Can Range Be Bought? —
Not with time, but with instruments Episode 53 concluded that range grows as $\log N$. ★ So how does one get more of it?
★ There are three ways — splice bands, use a different observable, change the system itself.
★★ And the price came out too — splicing costs "1 dB of calibration error across a 3-decade gap = 0.1 dB/oct." Exactly the size of Episode 42's target (0.081).

Tools needed: Episodes 8, 42, 51, 52, 53Computation: kenshou/kau.js

Episode 53 established that a "not enough range" problem cannot be solved by measuring more.
★ This episode counts, in this series' language, the ways experimentalists actually get around it.

01There are only three ways to buy it

Way	What it is	What it needs
① splice bands	join measurements from different bands	★ more instruments
② use a different observable	see the same physics through another quantity	★ find a form where corrections cancel
③ change the system	move to a subject with a wider inertial range	★ raise the Reynolds number

★ "Measure for longer" is not on the list — because of Episode 53's $\log N$.

02★ Splicing — a calibration error becomes a slope error

Join bands A and B and fit a line. If B's gain is off by $\delta$ dB, how far off is the slope?

band separation [decades]	$\delta=0.2$ dB	$\delta=0.5$ dB	$\delta=1$ dB	$\delta=2$ dB
1	0.04499	0.11247	0.22494	0.44989
2	0.02769	0.06923	0.13845	0.27690
3	0.01929	0.04821	★ 0.09643	0.19285
6	0.00991	0.02477	0.04954	0.09908

Analytic estimate (treating each band as a point) $$\text{slope error [dB/oct]}\approx\frac{0.3\times\delta\,[\mathrm{dB}]}{\text{separation in decades}}$$

The point of §02

At $S=3,\ \delta=1$ dB, measured 0.0964 against the predicted 0.1000 — the bands' own width makes the measurement slightly smaller.
★★★ 1 dB of calibration error across a 3-decade gap costs 0.1 dB/oct. The same size as Episode 42's 0.081.
★★ So buying 3 decades by splicing requires calibration better than 0.8 dB.
★ Read the other way — the further apart the bands, the looser the calibration requirement (6 decades apart, 1 dB costs only 0.05). A conclusion that looks backwards at first: distant instruments are easier to join.

03★★ A different observable — find where the corrections cancel

Assumption: every $S_p$ is a power of one common $\rho$ $$S_p(r)=\rho(r)^{\zeta_p},\qquad \rho(r)=r\,h(r)\ (h\ \text{the end correction})$$

In that form $\ln S_p$ against $\ln r$ bends, but $\ln S_p$ against $\ln S_3$ is a straight line of slope $\zeta_p/\zeta_3$, whatever $h$ is.

$p$	$\zeta_p$ (true)	$\ln S_p$ vs $\ln r$	★ $\ln S_p$ vs $\ln S_3$
1	0.360919	4.45 decades	★ 9.00
2	0.693721	4.16	★ 9.00
3	1.000000	4.00	★ 9.00
6	1.785875	3.75	★ 9.00 (all of it)

The point of §03

★★★ Put $\ln S_3$ on the horizontal axis and the correction $h(r)$ cancels completely — 4 decades become 9.
★★ This is the idea behind extended self-similarity (ESS), used in experiments.
★★★ But the cancellation happens because we assumed "every $S_p$ is a power of a common $\rho$" — which is not itself proved.
★ What this article showed is only that "if that assumption holds, range can be bought." Per Episode 43, an assumption is written as an assumption.

04★ Changing the system — the inertial range is $Re^{3/4}$

Kolmogorov length $\eta$ against outer scale $L$ $$\frac{L}{\eta}=Re^{3/4}\qquad\Longrightarrow\qquad \text{inertial decades}=\frac34\log_{10}Re$$

Situation	$Re$	★ inertial decades	Episode 51's best error
laboratory turbulence	$10^4$	3.00	≈ 0.1793 dB/oct
wind tunnel	$10^6$	4.50
around a car	$10^7$	5.25
large wind tunnel	$10^9$	6.75	< 0.06 dB/oct
Earth's atmosphere	$10^{12}$	9.00	< 0.06 dB/oct

The point of §04

★★ Gaining one decade of inertial range requires $Re$ to rise by $10^{4/3}=21.5$.
★★★ A logarithm again — but this time changing the subject moves the decades.
★ From the laboratory ($Re=10^4$, 3 decades) to the atmosphere ($Re=10^{12}$, 9) is a span of 6 — and Episode 51's table gave 0.18 dB/oct at 3 decades and 0.059 at 6.

05Putting the three together

Means	What you gain	What it costs
measure longer	precision $N^{-1/2}$	range grows only as $\log N$
★ splice instruments	★ range ∝ number of instruments	★ calibration within 0.8 dB (§02)
★ another observable (ESS)	★ range becomes nearly the whole span	★ one assumption (§03)
★ raise $Re$	★ inertial range $\propto\frac34\log Re$	★ the size of the apparatus

The point of §05 — ★★★ the core of this episode

All three buy range, but they buy it differently —
splicing in proportion to the number of instruments (linear),
another observable all at once, in exchange for an assumption,
changing the system as the logarithm of $Re$.

★★ Episode 53's "range goes as $\log N$" was about measuring the same system, with the same instrument, for longer.
★★★ Add instruments and range is bought linearly — that is the way out.

06What cannot be bought

Goal	Verdict	Why
Buy range by measuring the same system longer	× no	★ $\log N$ (Ep. 53)
Splice without calibration	× no	★ 1 dB / 3 decades = 0.1 dB/oct
Use ESS without an assumption	× no	★ a common $\rho$ is being assumed
Reach the ends of $f(\alpha)$	× no	★ Ep. 52; set by the log of the sample
Create inertial range the system does not have	× no	★ $Re$ decides it

07What wave language can and cannot say

Item	Verdict	Basis
★ The calibration requirement for splicing	◎ numerics	★ $0.3\delta/$separation
1 dB / 3 decades = 0.1 dB/oct	◎ numerics	★ same size as Episode 42's target
★★ ESS cancels the corrections	◎ numerics	★ 4 decades → 9 (all)
But one assumption is made	◎ stated	★ a common $\rho$ raised to powers
Inertial range $=\frac34\log_{10}Re$	◎ numerics	lab 3 decades, atmosphere 9
One more decade needs $Re\times21.5$	◎ numerics	a logarithm again
★★ Linear in the number of instruments	◎ logic	★ the way out of Episode 53
★ Validation on real data	× not done	★ synthetic data only, here too

Exercises

§02 found that bands further apart are easier to calibrate. That seems backwards — why is it true?
Show answer
Because a slope is "vertical offset ÷ horizontal distance." The same offset $\delta$ matters less the further apart the bands are.
★ 1 dB costs 0.22 dB/oct across one decade and only 0.05 across six.
★★ But separating them leaves a gap with nothing in it — and whether the same power law really continues across that gap has to be established separately.
★ So the further apart, the lighter the calibration and the heavier the extrapolation assumption — a trade, as always.
Does §03's ESS really "buy range"?
Show answer
Conditionally, yes. What was bought was paid for with the assumption that every $S_p$ is a power of one common $\rho$.
★ If that holds, $h(r)$ cancels exactly and the whole span becomes usable (4 decades → 9 in our numbers).
★★ If it does not, the cancellation is incomplete — and, worse, the plot still looks straight, which is the dangerous part.
★★★ In this series' stance, a $\zeta_p$ obtained via ESS must be labelled "conditional on an assumption." The same discipline as Episode 43's "do not call organizing a discovery" — do not call an assumption a measurement.
§04 says raising $Re$ widens the inertial range. Can $Re$ be raised without limit?
Show answer
In the laboratory it very nearly saturates. $Re=UL/\nu$, so there are three handles — speed $U$, size $L$, kinematic viscosity $\nu$.
★ $U$ is capped by the speed of sound (compressibility changes the physics), $L$ by the size of the building, $\nu$ by the choice of fluid (cryogenic helium buys a few decades).
★★ So $Re\approx10^7$ is the practical laboratory ceiling, and beyond that one "uses" the atmosphere or the ocean ($Re=10^{12}$).
★★★ But using nature costs control — neither isotropy nor stationarity is guaranteed. ★ Episode 22's "stationarity assumption" comes straight back.

Summary — not with time, but with instruments

Episode 53 concluded that range goes as $\log N$. Here we counted three ways around it — splice, use another observable, change the system.

★ The price of splicing: slope error $\approx0.3\delta[\mathrm{dB}]/$(separation in decades). 1 dB across a 3-decade gap costs 0.1 dB/oct — the same size as Episode 42's 0.081. Buying 3 decades needs calibration better than 0.8 dB.

★★ Another observable: if $S_p=\rho^{\zeta_p}$, then putting $\ln S_3$ on the horizontal axis makes the correction cancel exactly, turning 4 decades into 9. This is the ESS used in experiments — but the cancellation comes from an assumption, and the assumption is not proved.

★ Changing the system: inertial decades $=\frac34\log_{10}Re$. 3 in the laboratory ($10^4$), 9 in the atmosphere ($10^{12}$); one more decade needs $Re\times21.5$.

★★★ The core — all three buy range, but differently.
Splicing is linear in the number of instruments,
another observable buys it all at once for an assumption,
changing the system buys it as the logarithm of $Re$.

★★ Episode 53's "range goes as $\log N$" was about the same system, the same instrument, measured for longer. Add instruments and range is bought linearly.
★ Honestly: synthetic data only here too. ★ And all three have a price — calibration, an assumption, the size of the apparatus. There was no free range after all.

Next time Four episodes since 51 have been spent counting how far one may speak.
★ Next time closes Part XVIII — rewriting the handbook (Episode 49) as these four episodes have changed it, on one page.
★★ And we re-audit which of this series' own numbers were "conditional on an assumption." A finer-grained version of the scorecards of Episodes 43 and 50.

This document is Episode 54 of the "Thinking in Waves" series. Computations are in kenshou/kau.js. On the numbers here: the slope error from splicing, the comparison of flat ranges under ESS, and the inertial-range decades against Reynolds number were all computed here. Extended self-similarity, $L/\eta=Re^{3/4}$, and the general effect of a calibration error on a fitted slope are all well-known results and none are discoveries of this article — what this article did was place them under the single question "how is range bought," and price them in this series' unit of dB/oct. §03's ESS is computed under the assumption that all $S_p$ are powers of one common $\rho$; the validity of that assumption is not tested here, and by this series' standard a $\zeta_p$ obtained via ESS is "conditional on an assumption." The analytic estimate $0.3\delta/$separation in §02 treats each band as a point; with bands of finite width the measured value is smaller (0.0964 against 0.1000 at $S=3$). The $Re$-to-inertial-range correspondence in §04 is standard dimensional analysis; the real inertial range also depends on how isotropic and stationary the flow is. This article too used only synthetic data and was not applied to any real measurement. — To print, use your browser's Print and "Save as PDF."