Calculator
How to use this calculator
- Select a system type that matches your boundary conditions.
- Enter the length L and choose a mode index n.
- Set wave speed directly, or compute it from tension and density.
- Press Submit to show results under the header area.
- Download CSV or PDF to save your experiment calculations.
Formula used
- f = v / λ
- ω = 2πf
- k = 2π/λ
- T = 1/f
- Fixed–fixed or open–open: λ = 2L/n, f = nv/(2L)
- Fixed–free or open–closed: λ = 4L/(2n−1), f = (2n−1)v/(4L)
- String speed: v = √(T/μ)
- Air estimate: v ≈ 331 + 0.6T(°C)
Example data table
| System | L (m) | v (m/s) | n | λ (m) | f (Hz) |
|---|---|---|---|---|---|
| String fixed–fixed | 1.200 | 126.491 | 3 | 0.800 | 158.114 |
| Pipe open–open | 0.650 | 343.000 | 2 | 0.650 | 527.692 |
| Pipe open–closed | 0.450 | 343.000 | 2 | 0.600 | 571.667 |
FAQs
1) What does the mode index n represent?
It selects a specific standing-wave pattern. For fixed–fixed or open–open, n matches the harmonic number. For fixed–free or open–closed, n maps to odd harmonics only.
2) Why do open–closed systems only allow odd harmonics?
One end must be a node and the other an antinode. That boundary mismatch forces quarter-wave patterns, producing frequencies proportional to 1, 3, 5, and so on.
3) How accurate is the air speed estimate from temperature?
It is a practical approximation for dry air near standard pressure. Humidity and gas composition can shift the speed. For precision acoustics, measure speed or use calibrated references.
4) Should I include pipe end correction?
For open ends, the effective length can be slightly longer than the physical length. If your pipe is short or wide, applying an end correction improves accuracy, especially for higher modes.
5) What are nodes and antinodes in this output?
Nodes are positions of zero displacement. Antinodes are positions of maximum displacement. The listed x-values help you place sensors, clamps, or microphones along the length.
6) Why do I see many node positions?
Nodes repeat every half-wavelength. Higher modes have shorter wavelengths, so more nodes fit within the same length. The calculator caps lists to keep them readable.
7) Can I use this for pressure nodes in pipes?
Yes, but note pressure nodes and displacement nodes swap at boundaries. An open end is a pressure node, while a closed end is a pressure antinode. Use the positions carefully for your measurement type.