Question

An organ pipe has two successive harmonics with frequencies 400 and 560 Hz. The speed of sound in air is 344 m/s. What two harmonics are these ?

Answer 1

As we know that , in an open organ pipe we get all the harmonics (odd and even) and a closed organ pipe gives only odd harmonics . It is given that harmonics are successive therefore ,

for open pipe ,

let frequency of nth harmonic be ${\nu }_n=400Hz$ ,

and frequency of (n+1)th harmonic be ${\nu }_{n+1}=560Hz$ ,

therefore ,

${\nu }_{n+1}-{\nu }_n={\nu }_1$ (fundamental frequency) ,

or $560-400={\nu }_1$ ,

or ${\nu }_1=160Hz$ ,

for closed pipe ,

let frequency of nth harmonic be ${\nu }_n=400Hz$ ,

and frequency of (n+1)th harmonic be ${\nu }_{n+1}=560Hz$ ,

therefore ,

${\nu }_{n+1}-{\nu }_n=2{\nu }_1$ (fundamental frequency) ,

or $560-400=2{\nu }_1$ ,

or ${\nu }_1=80Hz$ ,

We can say that given pipe is a closed pipe because given harmonic frequencies $400Hz$ and $560Hz$ are multiples of $80Hz$ , which is the fundamental frequency of closed organ pipe and it is clear that frequency of $400Hz$ is the $400/80=5th$ harmonic and $560Hz$ is the $560/80=7th$ harmonic .