Thinking in Waves Episode 11 ／ Part V · Physical quantities as waves
Ground temperature and the skin of a wire obey the same equation
Heat does not travel as a wave ──
first order in time, second in space
The diffusion equation's dispersion relation is \(\omega=-iDk^2\) ── purely imaginary. It does not propagate; it decays in place.
Heat it periodically, though, and a frequency-dependent depth appears ── 12 cm for day/night, 2.2 m for the seasons.
And at 7 m underground, the seasons are shifted by half a year.
Part IV gave us three constraints on waves. From here we look at individual physical quantities with that eye.
First, heat ── and we find that the everyday phrase "heat travels" is not physically right.
01The dispersion relation goes purely imaginary
| Equation | Dispersion relation | Type of \(\omega\) | Behaviour |
|---|---|---|---|
| Wave equation | \(\omega=ck\) | real | propagates without decay |
| Klein–Gordon | \(\omega^2=c^2k^2+\omega_c^2\) | real (above cutoff) | propagates (Episode 6) |
| Diffusion equation | \(\omega=-iDk^2\) | ★ purely imaginary | does not propagate; decays in place |
Conclusion of §01
The everyday phrase "heat travels" is not physically correct.
Heat does not propagate. It merely spreads and thins out with time.
02But "heat it periodically" changes the story
Drive the surface at frequency \(\omega\) (day/night, seasons, alternating current) and you fix \(\omega\) real and solve for \(k\).
The heart of §02
Real and imaginary parts come out equal in size. Therefore ──
at depth \(\delta\) the amplitude falls to \(1/e\) and the phase lags by exactly one radian.
★ "Decay" and "lag" always happen over the same length. That is diffusion's fingerprint.
03Ground temperature ── the daily wave and the seasonal wave
Using a soil thermal diffusivity of \(D=5\times10^{-7}\) m²/s.
| Period | \(\omega\) [rad/s] | Penetration \(\delta\) [m] | Lag of 1 rad |
|---|---|---|---|
| 1 hour | \(1.7\times10^{-3}\) | 0.024 | 0.01 d |
| 1 day | \(7.3\times10^{-5}\) | 0.117 | 0.16 d |
| 1 month | \(2.4\times10^{-6}\) | 0.642 | 4.77 d |
| 1 year | \(2.0\times10^{-7}\) | 2.241 | 58.1 d |
| 100 years | \(2.0\times10^{-9}\) | 22.41 | 5813 d |
The daily wave reaches 12 cm, the seasonal wave 2.2 m. So at 1 m depth the daily swing has fallen to \(e^{-8.5}\approx0.02\ \%\) and only the seasonal one survives.
And the phase lag does something interesting.
| Depth [m] | Amplitude (vs surface) | Phase lag [rad] | Lag [days] | |
|---|---|---|---|---|
| 0.00 | 1.0000 | 0.000 | 0.0 | |
| 1.00 | 0.6401 | 0.446 | 25.9 | |
| 2.24 | 0.3679 | 1.000 | 58.1 | ★ amplitude \(1/e\) |
| 5.00 | 0.1074 | 2.231 | 129.7 | |
| 7.04 | 0.0432 | 3.142 | 182.6 | ★★ half a year out of phase |
| 10.00 | 0.0115 | 4.462 | 259.4 |
The heart of §03
At a depth of \(\pi\delta=7.0\) m the seasons are shifted by half a year.
Seven metres down it is warmest in winter and coldest in summer.
★ Traditional ice houses and root cellars work because of this phase lag.
Both effects matter — the amplitude has fallen to \(1/23\), and the phase is half a year off.
04The very same equation appears in electromagnetism
Inside a good conductor, Maxwell's equations become a diffusion equation.
★ Word for word the heat-conduction form. Only \(D\) differs
| Frequency | Skin depth in copper \(\delta\) |
|---|---|
| 50 Hz (mains) | 9.23 mm |
| 1 kHz | 2.06 mm |
| 1 MHz | 65.2 μm |
| 1 GHz | 2.06 μm |
| 10 GHz | 652 nm |
Copper's "electromagnetic diffusivity" is \(2.7\times10^4\) times soil's thermal one
\(D_{\text{em}}=\rho/\mu=1.34\times10^{-2}\) m²/s versus \(5\times10^{-7}\) m²/s for soil.
So electromagnetic fields act shallowly and fast ── which is why high-frequency conductors can be hollow, and why gold plating on a connector works.
★ And it comes out of exactly the same equation as the temperature of the ground.
05What is happening, in the language of calculus
| Equation | Order in time | Order in space | Character |
|---|---|---|---|
| Wave equation | 2 | 2 | ○ reversible; propagates |
| Diffusion equation | 1 | 2 | ★ mismatched; irreversible, does not propagate |
| Schrödinger | 1 | 2 | ○ reversible (thanks to \(i\), Episode 10) |
The heart of §05
First order in time and second in space gives \(\omega\propto k^2\), making \(\omega\) purely imaginary.
The origin of "does not propagate" was a mismatch of orders.
And the Schrödinger equation has exactly the same combination of orders, yet one \(i\) makes it reversible and lets it propagate as a wave ──
★ which is why the diffusion equation is called "the Schrödinger equation in imaginary time". Indeed \(t\to-it\) maps one onto the other.
06So can heat be made to travel as a wave? ── yes
Add a second-order time derivative and the equation becomes hyperbolic (Cattaneo).
| Medium | Relaxation time \(\tau\) | Cutoff \(1/\tau\) | Second-sound speed \(\sqrt{D/\tau}\) |
|---|---|---|---|
| Metal (electrons) | \(10^{-11}\) s | \(1.6\times10^{10}\) Hz | 3162 m/s |
| Dielectric | \(10^{-12}\) s | \(1.6\times10^{11}\) Hz | 1000 m/s |
| Superfluid helium | \(10^{-5}\) s | \(1.6\times10^{4}\) Hz | 3.2 m/s |
The heart of §06 ── a cutoff pointing the other way from Episode 6
At the cutoff \(\omega_c=1/\tau\), diffusion switches over to wave propagation.
Episode 6's mass was a low-frequency cutoff; this one turns into a wave at high frequency ── the opposite direction.
In ordinary matter the cutoff is \(10^{10}\sim10^{11}\) Hz ── out of everyday reach, so heat looks like something that diffuses.
But in superfluid helium \(\tau\) is \(10^6\) times longer and the cutoff is around 10 kHz. There, heat really does travel as a wave ── second sound, observed in 1944.
Exercises
- Give two reasons why a cellar is cool in summer and warm in winter.
Show answer
① The amplitude has decayed (\(1/e\) at 2.24 m, \(1/23\) at 7 m) ── the temperature swing itself is small.
② The phase has lagged (half a year at 7 m) ── when the surface is in summer, the ground is at "last winter's temperature".
A shallow cellar gets only ①; a deep one gets both. Go deeper than 7 m and the anti-phase breaks down again, leaving simply a constant temperature. - Give the numbers for why a high-frequency conductor can be hollow.
Show answer
At 1 GHz copper's skin depth is 2.06 μm. Current flows only in about that thickness.
For a 1 mm wire that is \(\approx4\delta/d=0.8\ \%\) of the cross-section ── 99 % of the copper inside is wasted.
Hence waveguides may be hollow, and only the surface finish (and gold plating) matters. - If "heat does not travel", why is it hot near a fire?
Show answer
Because that heat arrives by radiation and convection, not conduction.
Radiation is an electromagnetic wave (\(\omega=ck\), real) and literally propagates ── at the speed of light. Convection is bulk motion of matter.
This episode dealt only with conduction, and where conduction dominates (inside a metal rod, the ground, a circuit board) it really does "thin out rather than travel".
★ Of the three modes of heat transfer, only conduction obeys the diffusion equation.
Summary Whether it travels or thins out is set by the orders
The diffusion equation's dispersion relation is \(\omega=-iDk^2\) ── purely imaginary. It does not propagate; it decays in place. "Heat travels" is not physically right.
Heat it periodically, though, and a penetration depth \(\delta=\sqrt{2D/\omega}\) appears. Real and imaginary parts are equal, so decay and phase lag always occur over the same length ── diffusion's fingerprint.
In soil, 12 cm for day/night and 2.2 m for the seasons. And at 7.04 m the seasons are half a year out of phase ── why ice houses and root cellars work.
The same equation appears in electromagnetism ── the skin effect. Replace \(D\) by \(\rho/\mu\): copper gives 9.2 mm at 50 Hz and 2.1 μm at 1 GHz. The same mathematics as the temperature of the ground.
Irreversibility originates in the mismatch: first order in time, second in space. With the same orders but an \(i\), it becomes reversible (Schrödinger) ── the diffusion equation is "Schrödinger in imaginary time".
And add a second-order time term and heat turns into a wave above a cutoff frequency.
── In superfluid helium, that has actually been observed.
And the question is: why does radiation require acceleration?
The answer is in the order ── only the far field picks up the second derivative. And from there comes the reason an electron does not fall into the nucleus.
This document is Episode 11 of the series "Thinking in Waves". The computation is in kenshou/netsu.js. On the numbers: the soil thermal diffusivity \(D=5\times10^{-7}\) m²/s is a representative value and varies by a factor of 2–3 with moisture and density (about \(2\times10^{-7}\) for dry sand, \(6\times10^{-7}\) for wet clay) ── so "2.2 m for the seasons" and "7 m for the half-year shift" carry the same spread. Real ground temperature also sits on a geothermal gradient (\(\approx25\) K/km) and is affected by snow cover and vegetation. The skin-depth formula is the good-conductor approximation (\(\sigma\gg\omega\varepsilon\)) and fails at high enough frequency. The relaxation time \(\tau\) in the Cattaneo equation is defined differently in different materials, and the table gives orders of magnitude ── the measured second-sound speed in superfluid helium is about 20 m/s, an order off our naive \(\sqrt{D/\tau}\) estimate of 3.2 m/s (it really needs the two-fluid model). ── To print, use your browser's Print → Save as PDF.