Two-Microphone Intensity Probe

What is modelled

Sound intensity is the average of pressure times particle velocity. A two-microphone probe estimates the pressure as the mean of the two signals and the particle velocity from their difference through the momentum equation rho du/dt = -dp/dx. The finite difference underestimates the gradient when the spacing is not small against the wavelength, and a phase mismatch between the channels adds an error that grows at low frequency. A reflected wave carries energy back, so the net intensity falls by 1 - R^2 even though the pressure is louder.

Equations

• Pressure phasors P(x) = A e-jk x cosθ + R A ejk x cosθ + jφ, with k = 2 π f / c and c = 343 m/s, ρ = 1.204 kg/m³.
• Velocity U = j (P2 - P1) / (ρ ω d), intensity I = ½ Re(P̄ U*) with P̄ = (P1 + P2)/2.
• Ideal axial intensity A2 (1 - R2) cosθ / (2 ρ c).
• Spacing bias Iest / I = sin(k d cosθ) / (k d cosθ); the estimate is zero when d = λ/2.

Checks run in the page

• Phasor estimate against the ideal intensity times the sinc factor.
• A time-domain run of two simulated microphones against the phasor formula.
• The 3D page analyticIntensity and evaluate() model intensity.
• The d = lambda/2 zero and the low-frequency series 1 - (kd)^2/6.

Honest simplifications

Plane waves in a free field with one specular reflection, point microphones, no flow, no scattering by the probe body, an assumed phase mismatch applied to channel 2 only, and air at 20 C. Real p-p probes also need the gain mismatch and noise terms that the 3D page models.

• Press Spacing too wide and watch the measured arrow shrink: at 3.5 kHz with 60 mm the probe reads far too low.
• Press Reflective room and see the true intensity fall by 1 - R^2 while the signals stay loud.
• Press Low frequency, mismatch and read the phase-mismatch error graph: small phase errors matter most at low frequency.