JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Waves & Sound

For a certain organ pipe, the first three resonance frequencies are in the ratio of

1 : 3 : 5

respectively. If the frequency of fifth harmonic is 405 Hz and the speed of sound in air is

324 ms^{- 1}

the length of the organ pipe is ____ m.

Answer: 1

Solution

A resonance ratio of

1 : 3 : 5

contains only odd harmonics, the signature of a pipe closed at one end. For such a pipe the

n

-th harmonic (odd

n

) has frequency

f_{n} = \frac{n v}{4 ℓ}

, so the fifth harmonic is

f_{5} = \frac{5 v}{4 ℓ}

. Substituting

f_{5} = 405 Hz

and

v = 324 m s^{- 1}

gives

ℓ = \frac{5 v}{4 f _{5}} = \frac{5 \times 324}{4 \times 405} = \frac{1620}{1620}

. The length of the organ pipe is

1 m

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Integer answer