of the blue one. Ask the class what changed. The answer is the characteristics, not the look of the solution.
Make it Heaviside’s cable

At τ = 10, press Play from t = 0. The pulse splits into two halves that travel at v ≈ 0.32 and damp slowly as e^−t/2τ, leaving a wake between them. Outside the cone u is exactly zero, while the blue diffusion reference has already leaked everywhere.
Watch the reflections

At τ = 0.1 the fronts reach the grounded ends near t ≈ 6. On the space–time map the characteristics bounce off the walls. Change the speed to 0.5× to follow them.
Let diffusion come back

Set τ = 1 and drag t well past τ. The waves damp out and the yellow curve settles onto the blue one. This leads into the slide on diffusion via strong scattering: at times long compared with the scattering time, wave energy obeys a diffusion law.

Points to draw out

Type is about characteristics. The discriminant b² − 4ac = 4Dτ jumps from 0 to positive as soon as τ > 0, but the solution changes continuously. The type tells you how information propagates, not what a snapshot looks like.
Initial data. Parabolic needs only u(x,0). Hyperbolic also needs u_t(x,0); here it is zero, which is why the pulse splits symmetrically.
Singular limit. Setting τ = 0 lowers the order in time. That is why the limit is continuous for t ≫ τ but not near t = 0.
Reading the panels. Left: the telegraph solution (yellow), the τ = 0 reference (blue dashed) and the fronts x₀ ± (w + vt) (coral). Right: u over (x, t), coloured by √|u| so weak tails show, with the characteristics dx/dt = ±v overlaid.

Why “telegraph”?

A cable with resistance R, inductance L, capacitance C and leakage G per unit length obeys the telegrapher’s equations. Set G = 0 and eliminate the current:

LC∂²V∂t²+RC∂V∂t=∂²V∂x²

Dividing by RC gives exactly the equation above, with

τ = L/RD = 1/(RC)v = 1/√(LC)

So the slider holds the cable fixed (R and C, hence D) and switches on its inductance.

1855
W. Thomson (Kelvin), “On the theory of the electric telegraph”. He neglects L: the cable is an RC line and the voltage diffuses. Signal delay grows with the square of the length, his “law of squares”. This is τ = 0.
1857
G. Kirchhoff includes self-induction for an overhead wire and finds that signals propagate at a speed close to that of light.
1858
The first transatlantic cable carries messages for about three weeks. Pulses arrive smeared out, as Kelvin’s diffusion predicts, and the cable fails after very high voltages are applied to push signals through.
1876–87
O. Heaviside writes the equations with R, L, C, G. He shows that inductance turns diffusion into damped waves, and that a line with R/L = G/C transmits without distortion. This is τ > 0.
~1900
Pupin and Campbell put Heaviside’s idea to work: loading coils add inductance to long telephone lines.
1948
C. Cattaneo, “Sulla conduzione del calore” (Modena): a heat flux that relaxes over a time τ gives the same equation for temperature, removing the infinite speed of Fourier’s law.
1951 · 1974
S. Goldstein and M. Kac: a particle moving at speed v that reverses direction at random, with mean time 2τ between reversals, has a density obeying this equation. This is the scattering picture behind “diffusion via strong scattering”.

Model: D = 1, line 0 ≤ x ≤ 40 with u = 0 at both ends, initial pulse u = cos²(π(x − x₀)/2w) for |x − x₀| < w (x₀ = 20, w = 1) and zero elsewhere, u_t(x,0) = 0. Exact solution by expansion in 400 sine modes; each mode is a damped oscillator τ a″ + a′ + D k² a = 0 (overdamped for long wavelengths, underdamped for short ones). Keys: Space play/pause, ←→ on a slider to step.

τ = 0: parabolic

τ > 0: hyperbolic

t ≫ τ: diffusion returns

How to use it in class

Start with Kelvin’s cable

Add a little inductance

Make it Heaviside’s cable

Watch the reflections

Let diffusion come back

Points to draw out

Why “telegraph”?