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WAV-05

Characteristic acoustic impedance

Z = ρ c, and intensity I = p² / Z.

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AcousticsRayleigh

Governing equation

Z=ρc,I=p2ZZ=\rho c,\quad I=\dfrac{p^2}{Z}

p∈[2×10−5,20]p \in \left[2\times 10^{-5}, 20\right]

where

\rho
Density (kg/m³)
c
Sound speed (m/s)
p
Pressure amplitude (Pa)
Z
Impedance (Pa·s/m)
I
Intensity (W/m²)

The figure, part by part

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WAV-05 · wave
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①

Governing law — red

Z=ρc,I=p2ZZ=\rho c,\quad I=\dfrac{p^2}{Z}

p∈[2×10−5,20]p \in \left[2\times 10^{-5}, 20\right]

p=2×10−5p = 2\times 10^{-5}Z=9.72×10−13Z = 9.72\times 10^{-13}

②

Domain of the variable — blue

pmin⁡=2×10−5,pmax⁡=20p_{\min} = 2\times 10^{-5},\quad p_{\max} = 20

p∈[2×10−5,20]p \in \left[2\times 10^{-5}, 20\right]

p=2×10−5p = 2\times 10^{-5}p=2×10−5p = 2\times 10^{-5}

③

Response curve — green

Z=ρc,I=p2ZZ=\rho c,\quad I=\dfrac{p^2}{Z}

p∈[2×10−5,20]p \in \left[2\times 10^{-5}, 20\right]

p=2×10−5p = 2\times 10^{-5}Z=9.72×10−13Z = 9.72\times 10^{-13}

④

Rate of change — amber

dZdp\frac{\mathrm{d}Z}{\mathrm{d}p}

p∈[2×10−5,20]p \in \left[2\times 10^{-5}, 20\right]

p=0.126p = 0.126Z′=6.11×10−4Z^{\prime} = 6.11\times 10^{-4}

Lecture brief

Historical brief

d’Alembert’s wave equation, Snell, the thin-lens maker, Doppler and Bragg interference are the classical optics-and-sound toolkit. The lab is propagation, image and shift. This sheet (WAV-05 — Characteristic acoustic impedance) is the form associated with Rayleigh. Working symbols: ρ\rho, cc, pp →\rightarrow ZZ, II. The ratio of acoustic pressure to particle velocity in a travelling wave is the characteristic impedance ρ c of the medium.

Purpose

Purpose: compute ZZ, II from ρ\rho, cc, pp in Waves & optics via Z=ρc,I=p2ZZ=\rho c,\quad I=\dfrac{p^2}{Z} Z = ρ c, and intensity I = p² / Z. Use it when a real waves & optics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given ρ=1.200 kg/m3\rho = 1.200\,\mathrm{kg/m^{3}}, c=343.000 m/sc = 343.000\,\mathrm{m/s}, p=0.200 Pap = 0.200\,\mathrm{Pa}, the governing relation Z=ρc,I=p2ZZ=\rho c,\quad I=\dfrac{p^2}{Z} yields Z=411.600 Pa⋅s/mZ = 411.600\,\mathrm{Pa·s/m}, I=9.718e−5 W/m2I = 9.718e-5\,\mathrm{W/m^{2}}. Plane travelling wave, linear acoustics. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Impedance Z411.600 Pa·s/m
  • Intensity I0.000097 W/m²
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Narration of this film

Plane travelling wave, linear acoustics.

The ratio of acoustic pressure to particle velocity in a travelling wave is the characteristic impedance ρ c of the medium.

Reading speed

Watch on YouTube