The fundamental frequency in each of two vibrating systems is 400 Hz. For each ofthese systems, (below), calculate the two lowest frequencies (excluding the fundamental)at which standing waves can occur.(i) a string fixed at both ends and(ii) a cylindrical pipe open at both ends
Question
The fundamental frequency in each of two vibrating systems is 400 Hz. For each ofthese systems, (below), calculate the two lowest frequencies (excluding the fundamental)at which standing waves can occur.(i) a string fixed at both ends and(ii) a cylindrical pipe open at both ends
Solution
(i) For a string fixed at both ends, the frequencies at which standing waves can occur are integer multiples of the fundamental frequency. This is because the string can support waves with 1, 2, 3, etc. nodes.
The fundamental frequency (1st harmonic) is 400 Hz.
The second harmonic will have twice the frequency of the fundamental frequency. So, 400 Hz * 2 = 800 Hz.
The third harmonic will have three times the frequency of the fundamental frequency. So, 400 Hz * 3 = 1200 Hz.
So, the two lowest frequencies (excluding the fundamental) at which standing waves can occur are 800 Hz and 1200 Hz.
(ii) For a cylindrical pipe open at both ends, the frequencies at which standing waves can occur are also integer multiples of the fundamental frequency. This is because the pipe can support waves with 1, 2, 3, etc. nodes.
The fundamental frequency (1st harmonic) is 400 Hz.
The second harmonic will have twice the frequency of the fundamental frequency. So, 400 Hz * 2 = 800 Hz.
The third harmonic will have three times the frequency of the fundamental frequency. So, 400 Hz * 3 = 1200 Hz.
So, the two lowest frequencies (excluding the fundamental) at which standing waves can occur are 800 Hz and 1200 Hz.
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