It is the assumption that your gear behaves predictably, so doubling the input just doubles the output without weird surprises.
Top: a linear stage just scales the wave (same frequency, +6 dB). Bottom: overdrive clips the peaks flat and invents harmonics = distortion.
What it is
The assumption your gear behaves predictably: double the input, you get exactly double the output, with no new frequencies invented.
Key facts
A linear system obeys two rules: SCALING (homogeneity) and ADDING (superposition). Scale input by 2, output scales by 2. Sum two inputs, outputs sum.
It NEVER invents new frequencies. Feed in a pure 1 kHz tone, you only ever get 1 kHz out (maybe louder/softer/delayed, never 2 kHz or 3 kHz).
New frequencies (harmonics) = the signature of NON-linearity, i.e. distortion. THD (Total Harmonic Distortion) measures exactly this, in %.
+6 dB = double the VOLTAGE (amplitude x2). +10 dB = roughly double the perceived LOUDNESS. +3 dB = double the POWER (watts x2).
dB formula (voltage/SPL): dB = 20 x log10(V2 / V1), where V2/V1 is the ratio of new level to old level.
dB formula (power): dB = 10 x log10(P2 / P1), where P2/P1 is the ratio of new watts to old watts.
-3 dB = the 'half-power point', used to define a speaker/filter's frequency-response bandwidth (the -3 dB points).
Speed of sound in air = 343 m/s at 20 C (rises ~0.6 m/s per degree C hotter).
Hearing range = 20 Hz to 20,000 Hz (20 kHz). A linear PA should reproduce all of it proportionally.
1% THD is roughly the audible threshold for many signals; pro amps spec <0.05% THD; clipping can hit 10%+ THD.
How it works
Send a known signal in (a test tone or your mix).
If the system is linear, the output is just a scaled/delayed copy. Same frequencies, nothing added.
Double the input level. A linear system doubles the output level (+6 dB) and stays clean.
Push past the linear region (amp clips, speaker bottoms out). Now it adds harmonics = distortion.
Use a spectrum analyser (RTA): extra frequency spikes appearing means you've left the linear zone.
Real examples
Clean power amp at 50% gain: input up 6 dB, output up 6 dB, mix stays faithful = linear.
Same amp driven into clipping: square-wave edges add harsh harmonics = non-linear distortion.
A guitar overdrive pedal is non-linear ON PURPOSE: it adds harmonics for grunt and warmth.
EQ and faders are linear tools: they change level per band but invent no new frequencies.
A speaker pushed past its excursion (Xmax) goes non-linear: you hear crackle and breakup.
How it helps in live sound
Watch amp/console clip LEDs: first red = you've left the linear zone, sound gets harsh.
Gain-stage so peaks sit around -18 to -12 dBFS on a digital desk = stay in the clean linear region with headroom.
Keep ~10-20 dB of headroom before clipping so transients (kick, snare) don't distort.
If a subwoofer farts/crackles, it's gone non-linear past Xmax: pull level or high-pass it (e.g. 30-40 Hz).
Run an RTA on a sine sweep: extra harmonic spikes = non-linearity in that rig, hunt the weak link.
Want grit? Add a saturation/drive plugin deliberately. Don't rely on accidental clipping to do it.
Everyday analogy
A fair vending machine: put in twice the coins, get twice the snacks, with no random extra items spat out.
Watch out
Myth: 'louder always means just more of the same sound.' Wrong: push past the linear range and the gear ADDS harmonics (distortion), changing the tone, not just the volume.
Fun fact
A perfect square wave is just a 'too-loud' sine wave clipped flat, and it's mathematically a sine plus only its ODD harmonics (3rd, 5th, 7th...) stacked up forever.
Key takeaways
Linear = output is proportional to input: double in, double out (+6 dB).
Linear systems NEVER create new frequencies. New frequencies = distortion = non-linear.
Two rules: scaling (x2 in = x2 out) and superposition (signals add cleanly).
Headroom keeps you in the linear zone; clipping shoves you out of it.
Non-linearity is a tool (overdrive, saturation) AND a fault (clipping, blown cones) depending on intent.