Critical Frequency⚠ unverified
Physics / Acoustics · Compute the critical (coincidence) frequency for panel radiation
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | Pa | 1.0 | Young's modulus of the panel material |
| h | h | m | 1.0 | Panel thickness |
| rho | ρ | kg/m**3 | 1.0 | Panel material density |
| sigma | σ | — | 1.0 | Poisson's ratio of the panel material, dimensionless |
| c | c | m/s | 343.0 | Speed of sound in the surrounding fluid, in metres per second (m/s). Default is 343.0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | fc | Hz | Critical frequency, in hertz (Hz). Returns 0.0 when ``h`` is not positive |
The science & history
Understanding the Parameters
- $E$, $h$, $\sigma$ — enter bending stiffness $D = Eh^{3}/[12(1-\sigma^{2})]$ for isotropic plates.
- $\rho h$ — mass per unit area $\mu = \rho h$; heavier plates lower $f_c$ for fixed $D$.
- $c$ — fluid sound speed.
- $f_c$ — lower for thick stiff panels? Actually $f_c \propto 1/h$ for solid plates of given material (thicker → lower $f_c$).
Derivation (Approaching a Proof)
Bending-wave phase speed on a thin plate increases with frequency. Coincidence occurs when the bending wavelength matches the projected acoustic wavelength, leading to a critical frequency $f_c \propto c^{2}\sqrt{\mu/D}$. The factor $1/(2\pi)$ matches one common normalisation of $D$ and circular frequency; textbooks may write $f_c = (c^{2}/2\pi)\sqrt{\mu/D}$ for the plate critical frequency in air.
History
Coincidence and critical frequency are central to building acoustics and transmission-loss design (Cremer, Beranek, etc.).
Related Concepts: Radiation Efficiency, Transmission Loss, Acoustic Impedance
Notes: Registry calculator critical-frequency (unverified). Confirm $D$ is computed from $E,h,\sigma$
in code; latex shows $D$ explicitly.