Helmholtz Resonator Frequency Equation
Calculates the resonant cavity frequency of a bottle, acoustic bass trap, or ported loudspeaker enclosure from cavity volume V, neck cross-section S, and effective neck length.
\[ f_H = \frac{c}{2\pi} \sqrt{\frac{S}{V \cdot L_{eff}}} \quad \text{where } L_{eff} = L_{neck} + 0.61 \cdot d_{neck} \]
Plain Text Formula:
f_H = (c / (2 * pi)) * sqrt(S / (V * L_eff)) where L_eff = L_neck + 0.61 * d_neck
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ f_H \] | Resonant Frequency | Hertz (Hz) | Natural frequency of acoustic resonance |
| \[ c \] | Speed of Sound | 343 m/s | Acoustic velocity in air at room temperature |
| \[ S \] | Neck Port Area | m² | Cross-sectional area of opening or port tube: pi * r² |
| \[ V \] | Cavity Volume | m³ | Internal volume of the enclosed resonator body |
| \[ L_{eff} \] | Effective Port Length | meters (m) | Physical neck length plus end-correction flanging |
💡 Worked Engineering Example
Problem Statement:
A 0.05 m³ subwoofer box has a circular port (diameter 0.1 m, area 0.00785 m², length 0.2 m). Find tuning frequency.
Step-by-Step Solution:
- End correction = 0.61 * 0.1 = 0.061 m
- L_eff = 0.2 + 0.061 = 0.261 m
- Ratio S / (V * L_eff) = 0.00785 / (0.05 * 0.261) = 0.6015
- sqrt(0.6015) = 0.7756
- f_H = (343 / (2 * pi)) * 0.7756 = 54.59 * 0.7756 = 42.34 Hz
🎯 Result: The Helmholtz enclosure resonates at 42.34 Hz.