Calculate the length of a 1/4 wave resonator from frequency in air or RF cable, or find the resonant frequency of a quarter-wave tube or line.
Calculator Functions
The calculator relates a quarter-wave resonator’s length, its fundamental frequency and the wave speed in it, for sound in air, RF cable or any speed you enter. Select a function to open it in the calculator.
| Function | Use it when | You enter |
|---|---|---|
| Physical length | You know the frequency to tune for | Propagation medium, frequency and result unit |
| Fundamental frequency | You have a resonator of known length | Propagation medium, length and result unit |
| Required wave speed | You know the length and frequency and want the speed they imply | Length and frequency |
The medium sets the wave speed: dry air from a temperature between −50 and 50 °C, RF cable from a velocity factor, or a custom speed. An optional tuning allowance from −50% to 50% shortens or lengthens the ideal length. Results also list the full wavelength and, for the first two functions, the 3× and 5× frequencies.
1/4 Wave Resonator Formulas
| L | Physical resonator length (m) |
| v | Wave speed in the medium (m/s) |
| k | Tuning factor: 1 − reduction (%) ÷ 100. With no allowance, k = 1 |
| f | Fundamental frequency (Hz) |
A tube closed at one end and a shorted transmission line both resonate this way, with the wavelength four times the length [1] [2]. In air at 20 °C, 100 Hz needs 343.2 ÷ 400 = 0.858 m.
Fundamental frequency rearranges this to f = v × k ÷ (4 × L), and Required wave speed to v = 4 × L × f ÷ k.
Wave speed by medium
√(1 + T ÷ 273.15)
T is in °C, and c = 299,792,458 m/s. Speed in air depends on absolute temperature [3], and 331.3 m/s matches the modern value of 331.29 m/s for dry air at 0 °C [6]. Velocity factor is the fraction of the speed of light that a signal travels in the cable [5]. The field starts at 0.66, and your cable’s specification should replace it. At that factor, 146 MHz gives 0.339 m.
Odd harmonics
Only odd multiples of the fundamental resonate [1] [2], which is why the results list 3× and 5×. A 100 Hz resonator also responds near 300 Hz and 500 Hz.
Limits of This Estimate
- This is an ideal length. In a tube, the antinode sits slightly past the open end, and about 0.6 times the pipe radius should be added to the acoustic length [1]. The calculator does not add it, so use the tuning allowance and confirm the end correction for your geometry before fabrication.
- The air speed is a dry-air approximation, and it ignores humidity and changes in gas composition.
- The harmonics assume an ideal nondispersive resonator with the right terminations, and real designs may differ. A required speed above the speed of light gets a warning, because it is not a possible cable velocity.
Sources
- HyperPhysics (Georgia State University), Closed Cylinder Air Column: fundamental wavelength, odd harmonics and end correction in a tube
- University of Illinois, ECE 329 Lecture 32, Input impedance and microwave resonators: quarter-wave resonance in a shorted line
- NASA Glenn Research Center, Speed of Sound: speed of sound and absolute temperature
- NIST, CODATA Value: speed of light in vacuum: speed of light
- Times Microwave Systems, Shielding Effectiveness and Velocity of Propagation: cable velocity as a share of the speed of light
- Encyclopaedia Britannica, Speed of sound: speed of sound in dry air at 0 °C
