Presets
Playback
Sample
Flow resistivity: wool 5-30 kPa s/m2, dense felt 50-100 kPa s/m2.
Test tone and probe
Move the probe along the tube and read the pressure.
Display and sound
Animation runs at a slow visual rate; wavelengths are real.
Camera
Quality
Layers
Model checks
The science
What the tube measures
A loudspeaker sends plane waves down a rigid tube toward a sample on a rigid backing. Incident and reflected waves form a standing wave p(x) = ejkx + R e−jkx (x measured from the sample face). The pressure magnitude swings between 1 + |R| and 1 − |R|; the ratio s = (1 + |R|)/(1 − |R|) is the standing-wave ratio and the absorption coefficient is α = 1 − |R|2 = 4s/(1 + s)2 (the idea behind ISO 10534-1). ISO 10534-2 instead uses the transfer function between two fixed microphones; both routes are computed here and compared.
Porous layer: Delany-Bazley equivalent fluid
With X = ρf/σ (σ = flow resistivity), the empirical fits are Zc = ρc[1 + 0.0571 X−0.754 − j 0.087 X−0.732] and k = (ω/c)[1 + 0.0978 X−0.700 − j 0.189 X−0.595], valid for 0.01 < X < 1. A layer of thickness t on an air gap D and a rigid wall has the surface impedance Zs = Zc (Zg + Zc tanh(jkt)) / (Zc + Zg tanh(jkt)) with Zg = −j ρc cot(k0D) (and Zs = Zc coth(jkt) with no gap). Then R = (Zs − ρc)/(Zs + ρc).
Why thickness and air gaps matter
Porous absorbers work where the particle velocity is large: about a quarter of a wavelength from a rigid wall. A thin layer on the wall (where velocity is zero) absorbs little at low frequency; a spacer of air gap D moves the layer to the velocity maximum near f = c/(4D). Flow resistivity has an optimum: too low and the air flows through without loss, too high and the sample behaves like a hard wall that reflects.
Independent numerical checks
(1) The closed-form tanh/coth impedance is compared with a cascade of 300 thin transfer matrices through the sample and gap. (2) α from the standing-wave ratio, measured by scanning the whole wave pattern, is compared with 1 − |R|2. (3) R recovered from the two-microphone transfer function is compared in the same way.
Honest simplifications
Delany-Bazley is an empirical regression for fibrous materials, with errors of order 10 % and poor behaviour outside 0.01 < X < 1; foams, perforated plates and anisotropic materials need other models (Johnson-Champoux-Allard etc.). The sample is homogeneous and fills the tube cross-section with no edge gap, the walls are rigid and lossless, and the model is one-dimensional, valid below the tube's first cross mode (about 1.8 kHz for 110 mm diameter). No tube wall losses, no sample mounting resonances, no measurement noise.
What to try
- Press "Hard wall (no sample)": the standing wave is a perfect pattern with s very large and alpha = 0.
- Compare "Mineral wool 50 mm" with "Thick wool 100 mm" on the thickness graph.
- Use "Thin wool 25 mm + 75 mm air gap": absorption appears at low frequency without more material.
- Move the probe (or press "Sweep the probe") and read |p|; the maximum and minimum give s and then alpha.
- Raise sigma until the sample "reflects like a wall"; lower it until the air flows through unhindered.