A medium's resistance to sound is Z = ρc — density times sound speed. Air is ~415 rayls, water ~1.5×10&sup6; rayls: a mismatch of thousands. At the boundary the reflected fraction is R = ((Z₂−Z₁)/(Z₂+Z₁))², and for air→water that is 0.9989 — 99.9% bounces back. This is why your voice does not carry underwater, and why horns and the ear's ossicles exist: to match impedances and let sound cross.
SOURCE Specific acoustic impedance — Lord Rayleigh, The Theory of Sound, Vol. I, 1877 (Macmillan). Marked AMBER: the pressure/velocity ratio is Rayleigh's, but no single stable page anchor fixes the term. The rayl (rho·c) is named for him.
Rendered, not quoted. The engine below computes Z and R live from ρ, c, and the two media — no stored numbers.
Two half-spaces meet at a plane. Each carries a specific acoustic impedance Z = ρ·c in rayls (Pa·s/m): the ratio of sound pressure to particle velocity in a travelling wave.
At normal incidence the intensity reflection coefficient is R = ((Z₂−Z₁)/(Z₂+Z₁))², and by energy conservation T = 1 − R. R lives in [0,1]; the square makes it a true energy fraction, sign-blind to which side is denser.
Rayleigh, 1877: Z = ρc with reflection ((Z₂−Z₁)/(Z₂+Z₁))². The huge air–water mismatch bounces 99.9% back; impedance matching is the trick behind every horn and the ear's ossicles.
Closes the acoustics thread that opens at the-speed-of-sound (c = √(γRT/M)). That sphere sets c; this one multiplies by ρ to get Z — the quantity that decides what crosses a boundary and what returns.
A live re-check of the engine's laws, run again in your browser now. It confirms Z=ρc, the air/water ratio, R≈0.9989 for air→water, matched Z giving R=0, and R+T=1.
WITNESS: pending…If the Tamper (window 6) drops the square, this badge flips red — it re-derives R and catches the broken energy fraction.
Two media, each ρ (kg/m³) and c (m/s). Defaults: air ρ=1.204, c=343 and water ρ=1000, c=1480. Sound arriving from side 1 onto side 2.
For air→water: R ≈ 0.9989 — 99.9% reflected, ~0.1% crosses. Matched impedances give R=0 (full transmission). A large mismatch reflects nearly all sound either direction.
—"R is just the impedance difference — drop the square, it's simpler." No: a bare ratio (Z₂−Z₁)/(Z₂+Z₁) goes negative when side 1 is denser, and |ratio| is the pressure-amplitude reflection, not the energy. Only the square is a fraction in [0,1] that obeys R+T=1.
"Denser medium reflects more." No — R is symmetric: air→water and water→air reflect the same 99.9%. The mismatch reflects, not the density order.
"Loud enough sound will carry from air into water." — the ~1000× impedance jump reflects 99.9% no matter the level; only ~0.1% of intensity ever crosses.
"Impedance is a property of the wave (its loudness)." — Z=ρc is a property of the medium alone; it sets the ratio p/u independent of amplitude.
"Reflection depends on which medium is denser." — R squares the difference, so it is sign-blind; both directions give identical R.
Drop the square: compute R = (Z₂−Z₁)/(Z₂+Z₁) directly. Now R can be negative and R+T≠1 — a broken energy fraction. The Witness (7) catches it live.
Planted void (disclosed): the tamper is a known-bad edit built in so window 7 can be seen catching it. selfcheck() also flips the flag internally and asserts the value moves.