How do I calculate the fundamental frequency of a string fixed at both ends?
Quick answer
For an ideal string fixed at both ends, calculate the fundamental frequency with f₁ = v/(2L), where v is the wave speed along the string and L is its length. Put both measurements in consistent units first. For example, if L = 0.65 m and v = 130 m/s, then f₁ = 130/(2 × 0.65) = 100 Hz. This model assumes fixed ends and the stated wave speed. [S1]
The question, the details, the next step.
I know the length and wave speed of a string fixed at both ends. How do I get its fundamental frequency?
Use f₁ = v/(2L). Here, *f₁* is the fundamental frequency, *v* is the wave speed along the string, and *L* is the length between the fixed ends. The reason for the factor of 2 is that the fundamental standing-wave pattern has nodes at both ends and fits half a wavelength along the string: λ₁ = 2L. Then frequency is wave speed divided by wavelength, f = v/λ. [S1]
For a worked example, assume a string has length 0.65 m and wave speed 130 m/s. These are illustrative values, not measurements of a particular string. Substitute them:
f₁ = 130 m/s ÷ (2 × 0.65 m) = 130 ÷ 1.30 = 100 Hz
The metres cancel, leaving inverse seconds, or hertz. With these assumed inputs, the fundamental frequency is 100 Hz.
What if my string length is given in centimetres, or the answer doesn’t seem plausible?
Convert the length to metres before using the formula if your wave speed is in metres per second. For example, 65 cm = 0.65 m. Mixing centimetres with metres per second without converting gives inconsistent units and can make the numerical result wrong.
A quick reasonableness check: for the same wave speed, a longer string gives a lower fundamental frequency; for the same length, a higher wave speed gives a higher frequency. You can also check the units: (m/s)/m = 1/s = Hz. If you get a result in metres, seconds, or another unit instead of hertz, revisit the substitution.
Does “fundamental” mean the same thing as any frequency the string can vibrate at?
No. The fundamental is the lowest standing-wave frequency for this ideal fixed-end model. Higher modes also fit standing-wave patterns between the same ends; the source describes them as fₙ = n v/(2L). [S1] The calculation above uses n = 1, so it gives the fundamental, not a higher mode.
The result depends on having the wave speed for the string in question and on the idealized assumptions: a string of length L with both ends fixed. The formula alone does not tell you the wave speed from length, nor does it establish that a real string’s conditions match the model. If your problem supplies v and L, use those stated values; otherwise, identify what additional information the problem provides before calculating.
Can you give me a reusable setup so I can show the calculation clearly?
Write down the known quantities with units, convert them so they are consistent, substitute into f₁ = v/(2L), and state the result in hertz. For example:
- Given: L = [length in m]; v = [wave speed in m/s]
- Model: f₁ = v/(2L)
- Substitute: f₁ = [wave speed] ÷ (2 × [length])
- Calculate and report: f₁ = [result] Hz
For the illustrative values above, that setup becomes: f₁ = 130 m/s ÷ (2 × 0.65 m) = 100 Hz. Keep the assumption visible: this is the fundamental frequency predicted by the ideal string model with both ends fixed and the stated wave speed. [S1]
A practical checklist
Identify the length L between the two fixed ends.
Find the wave speed v along the string; do not infer it from this formula alone.
Convert length and speed to consistent units, such as metres and metres per second.
Calculate f₁ = v/(2L), using n = 1 for the fundamental.
Check that the units reduce to hertz and label the result as a model calculation.
A few more useful answers
Why does the fundamental use twice the string length?
With both ends fixed, the fundamental pattern has a node at each end and fits half a wavelength along the string, so λ₁ = 2L. Combining that wavelength with f = v/λ gives f₁ = v/(2L). [S1]
What information do I need if I know the length but not the wave speed?
Length alone is not enough to calculate the frequency with this formula; you also need the wave speed along the string. The cited model states the relationship but does not establish how to determine a particular string’s wave speed. [S1]
Does this calculation give the frequency of every vibration mode?
No. It gives the lowest mode when you use f₁ = v/(2L). Higher modes have frequencies fₙ = n v/(2L), where n is a positive mode number. [S1]
Sources
References used for this guide.
- [S1] OpenStax University Physics Volume 1, Section 16.6: Standing Waves and Resonance openstax.org · Checked