Open Pipe Resonance Calculator

The Open Pipe Resonance Calculator calculates the resonant (standing-wave) frequencies of an open–open pipe, where both ends are open to the air.
Such systems support all harmonic modes and are fundamental models in acoustics, wave physics, and musical instrument design (e.g., flutes and open organ pipes).

What Is Open Pipe Resonance?

An open pipe is a tube that is open at both ends. When air vibrates inside the pipe:

• Both ends behave as pressure nodes (atmospheric pressure)
• Standing waves form along the air column
• All harmonics are present: $n = 1, 2, 3, \dots$

This behavior contrasts with a closed (stopped) pipe, which only supports odd harmonics.

Open Pipe Resonance Formula

The resonant frequencies of an ideal open pipe are given by:

$$ f_n = \frac{n \cdot v}{2L} $$

Where:

• $f_n$ — nth resonant frequency (Hz)
• $n$ — harmonic number (1, 2, 3, …)
• $v$ — speed of sound in air (m/s)
• $L$ — length of the pipe (m)

Because the boundary conditions are symmetric, every integer harmonic is allowed.

Relationship to Harmonic Frequency

Mathematically, the open pipe resonance equation is identical to the general harmonic frequency formula.

The difference lies in the physical interpretation:

• Harmonic Frequency describes a general wave system with symmetric boundary conditions
• Open Pipe Resonance applies that same formula specifically to air columns open at both ends

For a general-purpose calculation, see:
👉 Harmonic Frequency Calculator

🧮 How to Use the Calculator

1. Enter the Harmonic Number ($n$)

   • $n = 1$ → Fundamental frequency (lowest pitch)
   • $n = 2, 3, \dots$ → Higher harmonics (overtones)

2. Enter the Speed of Sound ($v$)

   • Default: 343 m/s (air at 20 °C)
   • Sound speed increases with temperature:
     $v \approx 331 + 0.6T$ (m/s)

3. Enter the Pipe Length ($L$)

   • Use the full physical length of the tube
   • Units can be meters, centimeters, or feet (auto-converted)

The result is the resonant frequency at which a standing wave forms inside the open pipe.

Example Calculation

An open pipe has a length of 0.85 m.
Assume the speed of sound is 343 m/s.

For the fundamental mode ($n = 1$):

$$ f_1 = \frac{1 \cdot 343}{2 \cdot 0.85} \approx 202\ \text{Hz} $$

Higher harmonics occur at integer multiples of this frequency.

📊 Visualizing Standing Waves in an Open Pipe

The diagram below shows the pressure distribution inside an open–open pipe for the first two harmonics.

• Open ends: pressure nodes
• Middle regions: pressure antinodes (maximum variation)

🧠 Engineering Note: End Correction

In real pipes, the vibrating air column extends slightly beyond the physical openings.
A common approximation for open pipes is:

• End correction per open end: approximately $0.61,r$ (where $r$ is the pipe radius)

Therefore, the effective acoustic length is:

$$ L_{eff} \approx L_{physical} + 2 \times 0.61,r $$

Since the diameter $D = 2r$, this can also be written as:

$$ L_{eff} \approx L_{physical} + 0.61,D $$

Engineering approximation:

$$ L_{eff} \approx L_{physical} + 0.6,D $$

If high precision is required (e.g., instrument tuning), calculate $L_{eff}$ and enter it into the calculator.

Open Pipe vs Closed Pipe Resonance

Feature	Open Pipe	Closed Pipe
Open ends	Two	One
Pressure nodes	Both ends	One end
Harmonics	All (1, 2, 3…)	Odd only (1, 3, 5…)
Fundamental	$\frac{v}{2L}$	$\frac{v}{4L}$

Need the stopped-pipe case?
👉 Closed Pipe Resonance Calculator

🙋 Frequently Asked Questions

Why does an open pipe produce all harmonics?

Because both ends impose the same boundary condition (pressure nodes), the standing wave pattern supports an integer number of half-wavelengths.

Why does a flute sound sharper in hot air?

The speed of sound increases with temperature. Since $f \propto v$, higher sound speed leads to higher pitch.

Is this the same formula as for a stretched string?

Yes, mathematically. A string fixed at both ends and an open pipe both support standing waves with $n/2$ wavelengths.
However, the wave speed depends on tension for strings and air properties for pipes.

Related Calculators

• Harmonic Frequency Calculator
• Closed Pipe Resonance Calculator
• Speed of Sound Calculator (recommended)

📚 References

• Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics - Rossing, T. D. The Science of Sound