A string of a guitar 1.3 m long vibrates with 4 nodes, 2 of them at the two ends. Find the wavelength & speed of the wave in the string if it vibrates at 500 Hz. (𝝀=0.866m, v = 433 m s⁻¹) 


Given:

Length of the string = L = 1.3 m
Number of nodes = 4 (2 out of 4 nodes are at the ends) i.e the string is vibrating in three loops i.e third harmonic, so 
Frequency for third harmonic = f₃ = 500 Hz 

To Find:

The wavelength of third harmonic = λ₃ = ?
Speed of the waves = v = ?

Solution:

The general formula to find the wavelength of n number of loops

λₙ =  `\frac{2L}{n}`

for 3 loops  (n = 3)

λ₃ = `\frac{2L}{3}`

by putting values

λ₃ = `\frac{2 ｘ 1.3 m}{3}`

λ₃ = 0.867 m ------------Ans. 1


Now to find the speed of the wave in the string for the third harmonic (n = 3 loops):

v = f₃ λ₃

putting values

v = 50 Hz ｘ 0.867 m        [ Hz = s⁻¹]

v = 433.5 m s⁻¹ --------------Ans. 2

 


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