Question

Derive the equations for the first three resonant modes/harmonic for; i) a closed pipe ii) a vibrating string

Answer 1

To derive:

Equations for resonant modes of harmonics in

• Closed Pipe
• Vibrating String

For Closed Pipe:

Let length of pipe be "l" :

1st Harmonic :

$\therefore \: \dfrac{ \lambda}{2} = l$

$= > \lambda = 2l$

Let frequency be f ;

$\boxed{f = \dfrac{v}{ \lambda} = \dfrac{v}{2l} }$

2nd Harmonic :

$\therefore \: \dfrac{ \lambda}{2} + \dfrac{ \lambda}{2} = l$

$= > \: \lambda = l$

Let frequency be f ;

$\boxed{f = \dfrac{v}{ \lambda} = \dfrac{v}{l} }$

3rd Harmonic :

$\therefore \: \dfrac{ \lambda}{2} + \dfrac{ \lambda}{2} + \dfrac{ \lambda}{2} = l$

$= > \: \dfrac{ 3\lambda}{2} = l$

$= > \: \lambda = \dfrac{2l}{3}$

Let frequency be f ;

$\boxed{f = \dfrac{v}{ \lambda} = \dfrac{3v}{2l} }$

Same cases will be seen in vibrating string , because both closed pipe and vibrating strings have nodes at extreme ends.