Question

What is the ratio of the first three harmonics produced in a stretched string fixed at two ends ?

Answer 1

Answer:

To find:

Ratio of frequency of 1st 3 harmonics on a stretched string fixed at 2 ends.

Concept:

When the string is fixed at 2 ends , it means that there is formation of nodes(of standing waves) at the 2 ends.

Harmonics are integral frequencies of a reference frequency.

Diagram:

Please look at the attached photo to understand better.

Calculation :

Let string length be L and wavelengths be λ

For 1st harmonic : L = λ/2 => λ = 2L

$freq.1 = \dfrac{v}{ \lambda} = \dfrac{v}{2L}$

For 2nd harmonic : L = λ/2 + λ/2 = λ

$freq.2 = \dfrac{v}{ \lambda} = \dfrac{v}{L}$

For 3rd harmonic : L = λ/2 × 3 = 3λ/2

=> λ = 2L/3

$freq.3 = \dfrac{v}{ \lambda} = \dfrac{v}{ (\frac{2L}{3}) } = \dfrac{3v}{2L}$

So the ratio is as follows :

$freq1 : freq.2 : freq3$

$= 1 : 2 : 3$

So final answer :

$\boxed{ \huge{ \blue{1 : 2 : 3}}}$