PhysicsSolver360 logoPhysicsSolver360°
Powered by Solver360°

Closed Pipe Harmonics Calculator

Calculate resonant frequencies for a pipe closed at one end (odd harmonics)

Parameters

mⓘ
m/sⓘ
(odd)ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Resonant Frequency:
171.50;Hz171.50;Hz
Harmonic n (odd):
1.00;1.00;

Examples

Closed pipe fundamental

L = 0.5 m, n = 1.

    Third harmonic

    n = 3.

      Visualization

      Resonance in a Closed Organ Pipe

      A closed organ pipe is open at one end and closed at the other. The closed end is a rigid boundary: displacement node (air cannot move into the wall). The open end is a displacement antinode. Only certain quarter-wavelength patterns fit in the length L.

      Allowed wavelengths satisfy L = n(λ_n/4) but only odd n = 1, 3, 5, … produce valid standing waves (node at closed end, antinode at open end). Hence f_n = nv/(4L) with n odd only.

      Fundamental (n = 1): λ₁ = 4L — one quarter of a wavelength fits in the pipe. Example: L = 0.5 m, v = 343 m/s → f₁ = 343/(4×0.5) = 171.5 Hz. The same physical length as an open pipe would give f₁(open) = 343 Hz — one octave higher.

      The first overtone is n = 3 (third harmonic), not n = 2 — even harmonics are missing. This gives clarinets and stopped organ pipes a hollow, “hollow wood” timbre compared to flutes.

      End correction at the open end slightly lengthens the effective column. Closed end has less correction. Bottles blown over the lip are often modeled as closed pipes with large end correction.

      Pressure and displacement are 90° out of phase in standing sound waves: at a displacement node, pressure variation is maximum (closed end of a pipe is often a pressure antinode).

      Class 12 contrasts open and closed pipes in tables. JEE favorites: ratio of frequencies, which harmonic is heard, and combining with string formulas.

      Key Concepts

      • f_n = nv/(4L), n = 1, 3, 5, …
      • Node at closed end, antinode at open
      • λ₁ = 4L
      • Missing even harmonics
      • f₁(closed) = ½ f₁(open) same L
      • First overtone is n = 3

      Real-World Applications

      • Clarinet and stopped organ pipes
      • Bottle resonance and “blowing across” demos
      • Closed resonance tube experiments
      • Subwoofer and duct acoustics (some modes)
      • Class 12–JEE closed pipe problems

      Explore Further

      More waves tools

      • Wave Properties

        Calculate wavelength, frequency, amplitude, and wave velocity.

      • Doppler Effect

        Analyze frequency shifts due to relative motion of source and observer.

      • Wave Interference

        Analyze constructive and destructive interference patterns between two waves with interactive visualization.

      • Standing Waves

        Analyze standing wave patterns, nodes, antinodes, and resonance frequencies.

      • Wave Reflection

        Calculate reflection coefficients, phase shifts, and energy transfer at boundaries.

      • Wave Transmission

        Analyze wave transmission between different media and wavelength changes.

      Physics Equations

      Closed Pipe:
      fn=nv4Lf_n = \frac{nv}{4L}

      Step-by-Step Solution

      See how the main results are calculated.

      1

      Step 1: Closed Pipe Formula

      Equation:

      fn=nv4Lf_n = \frac{nv}{4L}

      Explanation:

      n = 1, 3, 5, … (odd harmonics only); closed end is a node.

      2

      Step 2: Given

      Result:

      L=0.5m,v=343m/s,n=1(odd)L = 0.5 m, v = 343 m/s, n = 1 (odd)
      3

      Step 3: Calculate

      Calculation:

      fn=1×3434×0.5=171.5000Hzf_n = \frac{1 \times 343}{4 \times 0.5} = 171.5000 Hz

      Result:

      fn=171.5000Hzfₙ = 171.5000 Hz
      4

      Step 4: Fundamental

      Calculation:

      f1=v/(4L)—quarter−waveinpipef₁ = v/(4L) — quarter-wave in pipe

      Explanation:

      Effective length ≈ L + 0.3d for end correction.

      5

      Step 5: Compare Open Pipe

      Same L: f₁(closed) = ½ f₁(open)

      Explanation:

      Closed pipe fundamental is one octave lower for same length.

      6

      Step 6: Application

      Clarinet behaves approximately as a closed cylindrical pipe.

      Explanation:

      Even harmonics suppressed — timbre differs from open pipe.

      Frequently Asked Questions (FAQ)

      Why no even harmonics?

      An even number of quarter-waves would force a displacement antinode at the closed end, which is impossible for a rigid closure.

      Is a clarinet exactly a closed pipe?

      Approximately for the lowest register; tone holes and bell change effective length and harmonic pattern at higher notes.

      What is the second resonant frequency?

      For n = 3: f₃ = 3v/(4L) = 3f₁ — not 2f₁.

      Pressure node or antinode at closed end?

      Closed end: displacement node, pressure antinode (air piles up). Open end: opposite.

      Can I use n = 2 in the formula?

      No for ideal closed pipe — n must be odd. Using n = 2 gives a mode that does not satisfy boundary conditions.

      Practice MCQs

      1. Closed pipe:
      2. Harmonics:
      3. Closed end is:
      4. n=3 is:
      5. Same L as open pipe, f₁ is:
      6. λ₁ =

      Related Calculators

      These tools connect to the same physics concepts used in this calculator.

      Beat FrequencyCalculate beat frequency from two superposing waves: f_beat = |f₁ − f₂|.Sound Intensity LevelConvert sound intensity to decibels using β = 10 log₁₀(I/I₀).String Fundamental FrequencyFind fundamental frequency of a string fixed at both ends: f₁ = v/(2L).Open Pipe HarmonicsResonant frequencies of an open organ pipe: fₙ = nv/(2L).Sound Wave PropertiesCalculate wavelength, period, and other properties of sound waves with interactive visualization.Related Acoustics topic.Snell's LawCalculate refraction angles and analyze light behavior at media boundaries.Related Optics topic.Error PropagationCombine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.Related Measurement & Uncertainty topic.Wave PropertiesCalculate wavelength, frequency, amplitude, and wave velocity.Part of the Class 11 — Waves learning path.