A pipe 20cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430Hz source? Will the same source be in resonance with the pipe if both ends are open? Take speed of sound in air `340m//s`.

Explanation

First (Fundamental), No
Length of the pipe, l = 20 cm = 0.2 m
Source frequency `= n^("th")` normal mode of frequency, `ν_(n)` = 430 Hz
Speed of sound, v = 340 m/s
In a closed pipe, the `n^(th)` normal mode of frequency is given by the relation:
`V_(n)=(2n-1)(v)/(4l)" ":n` is an integer=0,1,2,3...
`430=(2n-1)(340)/(4xx02)`
`2n-1=(430xx4xx0.2)/(340)=1.01`
`2n=2.01`
`n~1`
Hence, the first mode of vibration frequency is resonantly excited by the given source.
In a pipe open at both ends, the `n^(th)` mode of vibration frequency is given by the relation:
`v_(n)=(mv)/(2l)`
`n=(2lV_(n))/(v)`
`=(2xx0.2xx430)/(340)=0.5`
Since the number of the mode of vibration (n) has to be an integer, the given source does not produce a resonant vibration in an open pipe.