Four views of one harmonic solution of the sound wave equation, ∂²s/∂t² = (B/ρ₀) ∂²s/∂x²: molecules, displacement and pressure, speed, and intensity.
The piston moves with SHM. Each gas layer copies that motion with a delay x/v, so compressions and rarefactions travel down the tube while every molecule stays near home.
Watch the orange tracers: they only rock back and forth by sm. What travels at v is the pattern. Molecules crowd where s crosses zero going downhill (∂s/∂x most negative), and thin out where it crosses zero going uphill. At the crests of s the gas is displaced the most but not compressed at all.
s = sm cos(kx − ωt)
ΔP = −B ∂s/∂x = ΔPm sin(kx − ωt)
ΔPm = B k sm = ρ v ω sm
Same layout as Tipler fig. 15-10: (a) displacement, (b) molecules at rest positions, (c) displaced molecules, (d) density, (e) pressure. Dashed lines mark the zeros of s, where |ΔP| and |Δρ| peak. The small parameter of the derivation is the acoustic Mach number M = ωsm/v. The condensation equals it exactly, Δρm/ρ₀ = k sm = M, and so does ΔPm/B. Measured against the ambient pressure, ΔPm/P₀ = γM, larger by γ = 1.4 because B = γP₀ = ρ₀v².
Isothermal (Newton, 1687): v = √(P/ρ). Adiabatic (Laplace, 1816): v = √(γP/ρ) = √(γRT/M). The ratio is √γ at every temperature.
Pressure does not appear: at fixed T, raising P raises ρ in proportion, so v depends on temperature only. Newton's value for air was about 16% low. Laplace's fix says the compressions happen too fast for heat to diffuse across half a wavelength, so the gas behaves adiabatically.
I = ½ ρ v (ω sm)² = ΔPm² / (2ρv)
SIL = 10 log₁₀(I / I₀)
I₀ = 10⁻¹² W/m²
SPL = 20 log₁₀(prms / pref)
prms = ΔPm/√2, pref = 20 µPa
Use the buttons to see what doubling the amplitude or raising the frequency does.
At a fixed displacement amplitude, I ∝ ω², so ten times the frequency is +20 dB. Near 1 kHz the threshold of hearing corresponds to an air displacement of about 10⁻¹¹ m, a tenth of an atomic radius. The tone button plays at a fixed, comfortable volume; your speakers set the real level. Reference levels are the indicative ones from the slides, regardless of frequency and distance.
SIL vs SPL. Sound level meters measure pressure, so the level quoted in practice is SPL. With I = prms²/(ρ₀v), SPL − SIL = 10 log₁₀[ρ₀v I₀ / pref²] = 10 log₁₀(ρ₀v / 400). The two references were chosen to match for ρ₀v = 400 Pa·s/m; air at 1 atm has about 413 at 20 °C, so the two levels agree within a few tenths of a dB. In water (ρ₀v ≈ 1.5 × 10⁶ Pa·s/m) SPL would exceed SIL by about 36 dB, which is one reason underwater acoustics uses pref = 1 µPa.