11th Harmonic Calculator

Calculator Form

System Type Stretched String Open Pipe Closed Pipe

Input Mode Wave speed + length Known fundamental frequency

Fundamental Frequency, f1 (Hz)

Wave Speed, v (m/s)

Resonator Length, L (m)

Reference Amplitude

Phase Shift (degrees)

Amplitude Model Constant amplitude Inverse with harmonic number Inverse square with harmonic number

Graph Points

Calculate 11th Harmonic

Example Data Table

Formula Used

Core harmonic rule: f₁₁ = 11f₁

Period: T₁₁ = 1 / f₁₁

Angular frequency: ω₁₁ = 2πf₁₁

Wavelength from speed: λ₁₁ = v / f₁₁

String or open pipe fundamental: f₁ = v / 2L

Closed pipe fundamental: f₁ = v / 4L

String or open pipe 11th wavelength: λ₁₁ = 2L / 11

Closed pipe 11th wavelength: λ₁₁ = 4L / 11

Amplitude model choices: constant, A₁₁ = A₁ / 11, or A₁₁ = A₁ / 11²

The calculator switches formula paths based on your selected system and input mode. Closed pipes only allow odd harmonics, so the 11th harmonic is physically valid there.

How to Use This Calculator

1. Select the resonant system: stretched string, open pipe, or closed pipe.
2. Choose whether you know geometry values or the fundamental frequency.
3. Enter wave speed and resonator length for geometry mode.
4. Enter the fundamental frequency directly for frequency mode.
5. Set amplitude, phase shift, and amplitude model if needed.
6. Choose how many graph points you want for plotting.
7. Press the calculate button to show results above the form.
8. Use the CSV and PDF buttons to export the computed summary.

FAQs

1. What does the 11th harmonic mean?

The 11th harmonic is a resonant frequency eleven times the fundamental frequency. It represents a higher standing-wave pattern with more nodes and shorter wavelength.

2. Why is the 11th harmonic allowed in a closed pipe?

Closed pipes only support odd harmonics. Since 11 is odd, the 11th harmonic is valid and corresponds to a higher allowed resonance mode.

3. When should I use geometry mode?

Use geometry mode when you know the resonator length and wave speed. The calculator derives the fundamental frequency first, then computes the 11th harmonic values.

4. When should I use fundamental frequency mode?

Use it when the base resonant frequency is already known from measurement or another calculation. Optional length and wave speed can still check resonance consistency.

5. Why does wavelength decrease at higher harmonics?

If wave speed stays constant, frequency and wavelength are inversely related. A higher harmonic has a higher frequency, so its wavelength becomes shorter.

6. What does the amplitude model change?

It estimates how large the 11th harmonic should be relative to the base amplitude. This is useful for waveform comparison, signal synthesis, and rough energy interpretation.

7. What is the difference between open and closed systems?

Strings and open pipes allow all integer harmonics. Closed pipes support only odd harmonics because one end behaves like a node and the other like an antinode.

8. What does the Plotly graph show?

The graph compares the fundamental pattern with the 11th harmonic pattern. With length data, it shows standing-wave shape along the resonator. Otherwise, it shows a time-domain comparison.