Show that all harmonics are present in a streteched string under a tranverse mode of vibration.

Explanation

Let L be the length of the string . Then `L = ( lambda_(0))/( 2) ` or `lambda_(0) =2 L `
The fundamental frequency `f_(0) =( v)/( lambda_(0))`
i.e, `f_(0)= (v)/( 2L)`

where is velocity of transverse wave in the string.
and `v= sqrt((T)/( mu)),T-`tension in the string ,` mu =` linear density of the wire.
In the next mode of vibration `L = 2((lambda_(1))/(2))`
i.e, `lambda_(1) = L ` and II harmonic = I overtone
I overtone `f_(1) = ( v)/( lambda_(1))= ( v)/( L ) `

or `f_(1) = 2((v)/( 2L)) = 2f_(0)`
In the third mode of vibration ,
`L = ( 3lambda_(2))/( 2) ` so that `lambda_(2) = ( 2L)/( 3)`
II overtone `f_(2) =(v)/( lambda_(2)) =( 3V)/( 2L) = 3f_(0)`
Hence the harmonics are in the ratio
`f_(0) : f_(1) : f_(2) : f_(3) : "............." f_(n) : : 1:2:3: 4:".............":n`