Thinking in Waves Episode 23 / Part X — Breaking the Assumption

Breaking, head-on, the assumption this series has carried throughout

Nonlinearity —
Where the word "filter" stops working "Filter" and "transfer function," used since Episode 1, all presuppose linearity.
Break it and — frequencies you never put in are born, and the transfer function stops being definable.
★ And yet all three constraints of Part IV survived.

Tools needed: Episodes 1, 8, 9, 10, 14, 16Computation: kenshou/hisenkei.js

Episode 15 listed four limitations of the framework. Episode 22 plugged ① (stationarity).
The largest remaining hole is ②, linearity. This time we break it head-on.

01The assumption this series has carried

What the word "filter" actually contains $$L[a\,x+b\,y]=a\,L[x]+b\,L[y]\qquad\textbf{★ superposition}$$

Superposition is what let us think sinusoid by sinusoid — without it, the very idea of a "frequency response" does not exist

Property	Linear	Nonlinear	What it means
superposition	holds	★ breaks	the input cannot be split up
transfer function $H(\omega)$	definable	★ not definable	the response depends on the input's size
conservation of frequency	holds	★ they mix	frequencies you never put in come out
meaning of Fourier decomposition	complete	limited	splitting no longer makes the parts independent

02The simplest nonlinearity — new frequencies are born

Just add a squared term $$y=x+\varepsilon x^2,\qquad x=\sin(2\pi\cdot1000\,t)+\sin(2\pi\cdot1100\,t),\qquad \varepsilon=0.1$$

frequency [Hz]	amplitude	what it is
1000	1.000000	input $f_1$
1100	1.000000	input $f_2$
0	0.200000	★ DC (the rectified part)
100	0.100000	★ difference $f_2-f_1$
2100	0.100000	★ sum $f_1+f_2$
2000	0.050000	★ second harmonic $2f_1$
2200	0.050000	★ second harmonic $2f_2$

The point of §02

Two went in; seven came out.
★ A linear filter can never do this — all it can do is strengthen or weaken the frequencies you put in.

Look at the difference frequency, 100 Hz. ★ A frequency lower than either input has been born.
This is intermodulation, the most disliked distortion in audio — harmonics are octave-related and musical, but difference tones are dissonant.

03Measuring distortion — linearity turned out to mean smallness

$a_2$	$a_3$	THD [%]	Comparable to
0	0	0.0000	perfectly linear
$10^{-4}$	0	0.0050	high-end audio
$10^{-3}$	0	0.0500	an ordinary amplifier
$10^{-2}$	0	0.5000	a cheap loudspeaker
0.1	0	5.0000	★ the value used in §02
0.3	0.3	13.69	★ a guitar distortion pedal

The conclusion of §03

People are usually said to notice distortion from about 1 % upward.
★ So the everyday "linear world" was an approximation that ignored nonlinearity at the 0.1 % level.

Linearity is not a property of nature but a consequence of smallness.

04The transfer function stops being definable — the Duffing oscillator

Make the spring slightly stiffer at larger amplitude $$\ddot x+2\gamma\dot x+\omega_0^2x+\beta x^3=F\cos(\omega t),\qquad \gamma=0.05,\ \beta=0.3,\ F=0.3$$

We compare the response while sweeping the frequency slowly up and then down. If it were linear they would be identical.

$\omega$	sweeping up	sweeping down	difference
1.00	1.08409	1.08409	0.00000
1.20	1.62561	1.62561	0.00000
1.24	1.73233	0.65685	★ 1.07548
1.30	1.88817	0.45790	1.43028
1.40	2.12379	0.31664	1.80714
1.42	2.15049	0.29819	★ 1.85230 (max)
1.50	0.24082	0.24082	0.00000
2.00	0.09985	0.09986	0.00001

The point of §04 — this is decisive

Same frequency, same input, two different answers. Which one you get depends on how you arrived — hysteresis.
Between $\omega=1.24$ and $1.42$ the response is two-valued; sweeping up it drops at $1.44$, sweeping down it jumps at $1.22$.

★ A transfer function $H(\omega)$ is a function: put in a frequency, get a response.
With two answers, it cannot be a function at all.

── The word "filter," used since Episode 1, stops working here.

05★ What breaks and what survives — the real question

Tool	From	Under nonlinearity	Why
superposition	Ep. 1	★ breaks	it is the definition of nonlinear
transfer function / filter picture	Ep. 1	★ breaks	the hysteresis of §04
"tilting the spectrum" picture	Ep. 1	△ limited	frequencies mix
★ causality	Ep. 8	◎ survives	output still cannot precede input
★ the bandwidth theorem	Ep. 9	◎ survives	it is a property of the Fourier transform itself
★ the arrow of time (parity of order)	Ep. 10	◎ survives	$t\to-t$ behaves the same way
★ dimensional analysis	Ep. 14	◎ survives	if anything it stars here (turbulence)
★ information lives on the imaginary axis	Ep. 21	◎ survives	the second law holds for nonlinear systems too

§05 — the best find of this episode

What breaks is only Part I's "filter" picture.
All three constraints of Part IV survive.

The reason is clear — "filter" is a word from linearity, whereas

★ causality, bandwidth and the arrow of time come not from linearity
but from time running in one direction only.

★ And as Episode 21 showed, that one-directionality is the imaginary axis of the complex frequency plane.
Constraints that came from the imaginary axis survive nonlinearity.

06Measuring the strength of nonlinearity — always the same dimensionless form

Field	Quantity	Contents	Symbol
fluids	Reynolds number $Re$	$(u\cdot\nabla)u\ /\ \nu\nabla^2u$	$UL/\nu$
nonlinear optics	nonlinear phase $\varphi_{\rm NL}$	$n_2I/n_0$	$\gamma PL$
gravity	compactness	$GM/(rc^2)$	Ep. 15
acoustics	nonlinearity parameter	$(B/A)\cdot(u/c)$	grows with loudness
field theory	coupling constant	interaction / free term	$\alpha,\ \alpha_s$

All of them are "the nonlinear term divided by the linear term," dimensionless. Episode 7's "only dimensionless quantities mean anything" is at work here too.

Situation	$U$ [m/s]	$L$ [m]	$Re=UL/\nu$	Flow
a swimming sperm cell	$10^{-5}$	$5\times10^{-5}$	$5\times10^{-4}$	★ completely linear (viscosity rules)
blood in a capillary	$10^{-3}$	$8\times10^{-6}$	$2.7\times10^{-3}$	★ completely linear
a kitchen tap	1	0.01	$10^{4}$	turbulent
a car at 100 km/h	27.8	4	$7.4\times10^{6}$	turbulent
Earth's atmosphere	10	$10^{7}$	$6.7\times10^{12}$	★ completely nonlinear

§06 — living in a linear world changes how you swim

A sperm cell's world is $Re\sim10^{-4}$ — completely linear. It cannot coast on inertia; stop the flagellum and it stops instantly.

★ There is a deeper consequence — in a linear world, a time-reversible motion cannot propel you (the scallop theorem).
Opening and closing like a scallop just returns you to where you started. So microorganisms use motions that do not retrace under time reversal, such as a rotating helical flagellum.

★ Episode 10's "the arrow of time lives in the parity of the order" was deciding the shape of living things.

07What nonlinearity builds — solitons

Nonlinearity does not only break things. Balanced against dispersion, it preserves a shape.

Two effects cancelling $$\text{dispersion: the wave wants to spread (Ep. 16)}\qquad\text{nonlinearity: the strong parts are retarded in phase}$$ $$N^2=\frac{\gamma P_0T_0^2}{|\beta_2|}=1\qquad\textbf{★ the fundamental-soliton condition}$$

pulse width $T_0$ [ps]	peak power required $P_0$ [mW]	dispersion length $L_D$ [km]
1	15385	0.05
5	615	1.25
10	154	5.00
20	38	20.0
50	6.2	125

The conclusion of §07

A 20 ps pulse needs 38 mW — a practical figure. Soliton transmission was indeed studied for long-haul optical communication and used in places.

★ In wave language —
with linear dispersion alone, a pulse must spread (Ep. 16).
With nonlinearity alone, the waveform distorts and breaks (§02).
★ Balance the two and neither happens — that is a soliton.

08How long does perturbation theory hold?

System	expansion parameter	perturbation theory
QED (the electron's $g-2$)	$1/137$	◎ computed to five loops
QCD (high energy)	0.118	○ three to four loops
QCD (low energy)	$\sim1$	× unusable → lattice
graphene (bare)	2.19	× Episode 17. Strong coupling on a desk
gravity (Earth's surface)	$7\times10^{-10}$	◎ linear to ten digits
gravity (neutron-star surface)	0.345	△ nonlinearity matters
gravity (horizon)	0.5	× the whole framework fails

The conclusion of §08

Episode 15 said "on Earth we are linear to ten digits."
★ That is why "the language of waves" worked so well — which was luck, not a property of nature.

Exercises

Explain the difference between radio "intermodulation" and audio "harmonic distortion" using §02's table.
Show answer
Both come from the same squared term, but they land in different places.
Harmonics (2000, 2200 Hz) are integer multiples of the input — octave-related, so musically tolerable.
Intermodulation (100, 2100 Hz) is the sum and difference — frequencies unrelated to the input, so they sound dissonant.
★ In radio this is how a strong station leaks into a weak one, and §02's 100 Hz difference tone is exactly that.
Why does causality survive nonlinearity?
Show answer
Because causality is a condition on time ordering — output does not precede input — and has nothing to do with linearity.
That said, Episode 8's Kramers–Kronig relations are a theorem about linear response functions and do not carry over unchanged.
★ There are nonlinear extensions (dispersion relations for the nonlinear susceptibilities $\chi^{(2)}, \chi^{(3)}$) — the structure "causality = analyticity" is preserved order by order.
How does a sperm cell's inability to "swim like a scallop" relate to Episode 10?
Show answer
Both are about what happens under time reversal.
Episode 10: even-order terms are invariant under $t\to-t$, hence reversible.
Here: at $Re\ll1$ the inertial (nonlinear) term vanishes and only the viscous, linear equation remains. That equation is first order in time with time-independent coefficients, so running "open and close" backwards returns you exactly to the start.
★ To move forward, the shape change itself must fail to retrace under time reversal — turning a helix, sending a travelling wave.
★ The Reynolds number of the fluid was deciding the propulsion mechanism of living things.

Summary — what broke was only the vocabulary of linearity

"Filter" and "transfer function," used since Episode 1, all presupposed superposition. Add nothing but a squared term and — two go in, seven come out (DC, difference, sum, two harmonics). Something a linear filter can never do.

By distortion figures, people notice from about 1 %. The everyday "linear world" was an approximation ignoring nonlinearity at the 0.1 % level — linearity is not a property of nature but a consequence of smallness.

For the Duffing oscillator, the same frequency and the same input give two answers (two-valued from $\omega=1.24$ to $1.42$, with a maximum gap of 1.85). With no function to speak of, the word "transfer function" cannot be used.

★ And yet every constraint of Part IV survived —
causality, the bandwidth theorem, the arrow of time, and dimensional analysis.

The reason is clear — "filter" is a word from linearity, while the three constraints come from time running in one direction only. In Episode 21's language: constraints that came from the imaginary axis survive nonlinearity.

Across every field, the strength of nonlinearity is measured by the same dimensionless "nonlinear over linear" ($Re$, $n_2I/n_0$, $GM/rc^2$). A sperm cell's world is $Re\sim10^{-4}$, completely linear — so it cannot swim like a scallop and turns a helix instead. And when dispersion and nonlinearity balance, you get a soliton (38 mW for 20 ps).

Next time Episode 15 listed four limitations of the framework — stationarity, linearity, $\omega=0$, and small effects.
Episode 22 handled ①; this episode handled ②. ③ is a limit of principle, and ④ is a matter of measurement.
★ Next time we look, as a wave, at something this series has never faced directly — spacetime itself being the wave: gravitational waves. In the language of order, what order are they?

This document is Episode 23 of the "Thinking in Waves" series. Computations are in kenshou/hisenkei.js. On the numbers here: the nonlinearity of §02–03 is the simplest possible form, "output is a polynomial in the input" (memoryless nonlinearity); real amplifiers and loudspeakers have nonlinearity with memory (dependent on previous state), so their distortion is more complicated. The THD values are computed from assumed $a_2, a_3$, not measured from hardware (the "high-end audio" comparisons are order-of-magnitude labels). The Duffing oscillator of §04 is a numerical solution at $\gamma=0.05,\beta=0.3,F=0.3$, and the jump positions depend strongly on those values. Each sweep point discards 60 periods of transient before measuring the amplitude; a slower sweep moves the jump points slightly. The Reynolds numbers of §06 are estimates from representative values. The scallop theorem (Purcell's theorem) is a rigorous result about low-Reynolds-number flow, but this article only states its content and does not derive it. The soliton condition is the standard fundamental-soliton form of the nonlinear Schrödinger equation, using standard optical-fibre values $\gamma=1.3\times10^{-3}$/(W·m) and $\beta_2=-20$ ps²/km. The "survives nonlinearity" verdicts of §05 are this article's organization and are not rigorous proofs item by item — in particular, the nonlinear Kramers–Kronig relations require an order-by-order formulation and are not as simple as the linear case. — To print, use your browser's Print and "Save as PDF."