PhysicsSolver360 logoPhysicsSolver360°
Powered by Solver360°

String Fundamental Frequency Calculator

Find fundamental frequency for a string fixed at both ends

Parameters

mⓘ
m/sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Fundamental Frequency:
170.00;Hz170.00;Hz
Wavelength λ₁:
2.00;m2.00;m

Examples

Guitar string

L = 0.65 m, v = 260 m/s.

    Long string

    L = 1 m, v = 340 m/s.

      Visualization

      Standing Waves on a String — Fundamental and Harmonics

      A stretched string fixed at both ends cannot move at the ends. These fixed points are displacement nodes. Standing waves form when reflected waves interfere with incident waves; only certain wavelengths “fit” between the boundaries.

      For the fundamental (first harmonic, n = 1), the string length L equals half a wavelength: L = λ₁/2, so λ₁ = 2L. With wave speed v on the string, the fundamental frequency is f₁ = v/λ₁ = v/(2L). This is the lowest note the string can sustain as a standing wave.

      Higher harmonics (overtones) fit n half-wavelengths in L: L = n(λ_n/2), so λ_n = 2L/n and f_n = nv/(2L) for n = 1, 2, 3, … All integer harmonics are allowed for an ideal string fixed at both ends. The guitar harmonic at the 12th fret is the second harmonic (n = 2).

      Wave speed on a string is set by mechanical properties: v = √(T/μ), where T is tension (N) and μ is linear mass density (kg/m). Increase tension → higher v → higher pitch. Use a heavier (thicker) string → larger μ → lower v → lower pitch.

      Example: steel string L = 0.65 m, v = 260 m/s → f₁ = 260/1.3 = 200 Hz. Double the length → f₁ halves. Double the tension (v up by √2) → f₁ increases by √2.

      Energy is trapped on the string; nodes have zero displacement; antinodes have maximum motion. A real string has stiffness and the bridge/end pins add small corrections, so observed frequencies differ slightly from the ideal formula.

      Class 12 NCERT (Waves and Oscillations) covers sonometer experiments verifying f ∝ 1/L and f ∝ √T. JEE links string problems with beats, pipes, and Fourier ideas (timbre from harmonic content).

      Key Concepts

      • f_n = nv/(2L), n = 1, 2, 3, …
      • λ₁ = 2L (fundamental)
      • Nodes at fixed ends
      • v = √(T/μ)
      • Antinode at center for n = 1
      • f ∝ 1/L and f ∝ √T

      Real-World Applications

      • Guitar, violin, piano, and harp strings
      • Sonometer and resonance tube labs
      • String instrument design and intonation
      • Vibration analysis of cables and bridges
      • Class 12–JEE standing wave on string 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

      Fundamental:
      f1=v2Lf_1 = \frac{v}{2L}

      Step-by-Step Solution

      See how the main results are calculated.

      1

      Step 1: String Fixed at Both Ends

      Equation:

      fn=nv2Lf_n = \frac{nv}{2L}

      Explanation:

      n = 1, 2, 3, … for harmonics; fundamental n = 1.

      2

      Step 2: Given

      Result:

      L=1m,v=340m/sL = 1 m, v = 340 m/s
      3

      Step 3: Fundamental (n=1)

      Equation:

      f1=v2Lf_1 = \frac{v}{2L}

      Calculation:

      f1=3402×1=170.0000Hzf_1 = \frac{340}{2 \times 1} = 170.0000 Hz

      Result:

      f1=170.0000Hzf₁ = 170.0000 Hz
      4

      Step 4: Wavelength

      Calculation:

      λ1=2L=2.0000m\lambda_1 = 2L = 2.0000 m

      Explanation:

      One half-wavelength fits on the string for fundamental.

      5

      Step 5: Second Harmonic

      Calculation:

      f2=2f1=340.0000Hzf_2 = 2f_1 = 340.0000 Hz

      Explanation:

      Nodes at ends; antinodes at center and quarters.

      6

      Step 6: Wave Speed on String

      Equation:

      v=T/μv = \sqrt{T/\mu}

      Explanation:

      T = tension, μ = linear mass density — determines v for a given string.

      Frequently Asked Questions (FAQ)

      Why is the formula v/(2L) and not v/L?

      Only half a wavelength fits on the string for the fundamental — both ends are nodes, one antinode in the middle.

      What is the difference between harmonic and overtone?

      First harmonic = fundamental (n=1). First overtone usually means n=2 (second harmonic). Naming varies — read questions carefully.

      Does amplitude affect frequency?

      For ideal strings, no. Real strings show slight amplitude-dependent effects at large displacement.

      String vs open pipe same L?

      Same formula f_n = nv/(2L) for ideal models — both have antinodes at both ends (open pipe in displacement).

      How to measure v on a string?

      Use sonometer: find f₁ for known L, or compare two tensions/masses and use f ∝ √T.

      Practice MCQs

      1. String fundamental:
      2. Double length:
      3. Second harmonic n=2:
      4. v on string:
      5. Ends of string:
      6. λ₁ equals:

      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).Open Pipe HarmonicsResonant frequencies of an open organ pipe: fₙ = nv/(2L).Closed Pipe HarmonicsResonant frequencies of a closed pipe (odd harmonics): fₙ = nv/(4L).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.