Find the fundamental frequency and the first four overtones of a ` 15 cm` pipe (a) if the pipe is closed at one end , and (b) if the pipe is open at both ends. ( c) How many overtones may be heard by a person of normal hearing in each of the above cases ? velocity of sound in air ` = 330 m//s`.

Explanation

a. `n_(0) = ( c )/( 4 I) where n_(0) = "frequency of the fundamental"`
`n_(0) = ( 330)/( 4 xx 0.15) = 550 Hz`
The first four overtones are `3n_(0) , 5 n_(0) , 7 n_(0) and 9 n_(0)`.
So , the required frequencies are `550 , 1650 , 2750 , 3850 and 4950 Hz`.
b . `n_(0) = ( c )/( 2I) = (330)/( 2 xx 0.15) = 1100 Hz`
The first four overtones are `2 n_(0) , 3 n_(0) , 4 n_(0) and 5 n_(0)`.
So , the required frequencies are `1100 , 2200 , 3300 , 4400 and 5500 Hz`.
c. The frequency of the `n th` overtones overtones is `( 2n + 1)n_(0) or ( n + 1) n_(0)`.
`:. ( 2 n + 1)n_(0) = 20,000 or ( n + 1) 1100 = 20,000`
`rArr n = 17.18 or n = 17.18`
The acceptable value is `17`.