Standing Wave on String

f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}

Resonant frequencies for string fixed at both ends.

Variables

f_n

Resonant frequency

nth harmonic

In SI units: 1/s

Allowed values: greater than 0

Harmonic number

Integer 1, 2, 3...

Allowed values: greater than 0, whole number

unitless

String length

Fixed at both ends

Allowed values: greater than 0

Tension

String tension

In SI units: kg·m/s²

Allowed values: greater than 0

Linear density

Mass per length

Allowed values: greater than 0

kg/m

Derivation

λ_n = 2L/n, f_n = v/λ_n = (n/2L)√(T/μ).

Your notes

0/5000

Check value

Round inputs run through this page's calculator. Use it to test your own working or a rearranged form. It is a test case, not a real-world example.

n = 2 (Harmonic number)
L = 3 m (String length)
T = 4 N (Tension)
μ = 5 kg/m (Linear density)

Result: f_n = 0.2981424 Hz

Other waves formulas with the same quantities

Standing Wave (Open Pipe) (Superposition) shares Resonant frequency, Harmonic number
Standing Wave (Closed Pipe) (Superposition) shares Resonant frequency, Harmonic number

Where this formula is listed

Waves formulas, under Superposition.
In the waves list it comes after Doppler Shift (Light) and before Standing Wave (Open Pipe).
Waves formula sheet, one of 37 formulas printed with symbol keys and SI units.
The full formula index, grouped by category and topic.

Common questions

What is the Standing Wave on String formula?

Standing Wave on String is written f_n = (n/(2L))√(T/μ). Resonant frequencies for string fixed at both ends. It belongs to Waves, under Superposition.

What are the units in f_n = (n/(2L))√(T/μ)?

In SI units, f_n is resonant frequency, measured in 1/s; n is harmonic number (dimensionless); L is string length, measured in m; T is tension, measured in kg·m/s²; μ is linear density, measured in kg/m.

How is the Standing Wave on String formula derived?

λ_n = 2L/n, f_n = v/λ_n = (n/2L)√(T/μ). The result is f_n = (n/(2L))√(T/μ).